Stochastic Optimal Control in Inﬁnite Dimensions: Dynamic...

Stochastic Optimal Control in Infinite Dimensions:Dynamic Programming and HJB Equations

G. Fabbri1 F. Gozzi2 and A. Swiech3

with Chapter 6 by M. Fuhrman4 and G. Tessitore5

1Aix-Marseille University (Aix-Marseille School of Economics), CNRS and EHESS. e-mail:[email protected]

2Dipartimento di Economia e Finanza, Università LUISS - Guido Carli Roma, e-mail:[email protected]

3School of Mathematics, Georgia Institute of Technology, Atlanta, e-mail:[email protected]

4Dipartimento di Matematica, Politecnico di Milano, e-mail: [email protected] di Matematica, Università di Milano Bicocca, e-mail:

[email protected]

Contents

Preface 11

Chapter 1. Preliminaries on stochastic calculus in infinite dimensions 171.1. Basic probability 17

1.1.1. Probability spaces, σ-fields 171.1.2. Random variables 191.1.3. Bochner integral 231.1.4. Expectation, covariance and correlation 251.1.5. Conditional expectation and conditional probability 261.1.6. Gaussian measures on Hilbert spaces and Fourier transform 29

1.2. Stochastic processes and Brownian motion 321.2.1. Stochastic processes 321.2.2. Martingales 341.2.3. Stopping times 351.2.4. Q-Wiener process 361.2.5. Simple and elementary processes 40

1.3. Stochastic integral 421.3.1. Definition of stochastic integral 421.3.2. Basic properties and estimates 45

1.4. Stochastic differential equations 491.4.1. Mild and strong solutions 491.4.2. Existence and uniqueness of solutions 511.4.3. Properties of solutions of SDE 521.4.4. Uniqueness in law 55

1.5. Further existence and uniqueness results in special cases 581.5.1. SDE coming from boundary control problems 581.5.2. Semilinear SDE with additive noise 621.5.3. Semilinear SDE with multiplicative noise. 65

1.6. Transition semigroups 721.7. Itô’s and Dynkin’s formulae 731.8. Bibliographical notes 81

Chapter 2. Optimal control problems and examples 832.1. Stochastic optimal control problem: general formulation 84

2.1.1. Strong formulation 842.1.2. Weak formulation 86

2.2. Dynamic Programming Principle: setup and assumptions 872.2.1. The setup 87

3

4 CONTENTS

2.2.2. The general assumptions 882.2.3. The assumptions in the case of control problem for mild

solutions 912.3. Dynamic Programming Principle: statement and proof 94

2.3.1. Pullback to the canonical reference probability space 942.3.2. Independence of reference probability spaces 952.3.3. The proof of the abstract principle of optimality 96

2.4. Infinite horizon problems 1022.5. HJB equation and optimal synthesis in the smooth case 103

2.5.1. Finite horizon problem: Parabolic HJB equation 1042.5.2. Infinite horizon problem: Elliptic HJB equation 110

2.6. Some motivating examples 1122.6.1. Stochastic controlled heat equation: distributed control 1142.6.1.1. Setting of the problem 1142.6.1.2. The infinite dimensional setting and the HJB equation 1152.6.1.3. The infinite horizon case 117

2.6.2. Stochastic controlled heat equation: boundary control 1172.6.2.1. Setting of the problem: Dirichlet case 1172.6.2.2. Setting of the problem: Neumann case 1182.6.2.3. The infinite dimensional setting and the HJB equation 118

2.6.3. Stochastic controlled heat equation: boundary control andboundary noise 120

2.6.3.1. Setting of the problem 1202.6.3.2. The infinite dimensional setting 1212.6.3.3. The HJB equation 122

2.6.4. Optimal control of the stochastic Burgers equation 1232.6.4.1. Setting of the problem 1232.6.4.2. The infinite dimensional setting and the HJB equation 124

2.6.5. Optimal control of the stochastic Navier-Stokes equations 1252.6.5.1. Setting of the problem 1252.6.5.2. The infinite dimensional setting and the HJB equation 126

2.6.6. Optimal control of the Duncan-Mortensen-Zakai equation 1272.6.6.1. An optimal control problem with partial observation 1272.6.6.2. The separated problem 128

2.6.7. Super-hedging of forward rates 1312.6.8. Optimal control of stochastic delay equations 1332.6.8.1. Delay in the state variable only 1342.6.8.2. Delay in the state and control 136

2.7. Bibliographical notes 138

Chapter 3. Viscosity solutions 1433.1. Preliminary results 144

3.1.1. B-continuity and weak and strong B-conditions 1443.1.2. Estimates for solutions of stochastic differential equations 1493.1.3. Perturbed optimization 155

3.2. Maximum principle 1563.3. Viscosity solutions 161

3.3.1. Bounded equations 165

CONTENTS 5

3.4. Consistency of viscosity solutions 1663.5. Comparison theorems 168

3.5.1. Degenerate parabolic equations 1693.5.2. Degenerate elliptic equations 176

3.6. Existence of solutions: Value function 1803.6.1. Finite horizon problem 1823.6.2. Improved version of dynamic programming principle 1953.6.3. Infinite horizon problem 198

3.7. Existence of solutions: Finite dimensional approximations 2013.8. Singular perturbations 2143.9. Perron’s method and half-relaxed limits 2183.10. Infinite dimensional Black-Scholes-Barenblatt equation 2263.11. HJB equation for control of the Duncan-Mortensen-Zakai equation228

3.11.1. Variational solutions 2283.11.2. Weighted Sobolev spaces 2323.11.3. Optimal control of the Duncan-Mortensen-Zakai equation 2333.11.4. Estimates for the DMZ equation 2353.11.5. Viscosity solutions 2383.11.6. Value function and existence of solutions 243

3.12. HJB equation for a boundary control problem 2453.12.1. Definition of viscosity solution 2463.12.2. Comparison and existence theorem 2473.12.3. Stochastic control problem 253

3.13. HJB equations for control of stochastic Navier-Stokes equations 2623.13.1. Estimates for controlled SNS equations 2643.13.2. Value function 2693.13.3. Viscosity solutions and comparison theorem 2703.13.4. Existence of viscosity solutions 276


Chapter 4. Mild solutions in spaces of continuous functions 2894.1. The setting and an introduction to the methods 290

4.1.1. The method in the parabolic case 2914.1.2. The method in the elliptic case 293

4.2. Preliminaries 2944.2.1. G-derivatives 2944.2.2. Weighted spaces 299

4.3. Smoothing properties of transition semigroups 3034.3.1. The case of Ornstein-Uhlenbeck semigroup 3034.3.1.1. The equation 3034.3.1.2. The semigroup and the associated Kolmogorov equation 3044.3.1.3. Smoothing of Rt: assumptions and null controllability 3054.3.1.4. Smoothing of Rt: results and estimates 3084.3.1.5. Smoothing of Rt: examples 3124.3.1.6. Joint space-time regularity of Rt 316

4.3.2. The case of perturbed Ornstein-Uhlenbeck semigroup 320

6 CONTENTS

4.3.3. The case of invertible diffusion coefficient 3234.4. Mild solutions of HJB equations 328

4.4.1. The parabolic case 3284.4.1.1. Formal derivation of the mild form 3294.4.1.2. Definition of mild solution and assumptions 3294.4.1.3. Existence and uniqueness of mild solutions 332

4.4.2. The elliptic case for big discount factor 3384.4.2.1. Formal derivation of the mild form 3384.4.2.2. Definition of mild solution and assumptions 3394.4.2.3. Existence and uniqueness of mild solutions 341

4.5. Approximation of mild solutions: strong solutions 3444.5.1. The parabolic case 3444.5.2. The elliptic case 349

4.6. HJB equations of Ornstein-Uhlenbeck type: Lipschitz Hamiltonian 3524.6.1. The parabolic case 3544.6.1.1. C2 regularity of the mild solutions 355

4.6.2. The elliptic case 3604.6.2.1. C2 regularity of the mild solutions 3604.6.2.2. Existence of mild solutions for all λ > 0 363

4.7. HJB equations of Ornstein-Uhlenbeck type: Locally LipschitzHamiltonian 370

4.7.1. The parabolic case 3704.7.1.1. Local existence and uniqueness of mild solutions 3714.7.1.2. Apriori estimates 3734.7.1.3. Global existence of mild and strong solutions 378

4.8. Stochastic Control: Verification Theorems and Optimal Feedbacks 3804.8.1. The finite horizon case 3804.8.1.1. The state equation 3804.8.1.2. The optimal control problem and the HJB equation 3824.8.1.3. The verification theorem 3844.8.1.4. Optimal feedbacks 386

4.8.2. The infinite horizon case 3884.8.2.1. The state equation 3884.8.2.2. The optimal control problem and the HJB equation 3894.8.2.3. The verification theorem 3914.8.2.4. Optimal feedbacks 392

4.8.3. Examples 3934.8.3.1. Diagonal cases 3944.8.3.2. Invertible diffusion coefficients 396

4.9. Mild solutions of HJB for two special problems 3974.9.1. Control of Stochastic Burgers and Navier-Stokes equations 3974.9.1.1. The case of Burgers equation 3984.9.1.2. Two-dimensional Navier-Stokes equation 4024.9.1.3. Three-dimensional Navier-Stokes equation 405

4.9.2. Control of reaction-diffusion equations 4084.9.2.1. The finite horizon problem 4084.9.2.2. The infinite horizon problem 413

4.10. Regular solution through “explicit” representations 4144.10.1. Quadratic Hamiltonians 415

CONTENTS 7

4.10.2. Explicit solutions in the homogeneous case 4164.11. Bibliographical notes 416

Chapter 5. Mild solutions in L2 spaces 4195.1. Introduction to the methods 4205.2. Preliminaries and the linear problem 422

5.2.1. Notation 4225.2.2. The reference measure m and the main assumptions on the

linear part 4225.2.3. The operator N 4255.2.4. The gradient operator DQ and the space W 1,2

Q (H,m) 4275.2.5. The operator R 4285.2.6. Two key lemmas 435

5.3. The HJB equation 4385.4. Approximation of mild solutions 4415.5. Application to stochastic optimal control 445

5.5.1. The state equation 4455.5.2. The optimal control problem and the HJB equation 4465.5.3. The verification theorem 4465.5.4. Optimal feedbacks 4525.5.5. Continuity of the value function and nondegeneracy of the

invariant measure 4535.6. Examples 456

5.6.1. Optimal control of delay equations 4565.6.2. Control of stochastic PDEs of first order 4575.6.3. Second order SPDE in the whole space 458

5.7. Results in special cases 4595.7.1. Parabolic HJB equations 4595.7.2. Applications to finite horizon optimal control problems 4615.7.3. Elliptic HJB equations 463


Chapter 6. HJB Equations through Backward Stochastic DifferentialEquations 467

6.1. Complements on forward equations with multiplicative noise 4676.1.1. Notation on vector spaces and stochastic processes. 4676.1.2. The class G. 4696.1.3. The forward equation: existence, uniqueness and regularity. 472

6.2. Regular dependence on data 4746.2.1. Differentiability. 4746.2.2. Differentiability in the sense of Malliavin. 476

6.3. Backward Stochastic Differential Equations in Hilbert spaces 4876.3.1. Well-posedness. 4876.3.2. Regular dependence on data. 4936.3.3. Forward-backward systems. 499

6.4. BSDEs and mild solutions to HJB and other Kolmogorov non linearequations 502

8 CONTENTS

6.5. Applications to optimal control problems 5066.6. Elliptic HJB equation with arbitrarily growing Hamiltonian 5106.7. The associated forward-backward system 511

6.7.1. Differentiability of the BSDE and a-priori estimate on thegradient 513

6.8. Mild Solution of the elliptic PDE 5176.9. Application to optimal control in infinite horizon 5206.10. Elliptic case with non constant diffusion: statements only 524

Appendix A. Notations and Function Spaces 531A.1. Basic Notation 531A.2. Function spaces 531

Appendix B. Linear operators and C0-semigroups 537B.1. Linear operators 537B.2. Dissipative operators 539B.3. Trace class and Hilbert-Schmidt operators 540B.4. C0-semigroups and related results 543

B.4.1. Basic definitions 543B.4.2. Hille-Yosida Theorem and Yosida approximation 544B.4.3. Analytic semigroups and fractional powers of generators 546

B.5. π-convergence, K-convergence, π- and K-continuous semigroups 547B.5.1. π-convergence and K-convergence 547B.5.2. π- and K-continuous semigroups and their generators 551B.5.2.1. The definitions 551B.5.2.2. The generators 553B.5.2.3. Cauchy problems for π-continuous and K-continuous

semigroups 554B.6. Approximation of continuous functions through K-convergence 556B.7. Approximation of solutions of Kolmogorov equations through

K-convergence 558B.7.1. Classical and strong solutions of (B.30) 559B.7.2. The Ornstein-Uhlenbeck semigroup associated to (B.30), the

mild solutions and their properties 562B.7.3. The approximation results 567

Appendix C. Parabolic equations with non-homogeneous boundary conditions573C.1. Dirichlet and Neumann maps 573C.2. Non-zero boundary conditions, the Dirichlet case 575C.3. Non-zero boundary conditions, the Neumann case 577C.4. Boundary noise, Neumann case 579C.5. Boundary noise, Dirichlet case 579

Appendix D. Functions, derivatives and approximations 583D.1. Continuity properties, modulus of continuity 583D.2. Fréchet and Gâteaux derivatives 584

CONTENTS 9

D.3. Inf-Sup convolutions 585D.4. Two versions of Gronwall’s Lemma 589

Appendix E. Viscosity solutions in RN 591E.1. Second order jets 591E.2. Definition of viscosity solution 593E.3. Finite dimensional maximum principles 594E.4. Perron’s method 595

Bibliography 597

Preface

The main objective of this book is to give an overview of the theory of Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDE) in infinite dimensionalHilbert spaces and its applications to stochastic optimal control of infinite dimen-sional processes and related fields. Both areas have been developing very rapidly inthe last few decades. While there exist several excellent monographs on this subjectin finite dimensional spaces (see e.g. [194, 195, 294, 349, 364, 382, 449]), much lesshas been written in infinite dimensional spaces. A good account of the infinite di-mensional case in the deterministic context can be found in [312]. Other books thattouch the subject are [23, 129, 364]. We attempt to fill this gap in the literature.Infinite dimensional diffusion processes appear naturally and are used to model phe-nomena in physics, biology, chemistry, economics, mathematical finance, engineer-ing and many other areas (see e.g. [93, 127, 130, 282, 444]. This book investigatesthe PDE approach to their stochastic optimal control, however infinite dimensionalPDE can also be used to study other properties of such processes as large deviations,invariant measures, stochastic viability, stochastic differential games for infinite di-mensional diffusions, etc. (see [60, 127, 129, 182, 184, 192, 361, 363, 425, 428]).

To illustrate the main theme of the book let us begin with a model distributedparameter stochastic optimal control problem. We want to control a process (calledthe state) given by an abstract stochastic differential equation in a real, separableHilbert space H

dX(s) = (AX(s) + b(s,X(s), a(s)))ds+ σ(s,X(s), a(s))dW (s), s > t ≥ 0X(t) = x ∈ H,

where A is the generator of a C0 semigroup in H, b,σ are certain bounded functionsand W is a so called Q-Wiener process1 in H. The functions a(·), called controls,are stochastic processes with values in some metric space Λ, which satisfy certainmeasurability properties. The above abstract stochastic differential equation is verygeneral and includes various semi-linear stochastic PDE, as well as other equationswhich can be rewritten as stochastic functional evolution equations, for instancestochastic differential delay equations. In a most typical optimal control problemwe want to find a control a(·), called optimal, which minimizes a cost functional

J(t, x; a(·)) = E T

tl(s,X(s), a(s))ds+ g(X(T ))

.

(for some T > t) among all admissible controls for some functions l : [0, T ]×H×Λ →R, g : H → R.

The dynamic programming approach to the above problem is based on studyingthe properties of the so called value function

V (t, x) = infa(·)

J(t, x; a(·))

1Q is a suitable self-adjoint positive operator in H, the covariance operator for W .

11

12 PREFACE

and characterizing it as a solution of a fully nonlinear PDE, the associated HJBequation. Since the state X(s) evolves in the infinite dimensional space H, thisPDE is defined in [0, T ] × H. The link between the value function V and theHJB equation is established by the Bellman principle of optimality known as thedynamic programming principle (DPP),

V (t, x) = infa(·)

E η

tl (s,X (s) , a (s)) ds+ V (η, X (η))

, for all η ∈ [t, T ].

Heuristically the DPP can be used to define a two parameter nonlinear evolutionsystem and the associated HJB equation

Vt + Ax,DV + infa∈Λ

12Tr

(σ(t, x, a)Q

12 )(σ(t, x, a)Q

12 )∗D2V

+ b(t, x, a), DV + l(t, x, a)= 0,

V (T, x) = g(x).(0.1)

is its generating equation. Such PDE is called infinite dimensional or a PDE ininfinitely many variables. We also call it unbounded since it has a term with anunbounded operator A which is well defined only on the domain of A. Other termsmay also be undefined for some values of DV and D2V , the Fréchet derivativesof V , which we may identify with elements of H and with bounded, self-adjointoperators in H respectively. In particular, the term Tr[(σQ

12 )(σQ

12 )∗D2V ] is well

defined only if (σQ 12 )(σQ

12 )∗D2V is of trace class.

The main idea is to use the HJB equation to study properties of the valuefunction, find conditions for optimality, obtain formulas for synthesis of optimalfeedback controls, etc. This approach turned out to be very successful for finitedimensional problems because of its clarity and simplicity and thanks to the devel-opments of the theory of fully nonlinear elliptic and parabolic PDE, in particularthe introduction of the notion of viscosity solution and advances in the regularitytheory. However even there many open questions remain, especially if the HJBequations are degenerate. We hope the dynamic programming approach will beequally valuable for infinite dimensional problems even though a complete theoryis not available yet.

Equation (0.1) is an example of a fully nonlinear second-order PDE of (degen-erate) parabolic type. In the book we will deal with more general and differentversions of such equations and their degenerate elliptic counterparts. If Λ is a sin-gleton, (0.1) is just a terminal value problem for a linear Kolmogorov equation. IfΛ is not a singleton but the diffusion coefficient σ is independent of the controlparameter a, (0.1) is semi-linear. The theory of linear equations (and some specialsemi-linear ones) has been studied by many authors and can be found in books[23, 80, 129, 453]. The emphasis of this book is on semi-linear and fully nonlinearequations.

There are several notions of solution applicable to PDE in Hilbert spaces whichare discussed in the book: classical solutions, strong solutions, mild solutions in thespace of continuous functions, solutions in L2(µ), viscosity solutions. Classicalsolutions are the most regular ones. This notion of solution requires C1,2 regularityin the Fréchet sense and imposes additional conditions so that all terms in theequation make sense pointwise for (t, x) ∈ [0, T ]×H. When classical solutions existwe can apply the classical dynamic programming approach to obtain verificationtheorems and the synthesis of optimal feedback controls. Unfortunately in almost allinteresting cases it is not possible to find such solutions, however they are very usefulas a theoretical tool in the theory. The notions of strong solutions, mild solutionsin the space of continuous functions, and solutions in L2(µ) are introduced and

PREFACE 13

studied only for semi-linear equations and define solutions which have at least firstderivative (in some suitable sense). Verification theorems and synthesis of optimalfeedback controls can still be developed within their framework. The notion ofviscosity solutions is the most general and applies to fully nonlinear equations,however at the current stage there are no results on verification theorems andsynthesis of optimal feedback controls.

Infinite dimensional problems present unique challenges, among them are thelack of local compactness and no equivalent of Lebesgue measure. This means thatstandard finite dimensional elliptic and parabolic techniques which are based onmeasure theory cannot be carried over to the infinite dimensional case. Moreover,the equations are mostly degenerate and contain unbounded terms which are singu-lar. So the methods to find regular solutions to PDE in infinite dimensions like ourstend to be global and are based on semigroup theory, smoothing properties of tran-sition semigroups (like the Ornstein-Uhlenbeck ones), fixed point techniques, andstochastic analysis. These methods are mostly restricted to equations of semi-lineartype. On the other hand, the notion of viscosity solution is perfectly suited for fullynonlinear equations. It is local and it does not require any regularity of solutionsexcept continuity. As in finite dimensions it is based on maximum principle throughthe idea of “differentiation by parts”, i.e. replacing the non existing derivatives ofviscosity subsolutions (respectively, supersolutions) by derivatives of smooth testfunctions at points where their graphs touch the graphs of subsolutions (respec-tively, supersolutions) from above (respectively, below). However as the readerswill see, this idea has to be carried out very carefully in infinite dimensions.

The book contains chapters on the most important topics in HJB equationsand the DPP approach to infinite dimensional stochastic optimal control.

Chapter 1 contains basic material on infinite dimensional stochastic calculuswhich is needed in the subsequent chapters. It is however not intended to bean introduction to stochastic calculus and the readers are expected to have somefamiliarity with it. Chapter 1 is included to make the book more self-contained.Most of the results presented there are well known, hence we only provide referenceswhere the reader can find the proofs and find more information about the concepts,examples, etc. We provide proofs only in cases where we could not find goodreferences in the literature.

In Chapter 2 we introduce a general stochastic optimal control problem andprove a key result in the theory, namely the dynamic programming principle. Weformulate it in an abstract and general form so that it can be used in many caseswithout the need of reproving it. Solutions of stochastic PDE must be interpretedin various ways (strong, mild, variational, etc.) and our formulation of the DPPtries to capture this phenomenon. Our proof of the DPP is based on standard ideashowever we tried to avoid heavy probabilistic methods regarding weak uniquenessof solutions of stochastic differential equations. Our proof is thus more analytical.

We also introduce many examples of stochastic optimal control problems whichcan be studied in the framework of the approach presented in the book. They shouldgive the readers an idea about the range and applicability of the material.

Chapter 3 is devoted to the theory of viscosity solutions. The reader shouldkeep in mind the following principle when it comes to unbounded PDE in infinitedimensions: there is no single definition of viscosity solutions that applies to allequations. This is due to the fact that there are many different PDE which containdifferent unbounded operators and terms which are continuous in various norms.Also the solutions have to be continuous with respect to weaker topologies. Howeverthe main idea of the notion of viscosity solutions is always the same as we describedbefore. What changes is the choice of test functions, spaces, topologies, and the

14 PREFACE

interpretation of various terms in the equation. In this book we focus on the notionof so called B-continuous viscosity solution which was introduced by Crandall andP. L. Lions in [103, 104] for first order equations and later adapted to secondorder equations in [422]. The key result in the theory is the comparison principlewhich is very technical. Its main component is the so called maximum principle forsemicontinuous functions. The proof of such result in finite dimensions was firstobtained in [280] and was later simplified and generalized [99, 100, 101, 270]. It isheavily based on measure theory and is not applicable to infinite dimensions. Thusthe theory uses a finite dimensional reduction technique introduced by P. L. Lionsin [321]. It restricts the class of equations which can be considered, in particularthey have to be highly degenerate in the second order terms. We present threetechniques to obtain existence of viscosity solutions. The first and most importantfor this book is the DPP and the stochastic optimal control interpretation, showingdirectly that the value function is a viscosity solution. This technique applies toHJB equations. The other techniques are finite dimensional approximations andPerron’s method. Both can be applied to more general equations, for instance Isaacsequations associated to two-player, zero-sum stochastic differential games, howeverthey have limitations of their own. Moreover we discuss other topics of the theoryof viscosity solutions like consistency, singular perturbations, etc.. Several specialequations are also studied in the book because of their importance and becausethey are good examples to show how the definition of viscosity solutions and sometechniques can be adjusted to particular cases. They are the HJB equations forthe optimal control of Duncan-Mortensen-Zakai equation, stochastic Navier-Stokesequations, and stochastic boundary control. In particular the last one also containsideas on how to handle HJB equations which may be non-degenerate, for instanceif Q is not of trace class. Finally we present applications to infinite dimensionalBlack-Scholes-Barenblatt equations of mathematical finance.

Chapter 4 is devoted to the theory of mild and strong solutions in spacesof continuous functions through fixed point techniques based on the smoothingproperties of transition semigroups such as Ornstein-Uhlenbeck ones. This theoryapplies only to semi-linear equations, i.e. when the coefficient σ does not depend onthe control parameter a, and historically it was the first approach introduced in theliterature. It was first introduced by Barbu and Da Prato [23] and later improvedand developed in various papers, see e.g [63, 64, 79, 81, 226, 231, 232, 235].

Chapter 4 is divided into four main parts. In the first one (Sections 4.2-4.3), wepresent the basic tools needed for the analysis: the theory of generalized gradientsand the smoothing of transition semigroups. In the second one (Sections 4.4 to 4.7),we develop the theory for a general type of semi-linear HJB equation (parabolic andelliptic) without connection with optimal control problems. The main idea behindthis approach is the following. Consider the HJB equation (0.1) in the semi-linearautonomous case:

Vt +AV + infa∈Λ

b(x, a), DV + l(t, x, a)

= 0,

V (T, x) = g(x),(0.2)

where A is the linear operator

Aϕ = Ax,Dϕ+ 1

2Tr

(σ(x)Q

12 )(σ(x)Q

12 )∗D2ϕ

.

If such operator generates a semigroup etA then, by the variation of constantsformula, one can rewrite (0.2) in the integral form as

V (t, x) = e(T−t)Ag(x) +

T

t

e(T−s)AF (s, ·)

(x) ds

PREFACE 15

where F (s, x) := infa∈Λ b(x, a), DV + l(s, x, a). The solution of this integralequation is called a mild solution and is obtained by fixed point techniques. Todefine it, the solution must at least have first order spatial derivative. Thus oneneeds suitable smoothing properties of the semigroup etA (which is the Ornstein-Uhlenbeck semigroup in the simplest case). Since this semigroup is not strongly con-tinuous, apart from very special cases, one needs to use the theory of π-semigroupsintroduced in [386] or the one of weakly continuous (or K-continuous) semigroups[75, 82, 225].

In the third part (Section 4.8), we develop a connection with stochastic optimalcontrol problems. The fact that mild solutions have first order spatial derivativeallows to give a meaning to formulae for optimal feedbacks. However the proofsof verification theorems and optimal feedback formulae cannot be done straight-forwardly as one needs to apply Ito’s formula in infinite dimensions which requiressmooth functions. For this reason (following [231]), one introduces the notion ofstrong solution of the HJB equation (0.2) as a suitable limit of classical solutionsand proves that any mild solution is also a strong solution.

The fourth and the last part of the chapter (Sections 4.9-4.10) deals with somespecial equations. In Section 4.9 we show how the techniques developed in theprevious sections can be adapted to HJB equations and analysis of optimal controlproblems for stochastic Burgers equation, stochastic Navier-Stokes equations andstochastic reaction diffusion equations. In Section 4.10 we discuss some equationsfor which explicit representations of the solutions can be found. Such cases arealways of interest in applications.

Chapter 5 is devoted to a relatively new and promising theory of mild and strongsolutions in spaces of L2 functions with respect to a suitable measure µ (see [223, 3,4, 94]). The contents of this chapter are similar to the previous one as the main ideasbehind the definition of mild and strong solutions of HJB equations are the same.The difference is in the fact that the reference space is not the space of continuousfunctions but the space of square integrable ones with respect to the measure µ. Theresults are similar: existence and uniqueness of solutions of HJB equations throughfixed point arguments, verification theorem through approximations, existence ofoptimal feedbacks. The advantage of this approach is that the results require weakerassumptions on the data, thus enlarging the range of possible applications, includingfor instance the control of delay equations, however at a cost of weaker statements,for example the first order spatial derivative is now defined in a Sobolev weak senseand is not in general a Fréchet derivative. The main tools used here are the theoryof invariant measures for infinite dimensional stochastic differential equations andthe properties of transition semigroups in the space of integrable functions withrespect to such measures.

Chapter 6 is devoted to a different and in many respects complementary tech-nique of Backward Stochastic Differential Equations (BSDE). It is written by twoleading expert in the field. BSDE are Itô type equations in which the initial con-dition is replaced by a final condition and a new unknown process appears corre-sponding to a suitable martingale term. In the nonlinear, finite dimensional caseBSDE have been introduced in [372] while their direct connection with optimalstochastic control was firstly investigated in [154] and [377]. Since then, the generaltheory of BSDE has developed considerably, see [56, 58, 152, 289, 325, 371]. Besidesstochastic control, applications were given to many fields, for instance to optimalstopping, stochastic differential games, nonlinear partial differential equations andmany topics related to mathematical finance. Infinite dimensional BSDE have alsobeen considered, see for instance [97, 211, 252, 264, 373]. The interest for us is thatBSDE provide an alternative way to represent the value function of an optimal

16 PREFACE

control problem and consequently to study the corresponding HJB equation and tosolve the control problem. It turns out that the most suitable notion of solution forthe HJB equation is, in this context, the one of mild solution on spaces of continuousfunctions but, unlike in Chapter 4, the BSDE method seems particularly adaptedto treat degenerate cases in which the transition semigroup has no smoothing prop-erties. The price to pay is that normally we need more regular coefficients and astructural condition (imposing, roughly speaking, that the control acts within theimage of the noise). If these requirements are satisfied the BSDE techniques revealto be very flexible. In particular, in Chapter 6 we will show how they allow to treatboth parabolic and elliptic HJB equations (see [55, 212, 265, 336]). Moreover wewill present extensions to the case of locally Lipschitz Hamiltonian, quadraticallygrowing with respect to the gradient (see [54, 55, 205]), to the case of HJB equationscorresponding to ergodic control problems ([206]) and the case of state equationswith noise and control on the boundary ([131, 338]). We will finally describe howthe regularity requirement of the coefficients can be partially removed introducinga suitable concept of “generalized gradient”, see [213].

It is impossible to cover all aspects of the theory of HJB equations in infinitedimensions and its connections to stochastic optimal control. In particular thetheory of integro-PDE is an emerging area which is not presented in the book. We donot discuss first order equations and extensions to Banach spaces. Equations in thespace of probability measures is another emerging topic. We have chosen a selectionof topics which give a broad overview of the field and enough information so thatthe readers can start exploring the subject on their own. There are already enoughimportant applications to justify the interest in the subject. The readers should notbe restricted to the boundaries drawn by the book. We hope the book will spur theinterest and research in the field among theoretical and applied mathematicians,and it will be useful to all kinds of scientists and researchers working in areas relatedto stochastic control.

CHAPTER 1

Preliminaries on stochastic calculus in infinitedimensions

1.1. Basic probability

We recall basic notions of measure theory and give a short introduction torandom variables and the theory of Bochner integral.

1.1.1. Probability spaces, σ-fields.

Definition 1.1 (π-system, σ-field) Consider a set Ω and denote by P(Ω) thepower set of Ω.

(i) A nonempty class of subsets of Ω, F ⊂ P(Ω), is called a π-system if itis closed under finite intersections.

(ii) A class of subsets of Ω, F ⊂ P(Ω) is called a σ-field in Ω if Ω ∈ F

and F is closed under complements and countable unions.(iii) A class of subsets of Ω, F ⊂ P(Ω) is called a λ-system if:

– Ω ∈ F ;– if A,B ∈ F , A ⊂ B, then B \A ∈ F ;– if Ai ∈ F , i = 1, 2, ..., Ai ↑ A, then A ∈ F .

If G and F are two σ-fields in Ω and G ⊂ F , we say that G is a sub-σ-fieldof F . Given a class C ⊂ P(Ω), the smallest σ-field containing C is called theσ-field generated by C . It is denoted by σ(C ). A σ-field F in Ω is said to becountably generated if there exists a countable class of subsets C ⊂ P(Ω) such thatσ(C ) = F .

If C ⊂ P(Ω) and A ⊂ Ω we denote C ∩ A := B ∩ A : B ∈ C . We denote byσA(C ∩ A) the σ-field of subsets of A generated by C ∩ A. It is easy to see thatσA(C ∩A) = σ(C ) ∩A (see for instance [12], page 5).

For A ⊂ Ω we denote its complement by Ac := Ω \ A, and for A,B ⊂ Ω wedenote their symmetric difference by A∆B := (A \ B) ∪ (B \ A). We will writeR+ = [0,+∞),R +

= [0,+∞) ∪ +∞,R = R ∪ ±∞.Theorem 1.2 Let G be a π-system and F be a λ-system in some set Ω, suchthat G ⊂ F . Then σ(G ) ⊂ F .

Proof. See [281], Theorem 1.1, page 2. Corollary 1.3 Let G be a π-system and F be the smallest family of subsets ofΩ such that:

– G ⊂ F ;– if A ∈ F then Ac ∈ F ;– if Ai ∈ F , Ai ∩Aj = ∅ for i, j = 1, 2, ..., i = j, then ∪∞

i=1Ai ∈ F .Then σ(G ) = F .

Proof. Since σ(G ) satisfies the three conditions for F , we obviously haveF ⊂ σ(G ). For the opposite inclusion it remains to notice that F is a λ-system.(For a self-contained proof see also [130], Proposition 1.4, pages 17.)

17

18 1. PRELIMINARIES ON STOCHASTIC CALCULUS IN INFINITE DIMENSIONS

Definition 1.4 (Measurable space) If Ω is a set and F is a σ-field in Ω, thepair (Ω,F ) is called a measurable space .

Definition 1.5 (Probability measure, probability space) Consider a mea-surable space (Ω,F ). A function µ : F → [0,+∞)∪ +∞ is called a measure on(Ω,F ) if µ(∅) = 0, and whenever Ai ∈ F , Ai ∩Aj = ∅ for i, j = 1, 2, ..., i = j, then

µ

∞

i=1

Ai

=

∞

i=1

µ(Ai).

The triplet (Ω,F , µ) is called a measure space. If µ(Ω) < +∞ we say that µ is abounded measure. If Ω =

∞n=1 An, where An ∈ F , µ(An) < +∞, n = 1, 2, ..., we

say that µ is a σ-finite measure. If µ(Ω) = 1 we say that µ is a probability measure.We will use the symbol P to denote probability measures. The triplet (Ω,F ,P) iscalled a probability space.

Thus a probability measure is a σ-additive function P : F → [0, 1] such thatP(Ω) = 1.

Given a measure space (Ω,F , µ), we denote N := F ⊂ Ω : ∃G ∈ F , F ⊂G,µ(G) = 0. The elements of N are called µ-null sets. If N ⊂ F , the measurespace (Ω,F , µ) is said to be complete. The σ field F := σ(F ,N ) is called thecompletion of F (with respect to µ). It is easy to see that σ(F ,N ) = A∪B : A ∈F , B ∈ N. If G ⊂ F is another σ-field then σ(G ,N ) is called the augmentationof G by the null sets of F . The augmentation of G may be different from itscompletion, as the latter is just the augmentation of G by the subsets of the sets ofmeasure zero in G . We also have σ(G ,N ) = A ⊂ Ω : A∆B ∈ N for some B ∈ G .Lemma 1.6 Let µ1, µ2 be two bounded measures on a measurable space (Ω,F ),and let G be a π-system in Ω such that Ω ∈ G and σ(G ) = F . Then µ1 = µ2 ifand only if µ1(A) = µ2(A) for every A ∈ G .

Proof. See [281], Lemma 1.17, page 9.

Let Ωt, t ∈ T be a family of sets. We will denote the Cartesian product ofthe family Ωt by ×t∈T Ωt. If T is finite (T = 1, ..., n) or countable (T = N), wewill also write Ω1 × ...× Ωn, respectively Ω1 × Ω2 × .... If each Ωt is a topologicalspace, we endow ×t∈T Ωt with the product topology. If each Ωt has a σ-field Ft,we define the product σ-field ⊗t∈T Ft in ×t∈T Ωt as the σ-field generated by theone-dimensional cylinder sets At×(×s =tΩs). If T = 1, ..., n (respectively, T = N)we will just write ⊗t∈T Ft = F1⊗ ...⊗Fn (respectively, ⊗t∈T Ft = F1⊗F2⊗ ...).

If S is a topological space, the σ-field generated by the open sets of S is calledBorel σ-field. It will be denoted by B(S). If S is a metric space, unless statedotherwise, its default σ-field will always be B(S). It is not difficult to see that ifS1, S2, ... are separable metric spaces, then

B(S1 × S2 × ...) = B(S1)⊗ B(S2)⊗ ....

If (S, ρ) is a metric space, A ⊂ S, and we consider (A, ρ) as a metric space, thenB(A) = A∩B(S). A complete separable metric space is called a Polish space. AlsoB(R +

) = σ(B(R+), +∞),B(R) = σ(B(R), −∞, +∞).A measurable space (Ω,F ) is called countably determined (or F is called count-

ably determined) if there is a countable set F0 ⊂ F such that any two probabilitymeasures on (Ω,F ) that agree on F0 must be the same. It follows from Lemma1.6 that if F is countably generated then F is countably determined. If S is aPolish space then B(S) is countably generated.

1.1. BASIC PROBABILITY 19

If (Ωi,Fi, µi), i = 1, ..., n are measure spaces, their product measure on (Ω1 ×...× Ωn,F1 ⊗ ...⊗ Fn) is denoted by µ1 ⊗ ...⊗ µ2. It is the measure such that

µ1 ⊗ ...⊗ µ2(A1 × ...×An) = µ1(A1) · ... · µn(An)

for all Ai ∈ Ωi, i = 1, ..., n.If S is a metric space, a bounded measure µ on (S,B(S)) is called regular if

µ(A) = supµ(C) : C ⊂ A,C closed = infµ(U) : A ⊂ U,U open ∀A ∈ B(S).Every bounded measure on (S,B(S)) is regular (see [374], Chapter II, Theorem1.2). A bounded measure µ on (S,B(S)) is called tight if for every > 0 thereexists a compact set K ⊂ S such that µ(S \K) < . If S is a Polish space thenevery bounded measure on (S,B(S)) is tight (see [374], Chapter II, Theorem 3.2).

We refer to [42, 44, 199, 281, 374] for more on the general theory of measureand probability.

1.1.2. Random variables.

Definition 1.7 (Random variable) A measurable map X between two mea-surable spaces (Ω,F ) and (Ω,G ) is a called a random variable. This means thatX is a random variable if X−1(A) ∈ F for every A ∈ G . We write it shortly asX−1(G ) ⊂ F . Sometimes we will just say that X is F/G measurable.If Ω = R (resp. R+) and G is the Borel σ-field B(R) (resp. B(R+)) then X is saidto be a real random variable (resp. positive random variable).If Ω, Ω are topological spaces and F ,G are the Borel σ-fields then X is said to beBorel measurable.

Given a random variable X : (Ω,F ) → (Ω,G ) we denote by σ(X) the smallestsub-σ-field of F that makes X measurable, i.e. σ(X) := X−1(G ). It is called theσ-field generated by X. Given a set of indices I and a family of random variablesXi : (Ω,F ) → (Ω,G ), i ∈ I, the σ-field σ

Xii∈I

generated by Xii∈I is the

smallest sub-σ-field of F that makes all the functions Xi :Ω,σ

Xii∈I

→

(Ω,G ) measurable, i.e. σXii∈I

= σ

X−1

i (G ) : i ∈ I.

Lemma 1.8 Let (Ω,F ) be a measurable space. Then:(i) If (Ω,G ) is a measurable space, X : Ω → Ω, and C ⊂ G is such that

σ(C ) = G , then X is F/G measurable if and only if X−1(C ) ⊂ F . Moreover,σ(X) = σ(X−1(C )).

(ii) If Xn : Ω → R, n = 1, 2, ... are random variables, then supn Xn, infn Xn,lim supn Xn, lim infn Xn are random variables.

(iii) Let Xn : Ω → S, n = 1, 2, ... be random variables, where S is a metricspace. Then:

– if S is complete then ω : Xn(ω) converges ∈ F ;– if Xn → X on Ω, then X is a random variable.

(iv) Let (Ωi,Fi), i = 1, 2 be measurable spaces, and X : Ω1 × Ω2 → Ω be(F1 ⊗ F2)/F measurable. Then, for every ω1 ∈ Ω1, Xω1(·) = X(ω1, ·) is F2/Fmeasurable, and, for every ω2 ∈ Ω2, Xω2(·) = X(·,ω2) is F1/F measurable.

Proof. See for instance [281], Lemmas 1.4, 1.9, 1.10, and [407], Theorem7.5, page 138. Theorem 1.9 Let (Ω,F ) and (Ω,G ) be two measurable spaces and (S, d) a Pol-ish space (i.e. a complete and separable metric space). Let X : (Ω,F ) → (Ω,G )and φ : (Ω,F ) → (S,B(S)) be two random variables. Then φ is measurable asa map from (Ω,σ(X)) to (S,B(S)) if and only if there exists a measurable mapη : (Ω,G ) → (S,B(S)) such that φ = η X.


Proof. See [281], Lemma 1.13, page 7, or [449] Theorem 1.7 page 5.

We refer to [42, 199, 281, 407] for more on measurability and for the generaltheory of integration.

Definition 1.10 (Borel isomorphism) Let (Ω,F ) and (Ω,G ) be two measur-able spaces. A bijection f from Ω onto Ω is called a Borel isomorphism if f is F/Gmeasurable and f−1 is G /F measurable. We then say that (Ω,F ) and (Ω,G ) areBorel isomorphic.

Definition 1.11 (Standard measurable space) A measurable space (Ω,F ) iscalled standard if it is Borel isomorphic to one of the following spaces:

(i) (1, .., n,B(1, .., n)),(ii) (N,B(N)),(iii)

0, 1N,B(0, 1N)

,

where we have the discrete topologies in 1, .., n and N, and the product topologyin 0, 1N.

The following theorem collects results that can be found in [374] (Chapter I,Theorems 2.8 and 2.12).

Theorem 1.12 If S is a Polish space, then (S,B(S)) is standard. If a Borelsubset of S is uncountable, then it is Borel isomorphic to 0, 1N. Two Borel subsetsof S are Borel isomorphic if and only if they have the same cardinality. If (Ω,F )is standard and A ∈ F , then (A,F ∩A) is standard.

In particular we have the following result.

Theorem 1.13 If (Ω,F ) is standard, then it is Borel isomorphic to a closedsubset of [0, 1] (with its induced Borel sigma field).

Definition 1.14 (Simple random variable) Let (Ω,F ) be a measurable space,and (S, d) be a metric space (endowed with the Borel σ-field induced by the distance).A random variable X : (Ω,F ) → (S,B(S)) is called simple (or simple function) ifit has a finite number of values.

Lemma 1.15 Let f : (Ω,F ) → S be a measurable function between a measurablespace (Ω,F ) and a separable metric space (S, d) (endowed with the Borel σ-fieldinduced by the distance). Then there exists a sequence fn : Ω → S of simple,F/B(S) measurable functions, such that d (f(ω), fn(ω)) is monotonically decreasingto 0 for every ω ∈ Ω.


Lemma 1.16 Let S be a Polish space with metric d. Let (Ω,F ,P) be a completeprobability space and let G1,G2 ⊂ F be two σ-fields with the following property:for every A ∈ G2 there exists B ∈ G1 such that P(A∆B) = 0. Let f : (Ω,G2) →(S,B(S)) be a measurable function. Then there exists a function g : (Ω,G1) →(S,B(S)) such that f = g, P a.e., and simple functions gn : (Ω,G1) → (S,B(S))such that d(f(ω), gn(ω)) monotonically decreases to 0, P-a.e..

Proof. The proof follows the lines of the proof of Lemma 1.25, page 13, in[281].Step 1: Let us assume first that f = x1A (the characteristic function) for someA ∈ G2 and x ∈ S. By hypothesis, we can find B ∈ G1 s.t. P(A∆B) = 0 and thenthe claim is proved if we choose gn ≡ g = x1B . The same argument holds for asimple function f .Step 2: For the case of a general f , thanks to Lemma 1.15 we can find a sequence


of simple, G2-measurable functions fn such that d(f(ω), fn(ω)) monotonically de-creases to 0. By Step 1, we can find simple, G1-measurable functions gn such thatfn = gn, P-a.e. Thus the claim follows taking g(ω) := lim gn(ω) if the limit existsand g(ω) = s (for some s ∈ S) otherwise.

Lemma 1.17 Let (Ω,F ) be a measurable space, and V ⊂ E be two real separableBanach spaces such that the embedding of V into E is continuous. Then:

(i) B(E) ∩ V ⊂ B(V ) and B(V ) ⊂ B(E).(ii) If X : Ω → V is F/B(V ) measurable, then it is F/B(E) measurable.(iii) If X : Ω → E is F/B(E) measurable, then X · 1X∈V is F/B(V )

measurable.(iv) X : Ω → E is F/B(E) measurable if and only if for every f ∈ E∗,

f X is F/B(R) measurable.

Proof. The embedding of V into E is continuous, so B(E) ∩ V ⊂ B(V ).Since the embedding is also one-to-one, it follows from [374], Theorem 3.9, page21, that B(V ) ⊂ B(E), which completes the proof of (i). Parts (ii) and (iii) aredirect consequences of (i). f(Ω) is separable because E is separable, so Part (iv) isa particular case of the Pettis theorem, see [381] Theorem 1.1.

Notation 1.18 If E is a Banach space we denote by | · |E its norm. Given twoBanach spaces E and F , we denote by L(E,F ) the Banach space of all continuouslinear operators from E to F . If E = F we will usually write L(E) instead ofL(E,F ). If H is a Hilbert space we denote by ·, · its inner product. We willalways identify H with its dual. If V,H are two real separable Hilbert spaces,we denote by L2(V,H) the space of Hilbert-Schmidt operators from V to H (seeAppendix B.3). The space L2(V,H) is a real separable Hilbert space with the innerproduct ·, ·2, see Proposition B.25.

Lemma 1.19 Let (Ω,F ) be a measurable space and V,H be real separable Hilbertspaces. Suppose that F : Ω → L2(V,H) is a map such that for every v ∈ V , F (·)vis F/B(H) measurable. Then F is F/B(L2(V,H)) measurable.

Proof. Since L2(V,H) is separable, by Lemma 1.17-(iv) it is enough toshow that for every T ∈ L2(V,H)

ω → F (ω), T 2 =+∞

k=1

F (ω)ek, T ek

is F/B(R) measurable, where ek is any orthonormal basis of V . But this is clearsince for every ω

F (ω), T 2 = limn→+∞

FTn (ω),

where

FTn (ω) =

n

k=1

F (ω)ek, T ek

and FTn (ω) is F/B(R) measurable because it is a finite sum of functions that are

F/B(R) measurable.

Let I be an interval in R, E, F be two real Banach spaces, and let E beseparable. If f : I × E → F is Borel measurable then for every t ∈ I the functionf(t, ·) : E → F is Borel measurable (by Lemma 1.8-(iv)).

Assume now that, for all t ∈ I and for some m ≥ 0, f(t, ·) ∈ Bm(E,F ) (thespace of Borel measurable functions with polynomial growth m, see Appendix A.2


for the precise definition). It is not true in general that the function

I → Bm(E,F ), t → f(t, ·)is Borel measurable. As a counterexample1 one can take the function

[0, 1]× L2(R) → L2(R), (t, x) → Stx,

where (St)t≥0 is the semigroup of left translations. Indeed the map

[0, 1] → L(L2(R)), t → St

is not measurable (see e.g. [130], Section 1.2). Since L(L2(R)) ⊆ B1(L2(R), L2(R))and the norm in L(L2(R)) is equivalent to the one induced by B1(L2(R), L2(R)),the claim follows in a straightforward way.

On the other hand, we have the following useful result.

Lemma 1.20 Let I and Λ be two Polish spaces. Let µ be a measure defined onthe the Borel σ-field B(I) and denote by B(I) the completion of B(I) with respectto µ. Let f : I × Λ → R be Borel measurable and such that for every t ∈ I, f(t, ·)is bounded from below (respectively, above). Then the function

f : I → R, t → infa∈Λ

f(t, a) (1.1)

(respectively, f : I → R, t → supa∈Λ f(t, a)) is F/B(R) measurable2.In particular if I is an interval in R, E, F are two real Banach spaces with E

separable, if ρ : I × E → F is Borel measurable and, for all t ∈ I and for somem ≥ 0, ρ(t, ·) ∈ Bm(E,F ), then the function

ρ1 : I → R, t → f(t, ·)Bm(E,F ) (1.2)

is Lebesgue measurable.

Proof. The first part is Example 7.4.2 in Volume 2 of [44] (recall that Polishspaces are Souslin spaces, see [44], Definition 6.6.1, and so I×Λ is a Souslin space).

For the second claim, observe that since f is Borel measurable, also the function

f : I × E → R, f(t, x) :=|ρ(t, x)|F

(1 + |x|2E)m/2

is Borel measurable (since it is the product of a continuous function with the com-position of a continuous function and a Borel measurable one). The result thusfollows from part one. Definition 1.21 (Independence) Consider a probability space (Ω,F ,P). Let Ibe a set of indices, and Ci ⊂ F for all i ∈ I. We say that the families Ci, i ∈ I, areindependent if, for every finite subset J of I and every choice of Ai ∈ Ci, (i ∈ J),we have

P

i∈J

Ai

=

i∈J

P(Ai).

If Ci ⊂ F is, for all i ∈ I, a π-system (resp. σ-field), the definition above givesin particular the notion of independent π-systems (resp. σ-fields). Random vari-ables are said to be independent if they generate independent σ-fields. A randomvariable X is independent of some σ-field G if σ(X) and G are independent σ-fields.

Lemma 1.22 Consider a probability space (Ω,F ,P). Let Ci ⊂ F , be a π-systemfor every i ∈ I. If Ci, i ∈ I are independent, then σ (Ci) , i ∈ I, are independent.

1This example has been suggested to us by Mauro Rosestolato.2Observe that f is not always Borel measurable, see [44] Volume 2, Exercice 6.10.42(ii), page

59.


Proof. See [281] Lemma 2.6 page 27.

1.1.3. Bochner integral. Throughout this section (Ω,F , µ) is a measurespace where µ is σ-finite, and E is a separable Banach space with norm | · |E . Weendow E with the Borel σ-field B(E).

Lemma 1.23 Let X : (Ω,F ) → E be a random variable. Then the real valuedfunction |X|E is measurable.

Proof. See [130] Lemma 1.2 page 16.

Let p ≥ 1. We denote by Lp(Ω,F , µ;E) the quotient space of the set

Lp(Ω,F , µ;E) :=

X : (Ω,F ) → (E,B(E)) measurable :

Ω|X(ω)|pE dµ(ω) < +∞

with respect to the equivalence relation of equality µ-a.e. Lp(Ω,F , µ;E) is a Banachspace when endowed with the norm

|X|Lp(Ω,F ,µ;E) =

Ω|X(ω)|pE dµ(ω)

1/p

(see e.g. [138] Theorem 7.17 page 104). We will often denote the norm by |X|Lp

when the context is clear. If H is a separable Hilbert space, then L2(Ω,F , µ;H)is a Hilbert space as well equipped with the scalar product X,Y L2(Ω,F ,µ;H) =Ω X(ω), Y (ω)H dµ(ω).

The space L∞(Ω,F , µ;E) is the quotient space of the space of boundedF/B(E) measurable functions with respect to the relation of being equal a.e..It is a Banach space equipped with the norm

|X|L∞(Ω,F ,µ;E) = ess supΩ

|X(ω)|E .

We will denote Lp(Ω,F , µ;R) simply by Lp(Ω,F , µ) or Lp(Ω, µ) when the σ-algebra F is clear from the context. In the special case when Ω = I is an intervalwith endpoints a and b with a < b (which may be ±∞), F is the Borel σ-field ofI, and µ is the Lebesgue measure on I, we will simply write Lp(I;E) or Lp(a, b;E)for Lp(I,F , µ;E). Finally we denote by Lp

loc(I;E) the set of measurable functionsf : I → E such that

K |f(s)|pEds is finite for every compact subset K of I.

Lemma 1.24 If F is countably generated apart from null sets then Lp(Ω,F , µ;E)is a separable Banach space.

Proof. See [140] p. 92.

Definition 1.25 (Bochner integral) Let X : (Ω,F , µ) → E be a simple ran-dom variable X =

Ni=1 xi1Ai

, where xi ∈ E, Ai ∈ F , µ(Ai) < +∞. The Bochnerintegral of X is defined as

ΩX(ω)dµ(ω) :=

N

i=1

xiµ(Ai).

Let X be in L1(Ω,F , µ;E). The Bochner integral of X is defined as

ΩX(ω)dµ(ω) := lim

n→+∞

ΩXn(ω)dµ(ω),

where Xn : (Ω,F , µ) → E are simple random variables such that

limn→+∞

Ω|X(ω)−Xn(ω)|Edµ(ω) = 0. (1.3)


Remark 1.26 It follows easily from Lemma 1.15, that for X ∈ L1(Ω,F , µ;E),there always exists a sequence of simple random variables Xn : (Ω,F , µ) → E asin Definition 1.25, satisfying (1.3).

Proposition 1.27 Let X ∈ L1(Ω,F , µ;E). Then the Bochner integral of X iswell defined and does not depend on the choice of the sequence. Moreover

ΩX(ω)dµ(ω)

E

≤

Ω|X(ω)|Edµ(ω). (1.4)

Proof. See [130] Section 1.1 (in particular inequality (1.6) page 19 and thepart below Lemma 1.5). The proof there is done for a probability measure µ, butthe general case is identical.

Proposition 1.28 Assume that (Ω,F , µ) is a complete measure space, E andF are separable Banach spaces and A : D(A) ⊂ E → F is a closed operator(see Definition B.3). If X ∈ L1(Ω,F , µ;E) and X ∈ D(A) a.s., then AX isan F -valued random variable, and X is a D(A)-valued random variable, whereD(A) is endowed with the graph norm of A (see Definition B.3). If moreoverΩ |AX(ω)|F dµ(ω) < +∞, then

A

ΩX(ω)dµ(ω) =

ΩAX(ω)dµ(ω).

Proof. The facts that X is a D(A)-valued random variable and AX is anF -valued random variable follow from Lemma 1.17-(ii). For the last part, see theproof of Proposition 1.6, Chapter 1 of [130].

Corollary 1.29 Assume that E and F are separable Banach spaces andT : E → F is a continuous linear operator. If X ∈ L1(Ω,F , µ;E), then

T

ΩX(ω)dµ(ω) =

ΩTX(ω)dµ(ω).

Proof. It is a particular case of Proposition 1.28.

Definition 1.30 (Lebesgue point) Assume that Ω is a metric space with dis-tance d. Let X be in L1(Ω,F , µ;E). A point ω ∈ Ω is said to be a Lebesgue pointfor X if

lim→0

1

µ(BΩ (ω))

BΩ(ω)

|X(ω)−X(ω)|E dµ(ω) = 0

where BΩ (ω) := ω ∈ Ω : d(ω, ω) ≤ .

Theorem 1.31 Let (Ω1,F1) and (Ω2,F2) be two measurable spaces and µ1 (re-spectively µ2) be a σ-finite measure on (Ω1,F1) (respectively on (Ω2,F2)). Thenthere exists a unique measure µ1⊗µ2 on F1⊗F2 such that, for every A ∈ F1 andB ∈ F2 with finite measure,

(µ1 ⊗ µ2)(A×B) = µ1(A)µ2(B).

The measure µ1 ⊗ µ2 is σ-finite.

Proof. See Theorem 8.2, page 160 in Chapter VI, Section 8 of [306].

Theorem 1.32 (Fubini Theorem) Let (Ω1,F1) and (Ω2,F2) be two measurablespaces and µ1 (respectively µ2) be a σ-finite measure on (Ω1,F1) (respectively on(Ω2,F2)). Let E be a separable Banach space with norm | · |E.


(i) Let X be in L1(Ω1 × Ω2,F1 ⊗ F2, µ1 ⊗ µ2;E). Then, for µ1-almostevery ω1 ∈ Ω1, the function X(ω1, ·) is in L1(Ω2,F2, µ2;E), and thefunction given by

ω1 →

Ω2

X(ω1,ω2)dµ2(ω2)

for µ1-almost all ω1 (and defined arbitrarily for other ω1) is inL1(Ω1,F1, µ1;E). Moreover, we have

Ω1×Ω2

X(ω1,ω2)d(µ1 ⊗ µ2)(ω1,ω2) =

Ω1

Ω2

X(ω1,ω2)dµ1(ω1)dµ2(ω2).

(ii) Let X : Ω1×Ω2 → E be an F1⊗F2-measurable map. Assume that, forµ1-almost every ω1 ∈ Ω1, the function X(ω1, ·) is in L1(Ω2,F2, µ2;E)and that the map given by

ω1 →

Ω2

|X(ω1,ω2)|dµ2(ω2)

for µ1-almost all ω1 (and defined arbitrarily for other ω1) is inL1(Ω1,R). Then X is in L1(Ω1×Ω2,F1⊗F2, µ1⊗µ2;E) and part (i)of the theorem applies.

Proof. See Theorem 8.4, page 162, and Theorem 8.7, page 165 in ChapterVI, Section 8 of [306].

Theorem 1.33 Let E be a Banach space and µ be a bounded measure on(E,B(E)). Then the set of uniformly continuous and bounded functions UCb(E) isdense in Lp(E,B(E), µ) for 1 ≤ p < +∞.

Proof. By Lemma 1.15 and the monotone convergence theorem it is enoughto prove that every characteristic function 1A for some A ∈ B(E) can be approx-imated by functions in UCb(E). Since µ is regular, for every > 0 we can find aclosed set C,C ⊂ A and an open set U,A ⊂ U such that µ(U \C) < p. Moreover,considering sets Un = x ∈ U : dist(x : A) > 1/n if necessary, we can assume thatdist(C,U) > 0. Then the function

f(x) :=dist(x, U)

dist(x,A) + dist(x, U)

belongs to UCb(E) and |1A − f|Lp < .

1.1.4. Expectation, covariance and correlation. Let (Ω,F ,P) be a prob-ability space and E be a separable Banach space with norm | · |E .

Definition 1.34 (Expectation) Given X in L1(Ω,F ,P;E), we denote by E[X]the (Bochner) integral

Ω X(ω)dP(ω). E[X] is said to be the expectation (or the

mean) of X.

To define the covariance operator, we recall first that if x ∈ E, y ∈ F , whereE,F are Hilbert spaces, the operator x⊗ y : F → E is defined by

(x⊗ y)h = xy, hF .Definition 1.35 (Covariance operator, correlation) Given a real, separableHilbert space H and X ∈ L2(Ω,F ,P;H), the covariance operator of X is definedby

Cov(X) := E(X − E[X])⊗ (X − E[X])

.


For X,Y ∈ L2(Ω,F ,P;H), the correlation of X and Y is the operator defined by

Cor(X,Y ) := E(X − E[X])⊗ (Y − E[Y ])

.

Remark 1.36 For X ∈ L2(Ω,F ,P;H), the operator Cov(X) is positive, sym-metric and nuclear (see [130] pages 26).

1.1.5. Conditional expectation and conditional probability.

Theorem 1.37 Consider a separable Banach space E, a probability space(Ω,F ,P) and a sub σ-field G ⊂ F . There exists a unique contractive linear oper-ator E[·|G ] : L1(Ω,F ,P;E) → L1(Ω,G ,P;E) such that

AE[ξ|G ](ω)dP(ω) =

Aξ(ω)dP(ω) for all A ∈ G and ξ ∈ L1(Ω,F ,P;E).

If E = H is a Hilbert space the restriction of E[·|G ] to L2(Ω,F ,P;H) is the or-thogonal projection L2(Ω,F ,P;H) → L2(Ω,G ,P;H).

Proof. See [130] Proposition 1.10 page 26 and [355] Proposition V-2-5 pages102-103.

Definition 1.38 (Conditional expectation) Given X ∈ L1(Ω,F ,P;E), therandom variable E[X|G ] ∈ L1(Ω,G ,P;E), defined by Theorem 1.37, is called theconditional expectation of X given G .

The following proposition collects various properties of conditional expectation(see e.g. [380] Proposition 3.15 page 25, and [446] Section 9.7 page 88 for similarproperties for real-valued random variables).

Definition 1.39 Let (Ω,F ,P) be a probability space and let E be a separableBanach space. A family H of integrable random variables X ∈ L1(Ω,F ,P;E) iscalled uniformly integrable if

limR→∞

supX∈H

|X|E≥R|X(ω)|EdP(ω) = 0.

Proposition 1.40 Let (Ω,F ,P) be a probability space and let E be a separableBanach space. The conditional expectation has the following properties:

(i) If X ∈ L1(Ω,F ,P;E) is G -measurable, then E[X|G ] = X P-a.s.(ii) Given X ∈ L1(Ω,F ,P;E) and two σ-fields G1 and G2 such that G1 ⊂

G2 ⊂ F ,

EEX|G1

G2

= E

EX|G2

G1

= E

X|G1

P-a.s.

(iii) Let X ∈ L1(Ω,F ,P;E). If X is independent of G , then E [X|G ] = E[X]P-a.s.. Moreover, X is independent of G , if and only if, for any bounded,Borel measurable f : E → R, E [f(X)|G ] = Ef(X) P-a.s.

(iv) If X is G -measurable and ζ is a real-valued integrable random variablesuch that ζX ∈ L1(Ω,F ,P;E), then

EζX|G

= XE

ζ|G

P-a.s.

(v) If X ∈ L1(Ω,F ,P;E), ζ is an integrable, real-valued, G -measurablerandom variable such that ζX ∈ L1(Ω,F ,P;E), then

EζX|G

= ζE

X|G

P-a.s.


(vi) If X ∈ L1(Ω,F ,P;E) and f : R → R is a convex function such thatE [|f(|X|E)|] < +∞, then

fE

X|G

E

≤ E

f (|X|E) |G

P-a.s.

(vii) If X,Xn ∈ L1(Ω,F ,P;E) for every n ∈ N, the family Xnn∈N isuniformly integrable and Xn

n→∞−−−−→ X, P-a.s., then

EXn|G

n→∞−−−−→ E

X|G

P-a.s.

(viii) Let X ∈ L1(Ω,F ,P;E). Assume that Gnn∈N is an increasing familyof σ-fields such that G = σ (Gn : n ∈ N) is a sub-σ-field of F . Then

EX|Gn

n→∞−−−−→ E

X|G

P-a.s.

(ix) Let Z be a separable Banach space and let T ∈ L(E;Z). Then

E[TX|G ] = TE[X|G ] P-a.s.

Proposition 1.41 Let (Ω,F ,P) be a probability space. Then:(i) If X,Y ∈ L1(Ω,F ,P;R) and X ≥ Y , then

E[X|G ] ≥ E[Y |G ].

(ii) (Conditional Fatou Lemma) If Xn ∈ L1(Ω,F ,P;R) and Xn ≥ 0, then

E[lim infn→∞

Xn|G ] ≤ lim infn→∞

E[Xn|G ] P− a.s.

Proof. See [446], Section 9.7, page 88.

Proposition 1.42 Let (E1,E1) and (E2,E2) be two measurable spaces and ψ :E1×E2 → R be a bounded measurable function. Let X1, X2 be two random variablesin a probability space (Ω,F ,P) with values in (E1,E1) and (E2,E2) respectively, andlet G ⊂ F be a σ-field. If X1 is G−measurable and X2 is independent of G , then

E(ψ(X1, X2)|G ) = ψ(X1), P a.s., (1.5)

whereψ(x1) = E(ψ(x1, X2)), x1 ∈ E1. (1.6)

Proof. See Proposition 1.12, p. 28 of [130].

Let (Ω,F ,P) be a probability space, and G be a sub σ-field of F . The condi-tional probability of A ∈ F given G is defined by

P(A|G )(ω) := E[1A|G ](ω).

Definition 1.43 Let (Ω,F ,P) be a probability space, and G be a sub σ-field ofF . A function p : Ω × F → [0, 1] is called a regular conditional probability givenG if it satisfies the following conditions:

(i) for each ω ∈ Ω, p(ω, ·) is a probability measure on (Ω,F );(ii) for each B ∈ F , the function p(·, B) is G -measurable;(iii) for every A ∈ F , P(A|G )(ω) = p(ω, A), P-a.s..

It thus follows that, if X ∈ L1(Ω,F ,P;E), where E is a separable Banachspace, then

E[X|G ](ω) =

ΩX(ω)p(ω, dω) P a.s..


Theorem 1.44 Let (Ω,F ,P) be a probability space, where (Ω,F ) is a standardmeasurable space. Then, for every sub σ-field G ⊂ F , there exists a regular condi-tional probability p(·, ·) given G . Moreover, if p(·, ·) is another regular conditionalprobability given G , then there exists a set N ∈ G ,P(N) = 0 such that, if ω ∈ Nthen p(ω, A) = p(ω, A) for all A ∈ F .

Moreover, if H is a countably determined sub σ-field of G , then there exists aP-null set N ∈ G such that, if ω ∈ N then p(ω, A) = 1A(ω) for every A ∈ H . Inparticular, if (Ω1,F1) is a measurable space, F1 is countably determined, x ∈F1 for all x ∈ Ω1 and ξ : (Ω,F ) → (Ω1,F1) is a G /F1-random variable, thenp (ω, ω : ξ(ω) = ξ(ω)) = 1 for P-a.e. ω.

Proof. See Theorem 8.1, page 147 in [374], or Theorems 3.1, 3.2, and thecorollary following them in [267] (see also [449] Proposition 1.9, page 11). Notation 1.45 If the regular conditional probability exists we will often writeP(·|G )(ω) or Pω for p(ω, ·).

Definition 1.46 (Law of a random variable) Given a probability space(Ω,F ,P), a measurable space (Ω1,F1), and a random variable X : (Ω,F ) →(Ω1,F1), the probability measure on (Ω1,F1) defined by

LP(X)(A) := P(ω ∈ Ω : X(ω) ∈ A)is called the law (or distribution)3 of X. We denote the law of X by LP(X).

Proposition 1.47 (Change of variables) Given a probability space (Ω,F ,P),a measurable space (Ω1,F1), a random variable X : (Ω,F ) → (Ω1,F1), and abounded Borel function ϕ : Ω1 → R we have

Ωϕ(X(ω))dP(ω) =

Ω1

ϕ(ω)dLP(X)(ω).

Definition 1.48 (Convergence of random variables) Consider a probabil-ity space (Ω,F ,P) and a Polish space (S, d) endowed with the Borel σ-field. LetXn : Ω → S and X : Ω → S be random variables. We say that:

(i) Xn converges to X P-a.s. (and we write Xn → X P-a.s.) iflimn→∞ d(Xn(ω), X(ω)) = 0 P-a.s..

(ii) Xn converges to X in probability if, for every > 0,limn→+∞ P ω ∈ Ω : d(Xn(ω), X(ω)) > = 0.

(iii) Xn converges to X in law if, for every bounded and continuous f : S →R,

S f(u)dLP(X)(u) = limn→∞

S f(u)dLP(Xn)(u) (i.e. if E [f(X)] =

limn→∞ E [f(Xn)]).

Lemma 1.49 Consider a probability space (Ω,F ,P) and a Polish space (S, d)endowed with the Borel σ-field. Let Xn : Ω → S and X : Ω → S be random variables.

(i) If Xn converges to X P-a.s. then Xn converges to X in probability.(ii) If Xn converges to X in probability then Xn converges to X in law.(iii) If Xn converges to X in probability then it contains a subsequence Xnk

such that Xnkconverges to X P-a.s..

(iv) (Egoroff’s theorem) If Xn converges to X P-a.s. then for every > 0,there exists Ω ∈ F such that P(Ω\ Ω) < , and Xn converges uniformlyto X on Ω.

(v) Let X,Xn ∈ Lp(Ω,F ,P;E), n ∈ N, p ≥ 1, and E be a separable Banachspace. If Xn converges to X in Lp(Ω,F ,P;E), then Xn converges toX in probability.

3In measure theory it is more often called the push-forward of P and denoted by X#P.


Proof. For (i), (ii), and (iii) see for instance [281] Lemma 4.2, page 63 andLemma 4.7, page 66. Part (iv) can be found for instance in [52] Theorem 2, page170, Section IV.5.4. Property (v) is straightforward.

Lemma 1.50 Let p > 1 and X,Xn ∈ Lp(Ω,F ,P;E), n ∈ N, for some separableBanach space E. Suppose that, for some M > 0, E [|Xn|pE ] ≤ M for all n ∈ N. IfXn → X in probability, then E [|X −Xn|E ] → 0.

Proof. Since the sequence Xn is bounded in Lp(Ω,F ,P;E), it is uni-formly integrable (see e.g. [446] p.127, Section 13.3). The claim follows e.g. fromTheorem 13.7, p.131 of [446].

1.1.6. Gaussian measures on Hilbert spaces and Fourier transform.In this section we recall the notions of Gaussian measure and Fourier transform forHilbert-space valued random variables. For an extensive treatment of the subjectwe refer to [130], Chapter 2, [110], Chapter 1 or [111], Chapter 1.

For a real separable Hilbert space H we denote by L1(H) the Banach space ofthe trace class operators on H, by L+(H) the subspace (of L(H)) of all bounded,linear, self-adjoint, positive operators, and we set L+

1 (H) := L1(H) ∩ L+(H) (seeAppendix B.3). We will denote by M1(H) the set of probability measures on(H,B(H)).

Proposition 1.51 Consider a real, separable Hilbert space H with the Borel σ-field B(H) and a probability measure P on (H,B(H)). If

H |y| dP(y) < +∞, then

we can define

m :=

Hy dP(y) ∈ H.

IfH |y|2dP(y) < +∞, then there exists a unique Q ∈ L+

1 (H) such that

Qx, y :=

Hx, h−m y, h−m dP(h).

Proof. See [110] Page 7.

Definition 1.52 (Mean and covariance of a measure on H) We call m andQ, defined by Proposition 1.51, respectively the mean and the covariance of P.In other words the mean (respectively covariance) of P is the mean (respectivelycovariance) of the identity random variable I : (H,B(H),P) → (H,B(H)).

Definition 1.53 (Fourier transform of a measure) Let H be a Hilbert spaceand B(H) be its Borel σ-field. Given a probability measure P on (H,B(H)) wedefine, for x ∈ H,

P(x) :=

Heiy,xdP(y).

We call P : H → C the Fourier transform of P.

Proposition 1.54 Let H be a real, separable Hilbert space, B(H) be its Borelσ-field, and P1 and P2 be two probability measures on (H,B(H)). If P1(x) = P2(x)for all x ∈ H, then P1 = P2.

Proof. See [110] Proposition 1.7, page 6 or [130], Proposition 2.5, page35.

Theorem 1.55 Let X1, ..., Xn be random variables in a real, separable Hilbertspace H. The random variables are independent if and only if for every y1, ..., yn ∈


H

Ee[i

n

i=1Xi,yi]=

n

i=1

Ee[iXi,yi]

. (1.7)

Proof. Obviously if X1, ..., Xn are independent then (1.7) holds. Also The-orem 1.55 is well known if H = Rk. Let now k ∈ N and yji ∈ H, i = 1, ..., n, j =1, ..., k, and consider random variables Xk

i = (Xi, y1i , ..., Xi, yki ), i = 1, ..., nin Rk. Therefore, if (1.7) holds then Xk

i , i = 1, ..., n, are independent for ev-ery k ∈ N and yji ∈ H, j = 1, ..., k. Since cylindrical sets of the form x :(x, y1i , ..., x, yki ) ∈ A ∈ B(Rk) generate B(H) and are a π-system, the col-lection of sets ω : (Xi, y1i , ..., Xi, yki ) ∈ A ∈ B(Rk) over all k ∈ N andyji ∈ H, i = 1, ..., n, j = 1, ..., k, A ∈ B(Rk) is a π-system generating σ(Xi). Thus,by Lemma 1.22, the sigma algebras σ(X1), ...,σ(Xn) are independent. Theorem 1.56 Let H be a real, separable Hilbert space, B(H) be its Borel σ-field, a ∈ H, and Q ∈ L+

1 (H). Then there exists a unique probability measure P on(H,B(H)) such that

P(x) = eia,x−12 Qx,x.

The measure P has mean a and covariance Q.

Proof. See [110] Theorem 1.12 page 12. Definition 1.57 (Gaussian measure on H) Let H be a real, separable Hilbertspace, B(H) be its Borel σ-field, a ∈ H, and Q ∈ L+

1 (H). The unique probabilitymeasure P identified by Theorem 1.56 is called the Gaussian measure with mean aand covariance Q, and is denoted by N (a,Q). When a = 0 we will denote it by NQ

and call it a centered Gaussian measure.

Proposition 1.58 Let Q ∈ L+1 (H). Then for all y, z ∈ H

Hx, y x, zNQ(dx) = Qy, z (1.8)

Define, for y ∈ Q1/2(H), Qy ∈ L2(H,NQ) as

Qy(x) :=Q−1/2y, x

, (1.9)

where Q−1/2 is the pseudoinverse of Q1/2 (see Definition B.1). The map (calledthe “white noise function”, see e.g. [111][Section 2.5])

y ∈ Q1/2(H) → Qy ∈ L2(H,NQ)

can be extended to H0 = Q1/2(H) = (kerQ)⊥ and it satisfies

HQy(x)Qz(x)NQ(dx) = y, z , y, z ∈ H0.

Moreover for all m > 0 we have

H|x|2mNQ(dx) ≤ K(m)[Tr(Q)]m

for some K(m) > 0, independent of Q.

Proof. Formula (1.8) follows from [129][Proposition 1.2.4].The second statement is proved, when kerQ = 0, in [111][Section 2.5.2] (see

also Section 1.2.4 of [129]). Since here we do not assume kerQ = 0, we provide aproof. First we observe that kerQ = kerQ1/2 and that Q1/2(H) is dense in (kerQ)⊥

since Q1/2 is selfadjoint. Moreover by Definition B.1, the pseudoinverse of Q1/2 isthe operator Q−1/2 : Q1/2(H) → (kerQ)⊥, hence the map y → Qy =

Q−1/2y, x


is well defined for all y ∈ Q1/2(H). Furthermore, thanks to formula (1.8), we have,for y1, y2 ∈ Q1/2(H)

H

Q−1/2y1, x

Q−1/2y2, x

NQ(dx) =

Q(Q−1/2y1), Q

−1/2y2= y1, y2 ,

where we used that Q1/2Q−1/2y = y for all y ∈ Q1/2(H). Hence, for y1, y2 ∈Q1/2(H),

HQy1(x)Qy2(x)NQ(dx) = y1, y2 . (1.10)

In view of the above the map y → Qy =Q−1/2y, x

is an isometry and can be

extended to Q1/2(H) = (kerQ)⊥ (endowed with the inner product inherited fromH) and (1.10) extends to all y1, y2 ∈ (kerQ)⊥.

We remark that as pointed out in [111][Section 2.5.2], for a generic y ∈ (kerQ)⊥

the image Qy is an element of L2(H,NQ), hence an equivalence class of randomvariables defined NQ-a.e.; in particular, writing Qy(x) =

y,Q−1/2x

, NQ-a.e.,

would be misleading since, as proved in [111][Proposition 2.22], NQ(Q1/2(H)) = 0.Concerning the third claim by Proposition 2.19 page 50 of [130], it holds for

m ∈ N. If k − 1 < m < k for k = 1, 2, ..., we use

H|x|2mNQ(dx) ≤

H|x|2kNQ(dx)

m/k

.

We recall a useful result concerning compactness of a family of measures inM1(H) (see e.g. [130][Section 2.1], [161, 374] for more on this).

Definition 1.59(i) A sequence Pn in M1(H) is said to be weakly convergent to some

P ∈ M1(H) if, for every φ ∈ Cb(H),

limn→+∞

Hφ(x)Pn(dx) =

Hφ(x)P(dx).

(ii) A family Λ ⊆ M1(H) is said to be compact (respectively, relatively com-pact) if an arbitrary sequence Pn of elements from Λ contains a sub-sequence Pnk

weakly convergent to a measure P ∈ Λ (respectively, toa measure P ∈ M1(H)).

(iii) A family Λ ⊆ M1(H) is said to be tight if for any ε > 0 there exists acompact set Kε such that, for every P ∈ Λ,

P(Kε) > 1− ε.

The following theorem (which also holds when H is a Polish space) is due toProkhorov.

Theorem 1.60 Let H be a real separable Hilbert space. A family Λ ⊆ M1(H) isrelatively compact if and only if it is tight.

Proof. See [130], the proof of Theorem 2.3.

The next theorem gives a useful sufficient condition for compactness.

Theorem 1.61 Let H be a real separable Hilbert space and let eii be an or-thonormal basis in H. A family Λ ⊆ M1(H) is relatively compact if

limN→+∞

supP∈Λ

H

+∞

i=N

x, ei2 P(dx) = 0.


Proof. See [374], the proof of Theorem VI.2.2.

Concerning Gaussian measures, we have the following result (see Proposition1.1.5 of [387].

Proposition 1.62 Let NQn(n ≥ 1), and NQ be centered Gaussian measures

on H. If limn→+∞ Qn −QL1(H) = 0, then the measures NQnconverge weakly to

NQ.

Proof. Observe that if eii is an orthonormal basis in H, it follows from(1.8) that for any N ∈ N,

H

+∞

i=N

x, ei2 NQn(dx) =

+∞

i=N

Qnei, ei .

Since limn→+∞ Qn − QL1(H) = 0, the above formula implies in particular thatTheorem 1.61 applies and thus the sequence NQn

is relatively compact.Moreover, from Theorem 1.56 and Definition 1.57 it is immediate that, as

n → +∞,NQn

(x) = e−12 Qnx,x −→ e−

12 Qx,x = NQ(x), ∀x ∈ H.

Take now a subsequence NQnk

weakly convergent to a probability measure P0. ByDefinition 1.53 we must have

NQn(x) → P0(x), ∀x ∈ H.

This implies that P0 = NQ and hence, by Proposition 1.54, that P0 = NQ. Sincethis is true for any convergent subsequence, the claim now follows by a standardcontradiction argument.

1.2. Stochastic processes and Brownian motion

1.2.1. Stochastic processes.

Definition 1.63 (Filtration, usual conditions) Let t ≥ 0. A filtrationF t

ss≥t in a complete probability space (Ω,F ,P) is a family of σ-fields such thatF t

s ⊂ F tr ⊂ F whenever t ≤ s ≤ r.

(i) We say that F tss≥t is right continuous if, for all s ≥ t, F t

s+ :=r>s F t

r = F ts .

(ii) We say that F tss≥t is left continuous if, for all s > t, F t

s− :=

σ

r<s F tr

= F t

s . We say that F tss≥t is continuous if it is both

left and right continuous.(iii) We say that F t

ss≥t satisfies the usual conditions if it is right contin-uous and complete, i.e. if F t

s contains all P-null sets of F for everys ≥ t.

We will often write F ts instead of F t

ss≥t. We also set F t+∞ :=

σ

r<+∞ F tr

.

Since we will mostly deal with filtrations satisfying the usual conditions we willassume from now on that this property holds unless explicitly stated otherwise. Forthis reason we include the usual conditions in the definition of a filtered probabilityspace.

Definition 1.64 (Filtered probability space) Let F ts be a filtration satisfy-

ing the usual conditions on a complete probability space (Ω,F ,P). The 4-tuple(Ω,F ,F t

s ,P) is called a filtered probability space.

1.2. STOCHASTIC PROCESSES AND BROWNIAN MOTION 33

Notation 1.65 We use the following convention in this section. When we writes ∈ [t, T ] we mean that s ∈ [t, T ] if T ∈ R, and s ∈ [t,+∞) if T = +∞. So [t, T ] isunderstood to be [t,+∞) if T = +∞.

Definition 1.66 (Stochastic process) Let T ∈ (0,+∞], t ∈ [0, T ) and (Ω,F )and (Ω1,F1) be two measurable spaces. A family of random variables X(·) =X(s)s∈[t,T ], X(s) : Ω → Ω1, is called a stochastic process in [t, T ]. If (Ω1,F1) =(R,B(R)) then X(·) is called a real stochastic process.

Definition 1.67 LetΩ,F , F t

ss≥t ,P

be a filtered probability space and(Ω1,F1) be a measurable space. A stochastic process X(s)s∈[t,T ] : [t, T ]×Ω → Ω1

is said to be:

(i) Measurable, if the map (s,ω) → X(s)(ω) is B([t, T ])⊗ F/F1 measur-able.

(ii) Adapted, if, for each s ∈ [t, T ], X(s) : Ω → Ω1 is an F ts/F

-measurablerandom variable.

(iii) Progressively measurable, if for all s ∈ (t, T ], the restriction of X(·) to[t, s]× Ω is B([t, s])⊗ F t

s/F1 measurable.(iv) Predictable, if the map (s,ω) → X(s)(ω) is P[t,T ]/F1 measurable,

where P[t,T ] is the σ-field in [t, T ]×Ω generated by all sets of the form(s, r]×A, t ≤ s < r ≤ T,A ∈ F t

s and t×A,A ∈ F tt .

(v) If E is a separable Banach space (endowed with its Borel σ-field), theprocess X(s)s∈[t,T ] : [t, T ]×Ω → E is called stochastically continuousat s ∈ [t, T ], if for every , δ > 0 there exists ρ > 0 such that

P (|X(r)−X(s)| ≥ ) ≤ δ, for all r ∈ (s− ρ, s+ ρ) ∩ [t, T ].

(vi) If (S, d) is a metric space (endowed with its Borel σ-field), the processX(s)s∈[t,T ] : [t, T ] × Ω → S is called continuous (respectively, right-continuous, left-continuous), if for P-a.e. ω ∈ Ω, the function s →X(s)(ω) is continuous (respectively, right-continuous, left-continuous).

(vii) If E is a separable Banach space (endowed with its Borel σ-field), theprocess X(s)s∈[t,T ] : [t, T ]×Ω → E is called integrable, if E[|X(s)|] <+∞ for all s ∈ [t, T ]. The process is called uniformly integrable if itis integrable and the family X(s)s∈[t,T ] is uniformly integrable (seeDefinition 1.39).

(viii) If E is a separable Banach space (endowed with the Borel σ-field in-duced by the norm), the process X(s)s∈[t,T ] : [t, T ] × Ω → E is saidto be mean square continuous if E[|X(s)|2] < +∞ for all s ∈ [t, T ] andlimr→s E[|X(r)−X(s)|2] = 0 for all s ∈ [t, T ].

The concepts of adapted, progressively measurable, and predictable processescan be defined for any filtration G t

s . In such cases, to emphasize the filtrationused, we will refer to the processes as G t

s -adapted, G ts -progressively measurable,

and G ts -predictable.

Progressive measurability can also be defined using the concept of progressivelymeasurable sets, see e.g. [344], p.4, or [161], page 71. We say that a set A ⊂ [t, T ]×Ωis F t

s -progressively measurable if the function 1A is a progressively measurableprocess. Equivalently this means that A ∩ ([t, s] × Ω) ∈ B([t, s]) ⊗ F t

s for everys ∈ [t, T ]. It can be proved that the F t

s -progressively measurable sets form a σ-fieldand that a process X(s) is progressively measurable if and only if it is measurablewith respect to the σ-field of F t

s -progressively measurable sets.


Definition 1.68 (Stochastic equivalence, modification) Let (Ω,F ,P) be aprobability space, and (Ω1,F1) be a measurable space. Processes X(·), Y (·) : [t, T ]×Ω → Ω1 are called stochastically equivalent, if for all s ∈ [t, T ], P(X(s) = Y (s)) =1. In this case Y (·) is said to be a modification of X(·). The processes X(·) andY (·) are called indistinguishable, if P(X(s) = Y (s) : ∀s ∈ [t, T ]) = 1.

Lemma 1.69 LetΩ,F , F t

ss≥t ,P

be a filtered probability space and letX(s)s≥t be a process with values in a Polish space (S, d), endowed with the Borelσ-field induced by the distance.

(i) If X(·) is B([t, T ]) ⊗ F/B(S) measurable and F ts -adapted, then X(·)

has an F ts -progressively measurable modification.

(ii) If X(·) is F ts -adapted and X(·) is left- (or right-) continuous for every

ω, then X(·) itself is F ts -progressively measurable.

Proof. Part (i): Since S is Borel isomorphic to a Borel subset A of R,without loss of generality we can consider X(·) to be an R-valued process with valuesin A. By [346], Theorem T46, page 68, X(·) has an R-valued, F t

s -progressivelymeasurable modification X(·). Let a ∈ A. We define a process Y (·) by Y (s) :=X(s)1X(s)∈A + a1X(s)∈(R\A). The process Y (·) is F t

s -progressively measurable.Moreover, if X(s) = X(s), then Y (s) = X(s), so Y (·) is a modification of X(·).Part (ii): See [346], Theorem T47, page 70, or [283], Proposition 1.13, page 5.

1.2.2. Martingales.

Notation 1.70 Unless specified otherwise, any Banach space E and any met-ric space (S, d) will be understood to be endowed with the Borel σ-field inducedrespectively by the norm and by the distance.

Definition 1.71 (Martingale) Let (Ω,F ,F ts ,P) be a filtered probability space,

and let M(·) be an F ts -adapted and integrable process with values in a separable

Banach space E. Then M(·) is said to be a martingale if, for all r, s ∈ [t, T ], s ≤ r,

EM(r)|F t

s

= M(s) P− a.s..

If E = R, we say that M(s) is a submartingale (respectively, supermartingale), if

EM(r)|F t

s

≥ M(s), (respectively, E

M(r)|F t

s

≤ M(s)) P− a.s..

If E|M(s)|2 < +∞ for all s ≥ t we say that M(·) is square integrable.

Theorem 1.72 (Doob’s maximal inequalities) Let T > 0, (Ω,F ,F ts ,P) be a

filtered probability space, and H be a separable Hilbert space. Let M(·) be a right-continuous H-valued martingale such that M(s) ∈ Lp (Ω,F ,P;H) for all s ∈ [t, T ].Then:

(i) If p ≥ 1, Psups∈[t,T ] |M(s)| > λ

≤ 1

λpE [|M(T )|p], for all λ > 0.

(ii) If p > 1, Esups∈[t,T ] |M(s)|p

≤

p

p−1

pE [|M(T )|p].

Proof. We observe that, if M(·) is a right-continuous H-valued martin-gale such that M(s) ∈ Lp (Ω,F ,P;H) , p ≥ 1, for all s ∈ [t, T ], then byProposition 1.40-(vi), |M(·)|p is a right-continuous R-valued submartingale with|M(s)| ∈ Lp (Ω,F ,P;R) for all s ∈ [t, T ]. The claims now easily follow from [283]Theorem 3.8 (i) and (iii), pages 13-14.

In particular we see that a right-continuous E-valued martingale M(·) is squareintegrable if and only if E|M(T )|2 < +∞.


Notation 1.73 (Square integrable martingales) Let T ∈ (0,+∞), t ∈ [0, T ),let (Ω,F ,F t

s ,P) be a filtered probability space, and E be a separable Banachspace. The class of all continuous square integrable martingales M : [t, T ]×Ω → Eis denoted by M2

t,T (E).

If H is a separable Hilbert space then M2t,T (H) endowed with the scalar product

M,NM2t,T

:= E [M(T ), N(T )] .

is a Hilbert space (see [220], page 22).

Theorem 1.74 (Angle bracket process, Quadratic variation process) LetT > 0, t ∈ [0, T ), H be a separable Hilbert space, and (Ω,F ,F t

s ,P) be a filteredprobability space. For every M ∈ M2

t,T (H) there exists a unique (real) increasing,adapted, continuous process starting from 0 at t, called the angle bracket process,and denoted by Mt, such that |Ms|2−Ms is a continuous martingale. Moreoverthere exists a unique L+

1 (H)-valued continuous adapted process starting from 0 att, called the quadratic variation of M , and denoted by Ms, such that, for allx, y ∈ H, the process

Ms, x Ms, y −Ms (x), y

, s ∈ [t, T ]

is a continuous martingale. Moreover Ms = Tr(Ms).Proof. See [220], Definition 2.9 and Lemma 2.1, page 22.

Theorem 1.75 (Burkholder-Davis-Gundy inequality) Let T > 0, t ∈ [0, T ),H be a separable Hilbert space, and (Ω,F ,F t

s ,P) be a filtered probability space. Forevery p > 0 there exists cp > 0 such that, for every M ∈ M2

t,T (H) with M(0) = 0,

c−1p E

Mp/2T

≤ E

sup

s∈[t,T ]|M(s)|p

≤ cpE

Mp/2T

.

Proof. See [380], Theorem 3.49, page 37.

1.2.3. Stopping times.

Definition 1.76 (Stopping time) Consider a probability space (Ω,F ,P) anda filtration F t

ss≥t on Ω. A random variable τ : (Ω,F ) → [t,+∞] is said to be anF t

s -stopping time if, for all s ≥ t,

τ ≤ s := ω ∈ Ω : τ(ω) ≤ s ∈ Fts .

Given a stopping time τ we denote with Fτ the sub-σ-field of F defined by

Fτ :=A ∈ F : A ∩ τ ≤ s ∈ F

ts for all s ≥ t

.

Proposition 1.77 Let (Ω,F ,F ts ,P) be a filtered probability space.

(i) If τ and σ are F ts -stopping times, so are τ ∧ σ, τ ∨ σ and τ + σ.

(ii) If σn (for n = 1, 2...) are F ts -stopping times, then

supn

σn, infn

σn, lim supn

σn, lim infn

σn

are F ts -stopping times.

(iii) For any F ts -stopping time τ there exists a decreasing sequence of

discrete-valued F ts -stopping times τn, such that limn→∞ τn = τ .


(iv) Let (S, d) be a metric space (endowed with the Borel σ-field induced bythe distance), and X : [t,+∞)×Ω → S be a continuous and F t

s -adaptedprocess. Let A ⊂ S be an open or a closed set. Then the hitting time

τA := infs ≥ t : X(s) ∈ Ais a stopping time. (It is understood that inf∅ = +∞.)

Proof. (i) and (ii): see [283], Lemmas 2.9 and 2.11, page 7. (iii): see [281],Lemma 7.4, page 122. (iv): see [449], Example 3.3, page 24, or [349], Proposition1.3.2., page 12 (there S = Rn, but the proofs are the same).

Proposition 1.78 LetΩ,F , F t

ss≥t ,P

be a filtered probability space,(Ω1,F1) be a measurable space, X : [t,+∞)×Ω → Ω1 be an F t

s -progressively mea-surable process, and τ be an F t

s -stopping time. Then the random variable X(τ),(where X(τ)(ω) := X(τ(ω),ω)), is Fτ -measurable and the process X(s ∧ τ) isF t

s -progressively measurable.

Proof. See [349], Proposition 1.3.5., page 13, or [449], Proposition 3.5, page25.

Theorem 1.79 (Doob’s optional sampling theorem) LetΩ,F , F t

ss≥t ,P

be a filtered probability space, X : [t,+∞) × Ω → R be a right-continuous F ts -

submartingale, and τ,σ be two F ts -stopping times with τ bounded. Then Xτ is

integrable andE[Xτ |F t

σ] ≥ Xτ∧σ, P a.s..

If X+ (the positive part of the process) is uniformly integrable then the statementextends to unbounded τ .


Definition 1.80 (Local martingale) LetΩ,F , F t

ss≥t ,P

be a filteredprobability space. An F t

ss≥t-adapted process X(s)s≥t with values in a sepa-rable Banach space E is said to be a local martingale if there exists an increasingsequence of stopping times τnn∈N with P(τn ↑ +∞) = 1, such that the processX(s ∧ τn)s≥t is a martingale for every n ∈ N.

1.2.4. Q-Wiener process.

Definition 1.81 (Real Brownian motion) Given t ∈ R a real stochastic pro-cess β : [t,+∞) × Ω → R on a complete probability space (Ω,F ,P) is a standard(one-dimensional) real Brownian motion on [t,+∞) starting at 0, if

(1) β is continuous and β(t) = 0;(2) for all t ≤ t1 < t2 < ... < tn the random variables β(t1), β(t2)− β(t1),

..., β(tn)− β(tn−1) are independent;(3) for all t ≤ t1 ≤ t2, β(t2)− β(t1) has a Gaussian distribution with mean

0 and covariance t2 − t1.

Consider a real, separable Hilbert space Ξ and Q ∈ L+(Ξ). Define Ξ0 :=Q1/2(Ξ) and let Q−1/2 be the pseudo-inverse of Q1/2 (see Definition B.1). Ξ0

is a separable Hilbert space when endowed with the inner product x, yΞ0:=

Q−1/2x,Q−1/2yΞ. Let Ξ1 be an arbitrary real, separable Hilbert space such that

Ξ ⊂ Ξ1 with continuous embedding and Ξ0 ⊂ Ξ1 with Hilbert-Schmidt embeddingJ : Ξ0 → Ξ1 (see Appendix B.3 on Hilbert-Schmidt operators). The operator


Q1 := JJ∗ belongs to L+1 (Ξ1) and Ξ0 is identical with the space Q

121 (Ξ1) (see

[130] Proposition 4.7, page 85).

Theorem 1.82 Consider the setting described above. Let gkk∈N be an or-thonormal basis of Ξ0 and βkk∈N be a sequence of mutually independent, standardone-dimensional Brownian motions βk : [t,+∞)×Ω → R on [t,+∞) starting at 0.Then for every s ∈ [t,+∞) the series

WQ(s) :=∞

k=1

gkβk(s) (1.11)

is convergent in L2(Ω,F ,P;Ξ1).

Proof. See [130] Proposition 4.3, page 82, and Proposition 4.7, page 85. Definition 1.83 (Q-Wiener process) The process WQ defined by (1.11) iscalled a Q-Wiener process on [t,+∞) starting at 0.

Remark 1.84 We will use the notation WQ to denote a Q-Wiener process. If Qis trace-class, Ξ1 = Ξ is a canonical choice and it will be understood that WQ is a Ξvalued process. If Q is not trace-class, writing WQ and calling it a Q-Wiener processis a slight abuse of notation as it would be more precise to write WQ1 and call ita Q1-Wiener process with values in Ξ1. However, even though the construction wehave described is not canonical if Tr(Q) = +∞, and the choice of Ξ1 is not unique,the class of the integrable processes is independent of the choice of Ξ1 (see [130]Section 4.1 an in particular Proposition 4.7). Moreover (see [130] Section 4.1.2) forarbitrary a ∈ Ξ the stochastic process

< a,W (s) >:=∞

k=1

a, gkβk(s), s ≥ t,

is a real valued Wiener process and

E < a,W (s1) >< b,W (s2) >= ((s1 − t) ∧ (s2 − t))Qa, b, a, b ∈ Ξ.

For these reasons, even when Tr(Q) = +∞, we will still use the notation WQ andin this case we will also call it a cylindrical Wiener process in Ξ.

Proposition 1.85 Let Ξ be a real, separable Hilbert space, Q ∈ L+(Ξ) and letΞ0, Ξ1 and J be as described above. Let (Ω,F ,P) be a complete probability spaceand B : [t,+∞) × Ω → Ξ1 be a stochastic process. Denote by F t,0

s the filtrationgenerated by B i.e.

Ft,0s = σ(B(r) : t ≤ r ≤ s),

and F ts := σ(F t,0

s ,N ), where N is the class of the P-null sets. Then B is aQ-Wiener process on [t,+∞) starting at 0 if and only if:

(1) B(t) = 0.(2) B has continuous trajectories.(3) For all t ≤ t1 ≤ t2 the random variable B(t2)−B(t1) is independent of

F tt1 .

(4) LP (B(t2)−B(t1)) = N (0, (t2 − t1)Q1), where Q1 = JJ∗.

Proof. The “only if” part follows from [130], Proposition 4.7, page 85 (ob-serve that in [130] a Wiener process is in fact defined using the four properties (1)-(4)). The “if” part is proved in [130] Proposition 4.3-(ii), page 81 (if Tr(Q) = +∞we apply the proposition in the space Ξ1).

The existence of a process satisfying conditions (1) − (4) above can also beproved using the Kolmogorov extension theorem (see [130], Proposition 4.4).


Remark 1.86 If WQ(s) =∞

k=1 gkβk(s) for some orthonormal basis gkk∈N ofΞ0, it is easy to see that regardless of the choice of Ξ1, F t,0

s = σβk(r) : t ≤ r ≤s, k ∈ N. Thus the filtration generated by WQ does not depend on the choice ofΞ1.

Definition 1.87 (Translated G ts -Q-Wiener process) Let 0 ≤ t < T ≤ +∞.

Let Ξ be a real, separable Hilbert space, Q ∈ L+(Ξ) and let Ξ0, Ξ1 and J be asdescribed above. Let (Ω,F ,G t

s ,P) be a filtered probability space. We say that astochastic process B : [t, T ]×Ω → Ξ1 is a translated G t

s -Q-Wiener process on [t, T ]if:

(1) B has continuous trajectories.(2) B is adapted to G t

s .(3) For all t ≤ t1 < t2 ≤ T , B(t2)−B(t1) is independent of G t

t1 .(4) LP (B(t2)−B(t1)) = N (0, (t2 − t1)Q1), where Q1 = JJ∗.

If we also have B(t) = 0 then we call B a G ts -Q-Wiener process on [t, T ].

We remark that if B is a translated G ts -Q-Wiener process, then it is also a

translated F ts -Q-Wiener process, where F t

s is the augmented filtration generatedby B. Moreover if WQ is a Q-Wiener process as in Definition 1.83 then it is also aF t

s -Q-Wiener process, where F ts is the augmented filtration generated by B.

Lemma 1.88 Let 0 ≤ t < T ≤ +∞. Let Ξ be a real, separable Hilbert space,Q ∈ L+(Ξ) and let Ξ0 and Ξ1 be as described above. Let (Ω,F ,P) be a completeprobability space. Let B : [t, T ] × Ω → Ξ1 be a continuous stochastic process suchthat B(t) = 0. Then B is a Q-Wiener process on [t, T ] if and only if, for all a ∈ Ξ1,t ≤ t1 ≤ t2 ≤ T , we have

Eeia,B(t2)−B(t1)Ξ1 |F t

t1

= e−

Q1a,aΞ12 (t2−t1). (1.12)

Proof. (The proof uses the same arguments as in the finite-dimensionalcase, see Proposition 1.2.7 of [349]).

The “only if” part: if B is a Q-Wiener process then, by Proposition 1.85-(4),Theorem 1.56 and Definition 1.57,

Eeia,B(t2)−B(t1)Ξ1

= e−

Q1a,aΞ12 (t2−t1).

Moreover, since B(t2)−B(t1) is independent of F tt1 ,

Eeia,B(t2)−B(t1)Ξ1

= E

eia,B(t2)−B(t1)Ξ1 |F t

t1

.

The “if” part: We have to prove the four conditions in Proposition 1.85: (1)and (2) are already in the assumptions of the lemma. Condition (4) follows easilyfrom (1.12), Theorem 1.56 and Definition 1.57. To prove condition (3), i.e. thatY := B(t2) − B(t1) is independent of F t

t1 , observe that, for all Z : Ω → Ξ1 whichare F t

t1 -measurable, one has, for all a, b ∈ Ξ1,

Eeia,Y Ξ1 eib,ZΞ1

= E

Eeia,Y Ξ1 |F t

t1

eib,ZΞ1

= e−Q1a,aΞ1

2 (t2−t1) Eeib,ZΞ1

= E

eia,Y Ξ1

Eeib,ZΞ1

. (1.13)

Since the above holds for all Z : Ω → Ξ1 which are F tt1 -measurable, and for all

a, b ∈ Ξ1, we conclude that Y is independent of F tt1 by Theorem 1.55.

Lemma 1.89 Let F t,0s and F t

s be the filtrations defined in Proposition 1.85 fora Q-Wiener process WQ. Then F t

s is right continuous. Moreover, for all T > t,


Ft,0T , and consequently F t

T , are countably generated up to sets of measure zero. Ifthe trajectories of WQ are everywhere continuous then

Ft,0T = F

t,0T− = σ (WQ(si) : i = 1, 2, ...) , (1.14)

where (si), i = 1, 2, ... is any dense sequence in [t, T ), and hence the filtration F t,0s

is countably generated and left continuous.

Proof. The proof follows arguments from [402] and [283] (Section 2.7-A).Consider τ > s and > 0. Since WQ(τ + ) − WQ(s + ) is independent of F

t,0s+ ,

for every A ∈ Ft,0s+ and f ∈ Cb(Ξ1)

E (1Af(WQ(τ + )−WQ(s+ ))) = P(A)Ef(WQ(τ + )−WQ(s+ )).

Letting → 0 we thus have by the dominated convergence theorem that

E (1Af(WQ(τ)−WQ(s))) = P(A)Ef(WQ(τ)−WQ(s)). (1.15)

Now if B = B ⊂ Ξ1 then there exist functions fn ∈ Cb(Ξ1), 0 ≤ fn ≤ 1, such thatfn(x) → 1B(x) as n → +∞ for every x ∈ Ξ1. Therefore (1.15) implies that

P(A ∩ WQ(τ)−WQ(s) ∈ B) = P(A)P(WQ(τ)−WQ(s) ∈ B)

and since the sets WQ(τ)−WQ(s) ∈ B : B = B ⊂ Ξ1 are a π-system generatingσ(WQ(τ)−WQ(s)), it follows from Lemma 1.22 that F

t,0s+ and σ(WQ(τ)−WQ(s))

are independent.Now let s = τ0 < τ1 < ... < τk ≤ T . We have σ(WQ(τi) − WQ(s) : i =

1, ..., k) = σ(WQ(τi) − WQ(τi−1) : i = 1, ..., k). Let now A ∈ Ft,0s+ and Bi ∈

σ(WQ(τi)−WQ(τi−1)), i = 1, ..., k. Since Bi is independent of A∩B1 ∩ ...∩Bi−1 ∈F t,0

τi−1, i = 1, ..., k and B1, ..., Bk are independent

P(A ∩B1 ∩ ... ∩Bk) = P(A ∩B1 ∩ ... ∩Bk−1)P(Bk) = ...

= P(A ∩B1)k

i=2

P(Bi) = P(A)k

i=1

P(Bi) = P(A)P(B1 ∩ ... ∩Bk).

Thereforeσ(WQ(τi) − WQ(s) : i = 1, ..., k) (where the union is taken over all

partitions s = τ0 < τ1 < ... < τk ≤ T ) is a π-system independent of Ft,0s+ and thus

Gs = σ(WQ(τ)−WQ(s) : s ≤ τ ≤ T ) is independent of Ft,0s+ .

Since Ft,0T = σ(F t,0

s ,Gs), the family As ∩ Bs : As ∈ F t,0s , Bs ∈ Gs is a π-

system generating Ft,0T . Let now A ∈ F

t,0s+ and let ξ be a version of 1A−E(1A|F t,0

s ).Since ξ is F

t,0s+ -measurable, it is independent of Gs, so if As ∈ F t,0

s , Bs ∈ Gs then

E (ξ1As∩Bs) = E (ξ1As

1Bs) = P(Bs)E (ξ1As

)

= P(Bs)

As

ξdP = P(Bs)

As

1AdP−

As

E(1A|F t,0s )dP

= 0

by the definition of conditional expectation. This implies thatD ξdP = 0 for every

D ∈ F tT and thus ξ = 0,P a.e.. Therefore 1A = E(1A|F t,0

s ),P a.e., i.e if A =E(1A|F t,0

s )−1(1) then A ∈ F t,0s and P(A∆A) = 0. This shows that F

t,0s+ ⊂ F t

s .Now let A ∈ F t

s+, which means that for every n ≥ 1, A ∈ F ts+1/n and there

exists Bn ∈ Ft,0s+1/n such that A∆Bn ∈ N . Set

B =+∞

m=1

+∞

n=m

Bn.


Then B ∈ Ft,0s+ ⊂ F t

s and

B \A ⊂

+∞

n=1

Bn

\A =

+∞

n=1

(Bn \A) ∈ N .

Moreover

A \B = A ∩

+∞

m=1

+∞

n=m

Bn

c

= A ∩

+∞

m=1

+∞

n=m

Bcn

=+∞

m=1

+∞

n=m

(A ∩Bcn) ⊂

+∞

m=1

(A ∩Bcm) =

+∞

m=1

(A \Bm) ∈ N .

Thus A∆B ∈ N which implies that A ∈ F ts , which completes the proof of the right

continuity.To show that F

t,0T is countably generated up to sets of measure zero we take

a dense sequence (si), i = 1, 2, ..., in [t, T ). Since B(Ξ1) is countably generated (forinstance by open balls with rational radii centered at points of a countable denseset), each σ(WQ(si)) is countably generated and so σ(WQ(si) : i ≥ 1) is countablygenerated. It remains to show that for every s ∈ (t, T ],σ(WQ(s)) ⊂ σ(N ,WQ(si) :si < s). Let Ω0 ⊂ Ω,P(Ω0) = 1 be such that WQ has continuous trajectories on[t, T ] for ω ∈ Ω0. Let A be an open subset of Ξ1 and set An = x ∈ A : dist(x,Ac) >1/n, n = 1, 2, .... Then An is open, An ⊂ An+1, and

+∞n=1 An = A. Let sik be

a sequence of si such that sik < s and sik → s as k → +∞ . Then, using thecontinuity of the trajectories of WQ, it is easy to see that

Ω0 ∩WQ(s)−1(A) = Ω0 ∩

+∞

n=1

+∞

k=n

WQ(sik)−1(An) ∈ σ(N ,WQ(si) : si < s).

Therefore WQ(s)−1(A) ∈ σ(N ,WQ(si) : si < s) and since the sets WQ(s)−1(A) :A is an open subset of Ξ1 generate σ(WQ(s)), the result follows. If Ω0 = Ω thenwe have above

WQ(s)−1(A) =

+∞

n=1

+∞

k=n

WQ(sik)−1(An) ∈ σ(WQ(si) : si < s).

The argument that σ(WQ(t)) ⊂ σ(WQ(si) : i = 1, 2, ...) is similar (or we can justassume that s1 = t). This yields (1.14).

In fact the above argument shows that if S is a Polish space, T > t, andX : [t, T ]× Ω → S is a stochastic process with everywhere continuous trajectories,then the filtration generated by X, FX

s := σ(X(τ) : t ≤ τ ≤ s) is countablygenerated and left-continuous.

1.2.5. Simple and elementary processes.

Definition 1.90 (F ts -simple process) Let E be a Banach space (endowed with

the Borel σ-field) and let (Ω,F , F tss∈[t,T ] ,P) be a filtered probability space. A

process X : [t, T ]× (Ω,F ,P) → E is called F ts -simple if:

(i) Case T = +∞: there exists a sequence of real numbers tnn∈N witht = t0 < t1 < ... < tn < ... and limn→∞ tn = +∞, a constant C < +∞,and a sequence of random variables ξn : Ω → E with supn≥0 |ξn(ω)|E ≤C for every ω ∈ Ω, such that ξn is F t

tn-measurable for every n ≥ 0, and

X(s)(ω) =

ξ0(ω) if s = tξi(ω) if s ∈ (ti, ti+1].


(ii) Case T < +∞: there exist t = t0 < t1 < ... < tN = T , a con-stant C < +∞, and random variables ξn : Ω → E for n = 0, ..., N − 1with sup0≤n≤N−1 |ξn(ω)|E ≤ C for every ω ∈ Ω, such that ξn is F t

tn-measurable, and

X(s)(ω) =

ξ0(ω) if s = tξi(ω) if s ∈ (ti, ti+1].

Definition 1.91 (F ts -elementary process) Let T ∈ (0,+∞), t ∈ [0, T ).

Let (S, d) be a complete metric space (endowed with the Borel σ-field), and(Ω,F , F t

ss∈[t,T ] ,P) be a filtered probability space. We say that a processX : [t, T ] × (Ω,F ,P) → S is F t

s -elementary, if there exist S-valued random vari-ables ξ0, ξ1, .., ξN−1, and a sequence t = t0 < t1 < .. < tN = T , such that

(1) ξi has a finite numbers of values for every i ∈ 0, ..N − 1.(2) ξi is F t

ti-measurable for every i ∈ 0, ..N − 1.(3) X(s)(ω) = ξi(ω) for s ∈ (ti, ti+1] for i ∈ 0, ..N − 1, and X(t) = ξ0.

Finally we say that a process X : [t,+∞)×(Ω,F ,P) → S is F ts -elementary if there

exists T1 > t such that the restriction of X to [t, T1] is F ts -elementary and X(s) = 0

for s > T1.

It is immediate from the definitions that simple and elementary processes areprogressively measurable and predictable.

Remark 1.92 In Definitions 1.14, 1.90 and 1.91 we introduced the concepts ofsimple random variable, F t

s -simple processes, and F ts -elementary process. The

reader should be aware that in the literature the use of these terms varies and thesame word is often used by different authors to mean different things.

Lemma 1.93 Let E be a separable Banach space endowed with the Borel σ-field,(Ω,F ,F t

s ,P) be a filtered probability space and X : [t, T ] × Ω → E be a bounded,measurable, F t

s -adapted process, where T ∈ [t,+∞) ∪ +∞. There exists a se-quence Xm(·) of F t

s -elementary E-valued processes on [t, T ] such that, for every1 ≤ p < +∞ and R > t,

limm→+∞

E R∧T

t|Xm(s)−X(s)|pE ds = 0. (1.16)

The same claim holds if, instead of the Banach space, we consider E to be aninterval [a, b] ⊂ R or a countable closed subset of [a, b].

Proof. It is enough to prove the result for a single p ≥ 1. To obtaina sequence of F t

s -simple processes Xm with the required properties, the prooffollows exactly the proof of Lemma 3.2.4, page 132, in [283] with obvious technicalmodifications as we now have to deal with Bochner integrals in E. We then useLemma 1.16 to approximate the random variables ξi defining Xm by simple randomvariables to obtain F t

s -elementary approximating processes.If E is a countable closed subset of [a, b], we first produce [a, b]-valued F t

s -elementary approximating processes Xm. We then construct an E-valued F t

s -elementary process Y m from Xm as follows. Let Xm(s) = ξi for s ∈ (ti, ti+1]for i ∈ 0, ..N − 1, and X(t) = ξ0. Let ξi be defined in the following way. Ifξi(ω) ∈ E, we set ξi(ω) = ξi(ω). If ξi(ω) ∈ E, we set ξi(ω) = argminx∈E |ξ(ω)− x|if argminx∈E |ξ(ω) − x| is a singleton. If argminx∈E |ξ(ω) − x| has two pointsx1 < x2, we set ξi(ω) = x1. Obviously ξi is a simple, F t

ti -measurable process. Wenow define Y m(s) = ξi for s ∈ (ti, ti+1] for i ∈ 0, ..N − 1, and X(t) = ξ0. Then,since X has values in E, it is easy to see that |Y m −X| ≤ 2|Xm −X|. Thereforethe result follows.


Lemma 1.94 Let F t,0s and F t

s be as in Proposition 1.85, T ∈ [t,+∞) ∪ +∞,and let a(·) : [t, T ] × Ω → S be an F t

s -progressively measurable process, where(S, d) is a Polish space endowed with the Borel σ-field. Then there exists an F t,0

s -progressively measurable and F t,0

s -predictable process a1(·) : [t, T ] × Ω → S, suchthat a(·) = a1(·), dt⊗ P-a.e. on [t, T ]× Ω.

Proof. In light of Theorem 1.13 we can assume that S = [0, 1] or S is acountable closed subset of [0, 1]. Using Lemma 1.93, we can find an approximatingF t

s -elementary processes an(·) on [t, T ] of the form

an(t)(ω) =

ξn0 (ω) if s = tξni (ω) if s ∈ (ti, ti+1].

such that

supR≥t

limn→∞

E R∧T

t|a(s)− an(s)|2R ds = 0.

Using Lemma 1.16, we can change every ξni on a null-set to obtain a sequence ofF t,0

s -elementary processes an1 (·), that still satisfy

supR≥t

limn→∞

E R∧T

t|a(s)− an1 (s)|

2R ds = 0.

Obviously the processes an1 (·) are F t,0s -progressively measurable. We can now

extract a subsequence (still denoted by an1 (·)) such that an1 (·) → a(·) dt ⊗ Pa.e. on [t, T ] × Ω, and define a1(·) := lim infn→+∞ an1 (·). The process a1(·) isF t,0

s -progressively measurable, F t,0s -predictable, and a(·) = a1(·), dt ⊗ P a.e. on

[t, T ]× Ω.

1.3. Stochastic integral

Let T ∈ (0,+∞), and t ∈ [0, T ). Throughout the whole section Ξ andH will be two real, separable Hilbert spaces, Q will be an operator in L+(Ξ),Ω,F , F t

ss∈[t,T ] ,P

will be a filtered probability space, and WQ will be a trans-lated F t

s -Q-Wiener process on Ω on [0, T ]. The following concept will be used inChapter 2.

Definition 1.95 A 5-tuple µ :=Ω,F , F t

ss∈[t,T ] ,P,WQ

described above is

called a generalized reference probability space.

A process X(·) will always be assumed to be defined on Ω, and the expressions“adapted” and “progressively measurable” will always refer to the filtration F t

s .

1.3.1. Definition of stochastic integral. In this section we will assume thatTr(Q) < +∞. If Tr(Q) = +∞, the construction of the stochastic integral is thesame, we just have to consider WQ as a Ξ1 valued Wiener process with nuclearcovariance Q1 (see Section 1.2.4). This way WQ is not uniquely determined butQ1/2

1 (Ξ1) = Ξ0 = Q1/2(Ξ), |x|Ξ0 = |Q−1/21 x|Ξ1 for all possible extensions Ξ1 and

the class of integrands and the value of the integrals are independent of the choiceof the space Ξ1 (see [130], Proposition 4.7 and Section 4.1.2).

We recall that we denote by L2(Ξ0;H) the space of Hilbert-Schmidt operatorsfrom Ξ0 to H (see Appendix B.3). It is equipped with its Borel σ field B(L2(Ξ0;H)).L2(Ξ0;H) is a real, separable Hilbert space (see Proposition B.25), and L(Ξ;H) isdense in L2(Ξ0;H).

1.3. STOCHASTIC INTEGRAL 43

Definition 1.96 (The space N p

Q(t, T ;H)) Given p ≥ 1, we denote by

N pQ(t, T ;H) the space of all L2(Ξ0;H)-valued, progressively measurable processes

X(·), such that

|X(·)|Np

Q(t,T ;H) :=

E T

tX(s)pL2(Ξ0;H)ds

1/p

< ∞.

N pQ(t, T ;H) is a Banach space if it is endowed with the norm | · |Np

Q(t,T ;H).

We remark that, as always, two processes in N pQ(t, T ;H) are identified if they

are equal P⊗ dt a.e..

Remark 1.97 In several classical references (see e.g. [130] or [385]), the theory ofstochastic integration is developed for predictable processes instead of progressivelymeasurable ones like in our case. However, it follows for instance from Lemma 1.94,that for every L2(Ξ0;H)-valued progressively measurable process X there exists apredictable process X1 which is P⊗ dt a.e. equal to X. Thus, since we are workingwith stochastic integrals with respect to Wiener processes (which are continuous),the two concepts coincide.

For an L(Ξ;H)-valued, F ts -simple process Φ on [t, T ], Φ(s) = Φ01t(s) +i=N−1

i=0 1(ti,ti+1](s)Φi, the stochastic integral with respect to WQ is defined by

T

tΦ(s)dWQ(s) :=

N−1

i=0

Φi(WQ(ti+1)−WQ(ti)) ∈ L2(Ω;H).

Note that if we take Φ to be L2(Ξ0;H)-valued, we cannot guarantee that theexpression above is well defined, since L2(Ξ0;H) contains unbounded operators inΞ.

We now extend the stochastic integral to all processes in N 2Q(t, T ;H) by the

following theorem.

Theorem 1.98 (Itô isometry) For every L(Ξ;H)-valued, F ts -simple process Φ

we have

E

T

tΦ(s)dWQ(s)

2

H

= E T

tΦ(s)2L2(Ξ0;H)ds.

Thus the stochastic integral is an isometry between the set of L(Ξ;H)-valued,F t

s -simple processes in N 2Q(t, T ;H) and its image in L2(Ω;H). Moreover,

since L(Ξ;H)-valued, F ts -simple (and in fact elementary) processes are dense in

N 2Q(t, T ;H), it can be uniquely extended to all processes in N 2

Q(t, T ;H). We denotethis unique extension by

T

tΦ(s)dWQ(s)

and call it the stochastic integral of Φ with respect to WQ.

Proof. See [220], Propositions 2.1, 2.2, and Definition 2.10. See also [130],Proposition 4.22 in the context of predictable processes.

Proposition 1.99 For Φ ∈ N 2Q(t, T ;H), consider the process

I(Φ) : [t, T ]× Ω → H

I(Φ)(r) := rt Φ(s)dWQ(s) :=

Tt Φ(s)1[t,r]dWQ(s).


I(Φ) is a continuous square integrable martingale and I : N 2Q(t, T ;H) → M2

t,T (H)is an isometry. Moreover,

I(Φ)s = s

t

Φ(s)Q

12

Φ(s)Q

12

∗ds,

I(Φ)s = s

tΦ(s)2L2(Ξ0;H)ds.

Proof. See [220] Theorem 2.3, page 34.

The definition of stochastic integral can be further extended to all L2(Ξ0;H)-valued progressively measurable processes Φ(·) such that

P T

tΦ(s)2L2(Ξ0;H)ds < +∞

= 1. (1.17)

Lemma 1.100 Let Φ(s)s∈[t,T ] be an L2(Ξ0;H)-valued progressively measurableprocess satisfying (1.17). Then there exists a sequence Φn of L(Ξ;H)-valued F t

s -simple processes such that

limn→∞

T

tΦ(s)− Φn(s)2L2(Ξ0;H)ds = 0 P− a.s.. (1.18)

Moreover there exists an H-valued random variable, denoted by I, such that

limn→∞

T

tΦn(s)dWQ(s) = I in probability.

I does not depend on the choice of approximating sequence, more precisely,given Φ1

n and Φ2n satisfying (1.18), if I1 := limn→∞

Tt Φ1

n(s)dWQ(s) and I2 :=

limn→∞ Tt Φ2

n(s)dWQ(s), then I1 = I2 P− a.s..

Proof. See [220], Lemma 2.3, page 39, and Lemma 2.6, page 41.

The process I defined by Lemma 1.100 is also called the stochastic inte-gral of Φ with respect to WQ, and is denoted by

Tt Φ(s)dWQ(s). We also set

rt Φ(s)dWQ(s) :=

Tt Φ(s)1[t,r]dWQ(s).

Proposition 1.101 Let Φ(s)s∈[t,T ] be an L2(Ξ0;H)-valued progressively mea-surable process satisfying (1.17). Then the process

I(Φ) : [t, T ]× Ω → HI(Φ)(r) :=

rt Φ(s)dWQ(s).

is a continuous local martingale.

Proof. See [220] pages 42-44.

Finally we may extend the definition of stochastic integral to all processes thatare equivalent to progressively measurable processes satisfying (1.17).

Definition 1.102 We say that two processes Φ1 and Φ2 are equivalent if Φ1 =Φ2, dt⊗ P-a.e.. If Φ belongs to the equivalence class of a progressively measurableprocess Φ1 satisfying (1.17), we set

T

tΦ(s)dWQ(s) :=

T

tΦ1(s)dWQ(s).

This definition is obviously independent of the choice of a representative processΦ1. Thus a representative process defines the stochastic integral for the wholeequivalence class.


Example 1.103 Every L2(Ξ0;H)-valued, F ts -adapted, and B([t, T ])⊗ F mea-

surable process Φ satisfying (1.17) is stochastically integrable, where B([t, T ])⊗ F

is the completion of B([t, T ]) ⊗ F with respect to dt ⊗ P . To see this we needto find a progressively measurable process Φ1 which is equivalent to Φ. First, letΦ2 be a B([t, T ])⊗F measurable process equivalent to Φ (which exists by Lemma1.16). Since F is complete, Φ2(s, ·) is F t

s measurable for a.e. s. Thus there existsA ∈ B([t, T ]) of full measure such that Φ2(s, ·) is F t

s measurable for s ∈ A. Wethen define Φ3 = Φ21A. Φ3 is B([t, T ])⊗F measurable and F t

s -adapted, and thusit has a progressively measurable modification Φ1 which is clearly equivalent to Φ.

Theorem 1.104 Let (E,G , µ) be a measure space with bounded measure. LetΦ : [t, T ]×Ω×E → L2(Ξ0;H), be (B([t, T ])⊗F t

T ⊗ G )/B(L2(Ξ0;H)) measurable.Suppose that, for any x ∈ E, Φ(s, ·, x)s∈[t,T ] is progressively measurable and

E|Φ(·, ·, x)|N 2

Q(t,T ;H)dµ(x) < +∞.

Then:

(i) T

tΦ(s, ·, x)dW (s) has a F t

T ⊗ G /B(H) measurable version.

(ii)

GΦ(·, ·, x)dµ(x) is progressively measurable.

(iii) The following equality holds P-a.s.:

G

T

tΦ(s, ·, x)dW (s)dµ(x) =

T

t

GΦ(s, ·, x)dµ(x)dW (s).

Proof. See Theorem 2.8, Section 2.2.6, page 57 of [220] and Theorem 4.33,Section 4.5, page 110 of [130].

1.3.2. Basic properties and estimates.

Lemma 1.105 Let T > 0 and t ∈ [0, T ). Assume that Φ is in N 2Q(t, T ;H) and

that τ is an F ts -stopping time such that P(τ ≤ T ) = 1. Then P-a.s.

T

t1[t,τ ](r)Φ(r)dWQ(r) =

τ

tΦ(r)dWQ(r).

Proof. See [220], Lemma 2.7, page 43 (also [130], Lemma 4.24, page 99).

As a consequence of Theorem 1.75 and Proposition 1.99 we obtain the followingtheorem (see also e.g. [127], Theorem 5.2.4, page 58).

Theorem 1.106 (Burkholder-Davis-Gundy inequality for stochastic inte-grals) Let T > 0 and t ∈ [0, T ). For every p ≥ 2, there exists a constant cp suchthat, for every Φ in N p

Q(t, T ;H),

E

sups∈[t,T ]

s

tΦ(r)dWQ(r)

p≤ cpE

T

tΦ(r)2L2(Ξ0,H)dr

p/2

≤ cp(T − t)p

2−1E T

tΦ(r)pL2(Ξ0,H)dr

.

Proposition 1.107 Let T > 0 and t ∈ [0, T ). Let A be the generator of aC0-semigroup erA, r ≥ 0 on H such that erA ≤ Meαr for every r ≥ 0 for


some α ∈ R, M > 0. Let p > 2 and Φ ∈ N pQ(t, T ;H). Let An be the Yosida

approximation of A. Then the stochastic convolution process

Ψ(s) :=

s

te(s−r)AΦ(r)dWQ(r) (1.19)

has a continuous modification,

E

sups∈[t,T ]

s

te(s−r)AΦ(r)dWQ(r)

p≤ CE

T

tΦ(r)pL2(Ξ0,H)dr

, (1.20)

where the constants c and C depend only on T − t, p, M , α, and

limn→∞

E

sups∈[t,T ]

s

t

e(s−r)An − e(s−r)A

Φ(r)dWQ(r)

p= 0. (1.21)

If moreover A generates a C0-pseudo-contraction semigroup (i.e. M = 1 above, seeAppendix B.4) then the claims are also true for p = 2.

Proof. See [220], Lemma 3.3, page 87. The claims for p=2 can be provedrepeating the arguments of the proof of Proposition 3.3 of [428] which uses theUnitary Dilation Theorem. Proposition 1.108 Let A be the generator of a C0-semigroup on H, T > 0,and t ∈ [0, T ). Assume that Φ : [t, T ]×Ω → L2(Ξ0;H) is a progressively measurableprocess such that Φ(s) ∈ L2(Ξ0;D(A)) P-a.s., for a.e. s ∈ [t, T ]. Assume that

P T

tΦ(s)2L2(Ξ0;D(A))ds < +∞

= 1.

Then

P T

tΦ(s)dWQ(s) ∈ D(A)

= 1 (1.22)

and

A

T

tΦ(s)dWQ(s) =

T

tAΦ(s)dWQ(s), P− a.s. (1.23)

Proof. We can assume without loss of generality that Q ∈ L+1 (Ξ). The

proof follows the proof of Proposition 3.1 (p.76) of [220], however we present ithere to clarify a measurability issue. Indeed we first need to show that Φ is anL2(Ξ0;D(A))-valued, progressively measurable process. To do this we take Ψn =JnΦ, where Jn = n(nI − A)−1 (see Definition B.40). Since Jn ∈ L(H;D(A)), Ψn

is an L2(Ξ0;D(A))-valued, progressively measurable process. Moreover it is easyto see that if Φ(s)(ω) ∈ L2(Ξ0;D(A)), then Ψn(s)(ω) → Φ(s)(ω) in L2(Ξ0;D(A)).Therefore, defining V := (s,ω) : Ψn(s)(ω) converges in L2(Ξ0;D(A)), it followsfrom Lemma 1.8-(iii) that Φ is equivalent to a progressively measurable processlimn→+∞ 1V Ψn. The proof is now done in two steps.Step 1 : The claim is true for F t

s -simple L(Ξ;D(A))-valued processes.Step 2 : If Φ is a L2(Ξ0;D(A))-valued progressively measurable process satisfyingthe hypotheses of this proposition, we take a sequence of F t

s -simple L(Ξ;D(A))-valued processes Φn approximating Φ in the sense of (1.18) so that

limn→+∞

T

tΦ(s)− Φn(s)2L2(Ξ0;D(A))ds = 0 P− a.s..

In particular we have T

tΦn(s)dWQ(s)

n→∞−−−−→ T

tΦ(s)dWQ(s),


A

T

tΦn(s)dWQ(s) =

T

tAΦn(s)dWQ(s)

n→∞−−−−→ T

tAΦ(s)dWQ(s)

in probability, so the claim follows since A is a closed operator.

Lemma 1.109 Let T > 0, t ∈ [0, T ), and 0 < α < 1. Let A be the generatorof a C0-semigroup erA, r ≥ 0 on H, and A1 be a linear, closed operator suchthat A1erA is bounded and A1erA = erAA1 for every r > 0. Let Φ : [t, T ] × Ω →L2(Ξ0;H) be progressively measurable. Assume that, for all s ∈ [t, T ],

s

t(s− r)α−1

r

t(r − h)−2αE

A1e(r−h)AΦ(h)

2

L2(Ξ0;H)

dh

1/2

dr < +∞.

(1.24)Then s

tA1e

(s−r)AΦ(r)dWQ(r) =sin(απ)

π

s

t(s− r)α−1e(s−r)AY Φ

α (r)dr P− a.s.

for all s ∈ [t, T ], where

Y Φα (r) :=

r

t(r − h)−αA1e

(r−h)AΦ(h)dWQ(h). (1.25)

Proof. The statement is similar to [127], Theorem 5.2.5, page 58, Section5.2.1. The proof is based on the factorization method and we give it for complete-ness.We use the identity

t

σ(t− s)α−1(s− σ)−αds =

π

sin πα, for all σ ≤ s ≤ t, 0 < α < 1 (1.26)

(which can be proved by a simple direct computation). We have s


α (r)dr

=

s

t(s− r)α−1e(s−r)A

r

t(r − h)−αA1e

(r−h)AΦ(h)dWQ(h)dr

=π

sin πα

s

tA1e

(s−h)AΦ(h)dWQ(h),

where in the last line we used the Stochastic Fubini Theorem 1.104 to change theorder of integration (see Theorem 4.33, p. 110 in [130] or Theorem 2.8, p. 57[220], observing that the required hypotheses are satisfied thanks to (1.24)), and(1.26).

Lemma 1.110 Let A be the generator of a C0-semigroup erA, r ≥ 0 on H,T > 0, t ∈ [0, T ) and f ∈ Lp(t, T ;H), p ≥ 1. Then:

(i) If either 1/p < α ≤ 1, or p = α = 1, then the function

Gαf(s) :=

s

t(s− r)α−1e(s−r)Af(r)dr

is in C([t, T ];H).(ii) If the semigroup etA is analytic, λ ∈ R is such that that (λI − A)−1 ∈

L(H), β > 0 and α > β + 1/p, then the function

Gα,βf(s) :=

s

t(s− r)α−1(λI −A)βe(s−r)Af(r)dr

is in C([t, T ];H).


Proof. Part (i): Let 1/p < α ≤ 1. Let t ≤ s1 ≤ s2 ≤ T and denoteh = s2 − s1. We have

s2

t(s2 − r)α−1e(s2−r)Af(r)dr −

s1

t(s1 − r)α−1e(s1−r)Af(r)dr

≤ I1 + I2 :=

t+h

t

(s2 − r)α−1e(s2−r)Af(r) dr

+

s2

t+h(s2 − r)α−1e(s2−r)Af(r)dr −

s1

t(s1 − r)α−1e(s1−r)Af(r)dr

.

Set q := pp−1 and let R > 0 be such that

esA ≤ R for all s ∈ [0, T ]. Then

I1 ≤ R

h

0(h− r)q(α−1)dr

1/q T

t|f(r)|pdr

1/p

→ 0 as h → 0

since 0 ≥ q(α− 1) > −1. As regards I2, after a change of variables we have

I2 ≤ s1

t(s1 − r)α−1e(s1−r)A|f(r + h)− f(r)|dr

≤ R

T

t(T − r)q(α−1)dr

1/q T−h

t|f(r + h)− f(r)|pdr

1/p

→ 0 as h → 0.

The proof in the case p = α = 1 is straightforward.Part (ii) follows from Proposition A.1.1 in Appendix A, p. 307 of [127].

Proposition 1.111 Let T > 0 and t ∈ [0, T ). Let A, A1, Φ be as in Lemma1.109. Assume that there exist 0 < α < 1, C > 0 and p > 1

α , p ≥ 2 such that T

tE s

t(s− r)−αA1e

(s−r)AΦ(r)2L2(Ξ0;H)dr

p/2

ds < C. (1.27)

Then

Ψ(s) :=

s

tA1e

(s−r)AΦ(r)dWQ(r)

has a continuous modification.

Proof. We follow the scheme of the proof of Theorem 5.2.6 in [127] (page59, Section 5.2.1). We give some details because our claim is slightly more general.Observe that using Hölder’s and Jensen’s inequalities we obtain

s

t(s− r)α−1

r

t(r − h)−2αE

A1e(r−h)AΦ(h)

2

L2(Ξ0;H)

dh

1/2

dr

≤ s

t(s− r)

(α−1)pp−1

p−1p

s

tE r

t(r − h)−2α

A1e(r−h)AΦ(h)

2

L2(Ξ0;H)dh

p/2 1

p

< +∞,

where we used (1.27) and that (1−α)pp−1 < 1 which follows from p > 1/α. Therefore

the hypotheses of Lemma 1.109 are satisfied and thus we have s

tA1e

(s−r)AΦ(r)dWQ(r) =sin(απ)

π

s


α (r)dr P− a.s.

for all s ∈ [t, T ], where Y Φα (r) is defined by (1.25). The claim will follow from

Lemma 1.110-(i) applied to a.e. trajectory. Thus we need to know that the process

1.4. STOCHASTIC DIFFERENTIAL EQUATIONS 49

Y Φα (r) has p-integrable trajectories a.s.. This is guaranteed if

E T

t

Y Φα (s)

p

ds < +∞.

However, from Theorem 1.106, we have T

tEY Φ

α (s)p

ds ≤ cp

T

tE s

t(s− r)−αA1e

(s−r)AΦ(r)2L2(Ξ0;H)dr

p/2

ds

(1.28)that is bounded thanks to (1.27).

1.4. Stochastic differential equations

In this section we take T > 0 and take H, Ξ, Q, and a generalized referenceprobability space µ = (Ω,F , Fss∈[0,T ],P,WQ) as in Section 1.3 (with t = 0). Ais the infinitesimal generator of a C0-semigroup on H, and Λ is a Polish space. Wewill look at stochastic differential equations (SDE) on the interval [0, T ], howeverall results are the same if, instead of [0, T ], we took an interval [t, T ], for 0 ≤ t < T .

1.4.1. Mild and strong solutions. Let b : [0, T ]×H×Ω → H and σ : [0, T ]×H × Ω → L2(Ξ0, H). We consider the following general stochastic differentialequation (SDE)

dX(s) = (AX(s) + b(s,X(s)))ds+ σ(s,X(s))dWQ(s) s ∈ (0, T ]X(0) = ξ,

(1.29)

where ξ is an H-valued F0-measurable random variable. To simplify the notationwe dropped the ω variable in (1.29) and we use this convention throughout thesection.

Definition 1.112 (Strong solution of (1.29)) An H-valued progressively mea-surable process X(·) is called a strong solution of (1.29) if:

(i) For dt⊗ P-a.e. (s,ω) ∈ [0, T ]× Ω, X(s)(ω) ∈ D(A).(ii)

P T

0|X(s)|+ |AX(s)|+ |b(s,X(s))|ds < +∞

= 1,

P T

0σ(s,X(s))2L2(Ξ0,H)ds < +∞

= 1.

(iii) For every s ∈ [0, T ]

X(s) = ξ +

s

0AX(r) + b(r,X(r))dr +

s

0σ(r,X(r))dWQ(s) P-a.e..

Definition 1.113 (Mild solution of (1.29)) An H-valued progressively mea-surable process X(·) is called a mild solution of (1.29) if:

(i)

P t

0|X(s)|+ |e(t−s)Ab(s,X(s))|ds < +∞

= 1, ∀t ∈ [0, T ]

P t

0e(t−s)Aσ(s,X(s))2L2(Ξ0,H)ds < +∞

= 1, ∀t ∈ [0, T ].

(ii) For every s ∈ [0, T ]

X(s) = esAξ +

s

0e(s−r)Ab(r,X(r))dr +

s

0e(s−r)Aσ(r,X(r))dWQ(r) P-a.e..


In order for the above definitions to be meaningful, all the processes involvedmust be well defined and have proper measurability properties so that the inte-grals that appear in the definitions make sense. We do not want to analyze herethe required measurability properties in the most generality. We discuss one casewhich will frequently appear in applications to optimal control in Remark 1.117 be-low. Moreover, note that if An is the Yosida approximation of A, since by Lemma1.17-(i) D(A) ∈ B(H), it follows that the processes 1X(s)∈D(A)AnX(s) are progres-sively measurable and they converge as n → +∞ to 1X(s)∈D(A)AX(s) for every(s,ω). Thus the process AX(s) (understood as 1X(s)∈D(A)AX(s)) is progressivelymeasurable.

Remark 1.114 In the definition of mild solution we assumed that b : [0, T ] ×H × Ω → H and σ : [0, T ] ×H × Ω → L2(Ξ0, H). However, Definition 1.113 maystill make sense even if b and σ do not have values in H and L2(Ξ0, H), providedthat the terms e(t−s)Ab(s,X(s)) and e(t−s)Aσ(s,X(s)) have values in these spaceswhen they are interpreted properly (see for instance Section 1.5.1 and also Remark1.117). Therefore in the future when we are dealing with such cases, we will notrepeat the definition of mild solution, instead we will just say how to interpret theabove terms.

Definition 1.115 (Weak mild solution of (1.29)) A weak mild solution of(1.29) is defined to be any 6-tuple (Ω,F ,Fs,WQ,P, X(·)), where (Ω,F ,Fs,P) isa filtered probability space, WQ is a translated Fs-Q-Wiener process on Ω, andX(·) is a mild solution for (1.29) in the generalized reference probability space(Ω,F ,Fs,WQ,P).

Notation 1.116 In the existing literature, often different authors call the samenotion of solution differently, and the same name does not always correspond tothe same definition. For instance, the weak mild solution introduced above is oftencalled a weak solution and in [130, Chapter 8] it is called a martingale solution.

Remark 1.117 Let Λ be a Polish space. Suppose that σ : [0, T ] × H × Λ →L(Ξ0, H) is such that for every u ∈ Ξ0, the map (t, x, a) → σ(t, x, a)u is B([0, T ])⊗B(H) ⊗ B(Λ)/B(H) measurable, and esAσ(t, x, a) ∈ L2(Ξ0, H) for every (t, x, a)and s > 0. It then follows from Lemma 1.19 that, after possibly redefining itat s = 0, the map (s, t, x, a) → esAσ(t, x, a) is B([0, T ]) ⊗ B([0, T ]) ⊗ B(H) ⊗B(Λ)/B(L2(Ξ0, H)) measurable. Now, if X(·) : [0, T ]×Ω → H, a(·) : [0, T ]×Ω → Λare Fs-progressively measurable, then for every t ∈ [0, T ],

(s,ω) → e(t−s)Aσ(s,X(s), a(s))

is an L2(Ξ0, H)-valued Fs-progressively measurable process on [0, t] × Ω. If thisprocess is in N 2

Q(0, t;H) for every t then the process

(t,ω) → t

0e(t−s)Aσ(s,X(s), a(s))dWQ(s)

is an H-valued Ft adapted process. Denote

G(t, s,ω) := 1[0,t](s)e(t−s)Aσ(s,X(s), a(s)).

G is B([0, T ])⊗ B([0, T ])⊗ FT /B(L2(Ξ0, H)) measurable. Suppose that

T

0

E T

0G(t, s,ω)2L2(Ξ0,H)ds

12

dt < +∞.


Then, by (i) of Theorem 1.104, the process

(t,ω) → T

0G(t, s,ω)dWQ(s)

has a B([0, T ]) ⊗ FT /B(H) measurable modification on [0, T ] × Ω. Therefore, byLemma 1.69, the process

(t,ω) → t

0e(t−s)Aσ(s,X(s), a(s))dWQ(s) =

T

0G(t, s,ω)dWQ(s)

has an Ft-progressively measurable modification.

1.4.2. Existence and uniqueness of solutions.

Definition 1.118 (The space Mpµ(t, T ;E)) In this definition T ∈ (0,+∞) ∪

+∞. Let p ≥ 1 and 0 ≤ t < T . Given a Banach space E, we denote byMp

µ(t, T ;E) the space of all E-valued progressively measurable processes X(·) suchthat

|X(·)|Mp

µ(t,T ;E) :=

E T

t|X(s)|pds

1/p

< +∞. (1.30)

Mpµ(t, T ;E) is a Banach space endowed with the norm | · |Mp

µ(t,T ;E).

Note that in the notation Mpµ(t, T ;E) we emphasize the dependence on the

generalized reference probability space µ. Processes in Mpµ(t, T ;E) are identified if

they are equal P⊗ dt a.e..Let a : [0, T ] → Λ be an Fs-progressively measurable process (a control pro-

cess). (Recall that Λ is a Polish space.) We consider the controlled SDE

dX(s) = (AX(s) + b(s,X(s), a(s))) ds+ σ(s,X(s), a(s))dWQ(s)X(0) = ξ.

(1.31)

This equation falls into the category of equations (1.29) with b(s, x,ω) :=b(s, x, a(s,ω)) and σ(s, x,ω) := σ(s, x, a(s,ω)). Thus strong, mild and weak mildsolutions of (1.31) are defined using the definitions for equation (1.29).

Hypothesis 1.119 The operator A is the generator of a strongly continuoussemigroup esA on H. The function b : [0, T ] × H × Λ → H is B([0, T ]) ⊗ B(H) ⊗B(Λ)/B(H) measurable, σ : [0, T ] × H × Λ → L2(Ξ0, H) is B([0, T ]) ⊗ B(H) ⊗B(Λ)/B(L2(Ξ0, H)) measurable, and there exists a constant C > 0 such that

|b(s, x, a)− b(s, y, a)| ≤ C|x− y| ∀x, y ∈ H, s ∈ [0, T ], a ∈ Λ, (1.32)σ(s, x, a)− σ(s, y, a)L2(Ξ0,H) ≤ C|x− y| ∀x, y ∈ H, s ∈ [0, T ], a ∈ Λ, (1.33)|b(s, x, a)| ≤ C(1 + |x|) ∀x ∈ H, s ∈ [0, T ], a ∈ Λ, (1.34)σ(s, x, a)L2(Ξ0,H) ≤ C(1 + |x|) ∀x ∈ H, s ∈ [0, T ], a ∈ Λ. (1.35)

Definition 1.120 (The space Hµp (t, T ;E)) Let p ≥ 1 and 0 ≤ t < T . Given

a Banach space E, we denote by Hµp (t, T ;E) the set of all progressively measurable

process X : [t, T ]× Ω → E such that

|X(·)|Hµ

p (t,T ;E) :=

sup

s∈[t,T ]E|X(s)|p

1/p

< +∞.

It is a Banach space with the norm | · |Hµ

p (t,T ;E).

Processes in Hµp (t, T ;E) are identified if they are equal P ⊗ dt a.e.. Therefore

the sup in the definition of Hµp (t, T ;E) must be understood as esssup. However we

will keep the notation sup here and in the all subsequent uses of this space. If the


generalized reference probability space µ is clear we will just write Mp(t, T ;E) andHp(t, T ;E) for simplicity.

Theorem 1.121 Let ξ ∈ Lp(Ω,F0,P) for some p ≥ 2, and let A, b and σsatisfy Hypothesis 1.119. Let a(·) : [0, T ] → Λ be an Fs-progressively measurableprocess. Then the SDE (1.31) has a unique, up to a modification, mild solutionX(·) ∈ Hp(0, T ;H). The solution is in fact unique, among all processes such thatP T

0 |X(s)|2ds < +∞

= 1, in particular among the processes in M2µ(0, T ;H).

X(·) has a continuous modification. Given two continuous versions X1(·), X2(·) ofthe solution, there exists Ω ⊂ Ω with P(Ω) = 1 s.t. X1(s) = X2(s) for all s ∈ [0, T ]and ω ∈ Ω.

Proof. The proof can be found for instance in [130], Theorem 7.2, page 188or [220], Theorems 3.3, page 97, and 3.5, page 105. For the last claim, we can take

Ω :=

s∈Q∩[0,T ]

ω ∈ Ω : X1(s)(ω) = X2(s)(ω) .

Since X1(·) is a modification of X2(·), we have P(Ω) = 1, and since X1(·) and X2(·)are continuous, it follows that X1(s)(ω) = X2(s)(ω) for all s ∈ [0, T ], ω ∈ Ω.

We will denote the solution of (1.31) by X(·; ξ, a(·)) if we want to emphasizethe dependence on the initial datum and the control.

Corollary 1.122 Let ξ ∈ Lp(Ω,F0,P) for some p ≥ 2, let A, b and σ satisfyHypothesis 1.119. If a1(·), a2(·) : [0, T ] × Ω → Λ are two progressively measurableprocesses such that a1(·) = a2(·), dt⊗ P-a.e. on [0, T ]× Ω, then, P− a.e.,

X (s; ξ, a1(·)) = X (s; ξ, a2(·)) for all s ∈ [0, T ].

Proof. Denote Xi(·) := X (·; ξ, ai(·)). Using Theorem 1.98, Jensen’s in-equality, and sups∈[0,T ] esA ≤ C for some C ≥ 0, it follows that, for suitablepositive C1 and C2:

E|X1(s)−X2(s)|2

≤ C1

s

0E|b(r,X1(r), a1(r))− b(r,X2(r), a2(r))|2dr

+

s

0Eσ(r,X1(r), a1(r))− σ(r,X2(r), a2(r))2L2(Ξ0;H)dr

≤ C2

s

0E|X1(r)−X2(r)|2dr, (1.36)

and the claim follows using Gronwall’s lemma and the continuity of the trajectories.

Remark 1.123 Above we assumed that the σ always takes values in L2(Ξ0, H).Existence and uniqueness results for SDEs with more general σ can be found forinstance in [220] Theorem 3.15, page 143, or in [130] Theorem 7.5, page 197. Totreat some specific examples we will also prove more general results in Section 1.5.

1.4.3. Properties of solutions of SDE.

Theorem 1.124 Let ξ ∈ Lp(Ω,F0,P) for some p ≥ 2, a : [0, T ] → Λ be Fs-progressively measurable, and let A, b and σ satisfy Hypothesis 1.119.


(i) Let X(·) = X(·; ξ, a(·)) be the unique mild solution of (1.31) (providedby Theorem 1.121). Then

sups∈[0,T ]

E [|X(s)|p] ≤ Cp(T )(1 + E|ξ|p) if p ≥ 2, (1.37)

E

sups∈[0,T ]

|X(s)|p≤ Cp(T )(1 + E|ξ|p) if p > 2, (1.38)

and

E

supr∈[0,s]

|X(r)− ξ|p≤ ωξ(s) if p > 2, (1.39)

where Cp(T ) is a constant depending on p, T , C (from Hypothesis 1.119)and M,α (where erA ≤ Merα for r ≥ 0), and ωξ is a modulus depend-ing on the same constants and on ξ (in particular they are independentof the process a(·) and of the generalized reference probability space).

(ii) If ξ, η ∈ Lp(Ω,F0,P) for p > 2, and X(·) = X(·; ξ, a(·)), Y (·) =Y (·; η, a(·)) are the solutions of (1.31), then

E

sups∈[0,T ]

|X(s)− Y (s)|2≤ CT (E [|ξ − η|p])

2p , (1.40)

where CT depends only on p, T , C, M , α.

Proof. Part (i): For (1.37) and (1.38) we refer for instance to [130] The-orem 9.1, page 235, or [220], Lemma 3.6, page 102, and Corollary 3.3, page 104.Regarding (1.39), we have that there is a constant c1 depending only on p andsupt∈[0,T ] etA, such that

E

supr∈[0,s]

|X(r)− ξ|p≤ c

E

supr∈[0,s]

erAξ − ξp

+ E

supr∈[0,s]

r

0|b(u,X(u), a(u))|du

p

+ E

supr∈[0,s]

r

0e(r−u)Aσ(u,X(u), a(u))dWQ(u)

p

.

Using Hypothesis 1.119, (1.38), Hölder inequality, and Theorem 1.107, we see that

E

supr∈[0,s]

|X(r)− ξ|p≤ c2

E

supr∈[0,s]

erAξ − ξp+

s

0(1 + E|ξ|p) dr

.

Since supr∈[0,s]

erAξ − ξp s→0+−−−−→ 0 a.e., and supr∈[0,s]

erAξ − ξp ≤ C1|ξ|p, we

conclude by the Lebesgue dominated convergence theorem.

Part (ii): See [130] Theorem 9.1, page 235.

Theorem 1.125 Let ξ ∈ Lp(Ω,F0,P) for some p > 2, and let A, b and σ satisfyHypothesis 1.119. Let a : [0, T ] → Λ be a progressively measurable process. Let X(·)be the unique mild solution of (1.31). Consider the approximating equations

dXn(s) = (AnXn(s) + b(s,Xn(s), a(s))) ds+ σ(s,Xn(s), a(s))dWQ(s)Xn(0) = ξ,

(1.41)


where An is the Yosida approximation of A. Let Xn(·) be the solution of (1.41).Then

limn→∞

E

sups∈[0,T ]

|Xn(s)−X(s)|p= 0. (1.42)

Proof. See [130] Proposition 7.4, page 196, or [220], Proposition 3.2, page101.

The next proposition is a simpler version of Theorem 1.125 which will be usefulto simplify the proofs of the results of Section 1.7.

Proposition 1.126 Let ξ ∈ Lp(Ω,F0,P), f ∈ Mpµ(0, T ;H), and Φ ∈

N pQ(0, T ;H) for some p ≥ 2. Let X(·) be the mild solution of

dX(s) = (AX(s) + f(s)) ds+ Φ(s)dWQ(s)X(0) = ξ

(1.43)

and Xn(·) be the solution of

dXn(s) = (AnXn(s) + f(s)) ds+ Φ(s)dWQ(s)Xn(0) = ξ,

(1.44)

where A generates a C0-semigroup and An is the Yosida approximation of A. Then,if p > 2,

limn→∞

E

sups∈[0,T ]

|Xn(s)−X(s)|p= 0. (1.45)

Moreover, for p ≥ 2, there exists M > 0, independent of n, such that

sups∈[0,T ]

E [|Xn(s)|p] ≤ M, sups∈[0,T ]

E [|X(s)|p] ≤ M. (1.46)

Proof. Observe first that the mild solution of (1.43) is well defined thanksto the assumptions on ξ, f and Φ, and

X(s) = esAξ +

s

0e(s−r)Af(r)dr +

s

0e(s−r)AΦ(r)dWQ(r).

The same is true for the mild solution of (1.44) (which is also a strong solution).To prove (1.45), we write

Xn(s)−X(s) =esAn − esA

ξ +

s

0


f(r)dr

+

s

0


Φ(r)dWQ(r) =: In1 (s) + In2 (s) + In3 (s). (1.47)

It is enough to show that limn→∞ Esups∈[0,T ] |Ini (s)|p

= 0 for i ∈ 1, 2, 3. For

i = 3 this follows from (1.21). To prove it for i = 2, we notice that (B.12) impliesthat if

ψn(r) := sups∈[r,T ]


f(r)

,

then ψn(r)n→∞−−−−→ 0 a.e. on Ω. Moreover, thanks to (B.11) there exists C1 such

that, for all t ∈ [0, T ] and all n,etAn

≤ C1, so ψn(r) ≤ 2C1|f(r)| for all n. Since Tt |f(r)|dr < +∞ for almost every ω ∈ Ω, by the Lebesgue dominated convergence


theorem we have

sups∈[0,T ]

s

0


f(r)

drp

≤ sups∈[0,T ]

s

0ψn(r)dr

p

≤

T

0ψn(r)dr

p

n→∞−−−−→ 0

for a.e. ω ∈ Ω. Now observe that

sups∈[0,T ]

s

0


f(r)

drp

≤ sups∈[0,T ]

s

0(2C1)

p |f(r)|p dr ≤ T

0(2C1)

p |f(r)|p dr,

and the last expression is integrable (on Ω), since f ∈ Mpµ(0, T ;H). There-

fore we can apply the Lebesgue dominated convergence theorem, obtaininglimn→∞ E

sups∈[0,T ] |In2 (s)|p

= 0. The claim for i = 1 follows again from (B.12)

and the Lebesgue dominated convergence theorem.Estimates (1.46) are easy consequences of (B.11) and the assumptions on ξ, f,Φ.

1.4.4. Uniqueness in law.

Definition 1.127 (Finite dimensional distributions) Let T > 0 and t ∈[0, T ). Consider a measurable space (Ω,F ), two probability spaces (Ωi,Fi,Pi) fori = 1, 2, and two processes Xi(s)s∈[t,T ] : (Ωi,Fi,Pi) → (Ω,F ). We say thatX1(·) and X2(·) have the same finite dimensional distributions on D ⊂ [t, T ] if forany t ≤ t1 < t2 < ... < tn ≤ T, ti ∈ D and A ∈ F ⊗ F ⊗ ...⊗ F

n times

, we have

P1 ω1 : (X1(t1), ..X1(tn))(ω1) ∈ A = P2 ω2 : (X2(t1), ..X2(tn))(ω2) ∈ A .

In this case we write LP1(X1(·)) = LP2(X2(·)) on D. Often we will just writeLP1(X1(·)) = LP2(X2(·)) which should be understood that the finite dimensionaldistributions are the same on some set of full measure.

Theorem 1.128 Let H be a separable Hilbert space. Let (Ωi,Fi,Pi) for i = 1, 2two complete probability spaces, and (Ω, F ) a measurable space. Let ξi : Ωi →Ω, i = 1, 2 be two random variables, and fi : [t, T ]× Ωi → H, i = 1, 2 two processessatisfying

P1

T

t|f1(s)|ds < +∞

= P2

T

t|f2(s)|ds < +∞

= 1

and, for some subset D ⊂ [t, T ] of full measure,

LP1 (f1(·), ξ1) = LP2 (f2(·), ξ2) on D.

Then

LP1

·

tf1(s)ds, ξ1

= LP2

·

tf2(s)ds, ξ2

on [t, T ]. (1.48)

Proof. See [368] Theorem 8.3, where the theorem was proved for a moregeneral case of Banach space valued processes.


Theorem 1.129 LetΩ1,F1,F 1,t

s ,P1,WQ,1

and

Ω2,F2,F 2,t

s ,P2,WQ,2

be

two generalized reference probability spaces. Let Φi : [t, T ]×Ωi → L2(Ξ0;H), i = 1, 2be two F i,t

s -progressively measurable processes satisfying

P1

T

tΦ1(s)2L2(Ξ0;H)ds < +∞

= P2

T

tΦ2(s)2L2(Ξ0;H)ds < +∞

= 1.

Let (Ω, F ) a measurable space and ξi : Ωi → Ω, i = 1, 2 be two random variables.Assume that, for some subset D ⊂ [t, T ] of full measure,

LP1 (Φ1(·),WQ,1(·), ξ1) = LP2 (Φ2(·),WQ,2(·), ξ2) on D.

Then

LP1

·

tΦ1(s)dWQ,1(s), ξ1

= LP2

·

tΦ2(s)dWQ,2(s), ξ2

on [t, T ]. (1.49)

Proof. See [368] Theorem 8.6.

Consider now an operator A and mappings b,σ satisfying Hypothesis 1.119,and x ∈ H. Let

Ω1,F1,F 1,t

s ,P1,WQ,1

and

Ω2,F2,F 2,t

s ,P2,WQ,2

be as in

Theorem 1.129. For i = 1, 2 consider an F i,ts -progressively measurable process

ai : [t, T ]× Ωi → Λ.Let p > 2 and let ζi ∈ Lp(Ωi,F

i,tt ,Pi), i = 1, 2. Denote by Hp,i the Banach

space of all F i,ts -progressively measurable processes Zi : [t, T ]× Ωi → H such that

sup

s∈[t,T ]Ei|Zi(s)|p

1/p

< +∞.

Let Ki : Hp,i → Hp,i be the continuous map (see [130] pages 189) defined as

Ki(Zi(·))(s) := e(s−t)Aζi +

s

te(s−r)Ab(r, Zi(r), ai(r))dr

+

s

te(s−r)Aσ(r, Zi(r), ai(r))dWQ,i(r). (1.50)

Lemma 1.130 Consider the setting described above, and let θi : [t, T ] × Ωi →H, i = 1, 2 be stochastic processes. If

LP1(Z1(·), a1(·),WQ,1(·), θ1(·), ζ1) = LP2(Z2(·), a2(·),WQ,2(·), θ2(·), ζ2)

on some subset D ⊂ [t, T ] of full measure, then

LP1(K1(Z1(·))(·), a1(·),WQ,1(·), θ1(·), ζ1)= LP2(K2(Z2(·))(·), a2(·),WQ,2(·), θ2(·), ζ2) on D. (1.51)

Proof. Observe that, since we only have to check the finite dimensionaldistributions, the claims of Theorems 1.128 and 1.129 hold even if ξ1 and ξ2 arestochastic processes, with (1.48) and (1.49) then being true on some set of fullmeasure. Let us choose a partition (t1, .., tn), with t ≤ t1 < t2 < ... < tn ≤ T, tk ∈D, k = 1, ..., n. We need to show that

LP1(K1(Z1(·))(tk), a1(tk),WQ,1(tk), θ1(tk), ζ1 : k = 1, ..., n)

= LP2(K2(Z2(·))(tk), a2(tk),WQ,1(tk), θ2(tk), ζ2 : k = 1, ..., n). (1.52)


We have, denoting f i(r) := 1[t,t1](r)e(t1−r)Ab(r, Zi(r), ai(r)) and Φi(r) :=

1[t,t1](r)e(t1−r)Aσ(r, Zi(r), ai(r)), i = 1, 2,

LP1(f1(·),Φ1(·), Z1(·), a1(·),WQ,1(·), θ1(·), ζ1)

= LP2(f2(·),Φ2(·), Z2(·), a2(·),WQ,2(·), θ2(·), ζ2) on D,

and thus, by Theorem 1.128 applied with

ξ1(·) = (f1(·),Φ1(·), Z1(·), a1(·),WQ,1(·), θ1(·), ζ1),ξ2(·) = (f2(·),Φ2(·), Z2(·), a2(·),WQ,2(·), θ2(·), ζ2),

LP1

t1

tf1(s)ds, f1(·),Φ1(·), Z1(·), a1(·),WQ,1(·), θ1(·), ζ1

= LP2

t1

tf2(s)ds, f2(·),Φ2(·), Z2(·), a2(·),WQ,2(·), θ2(·), ζ2

on D.

Now, applying Theorem 1.129 with

ξ1(·) = t1

tf1(s)ds, f1(·),Φ1(·), Z1(·), a1(·),WQ,1(·), θ1(·), ζ1

,

ξ2(·) = t1

tf2(s)ds, f2(·),Φ2(·), Z2(·), a2(·),WQ,2(·), θ2(·), ζ2

,

we obtain

LP1

t1

tf1(s)ds,

t1

tΦ1(s)dWQ,1(s), f

1(·),Φ1(·), Z1(·), a1(·),WQ,1(·), θ1(·), ζ1

= LP2

t1

tf2(s)ds,

t1

tΦ2(s)dWQ,2(s), f

2(·),Φ2(·), Z2(·), a2(·),WQ,2(·), θ2(·), ζ2

on D. (We recall that the stochastic convolution terms in (1.50) and the stochasticintegrals above have continuous trajectories a.e.). In particular this implies that

LP1(K1(Z1(·))(t1), f1(·),Φ1(·), Z1(·), a1(·),WQ,1(·), θ1(·), ζ1)= LP2(K2(Z2(·))(t1), f2(·),Φ2(·), Z2(·), a2(·),WQ,2(·), θ2(·), ζ2) on D.

We now repeat the above procedure for t2, ..., tn which will yield (1.52) as its con-sequence. Proposition 1.131 Let the operator A and the mappings b,σ satisfy Hypoth-esis 1.119. Let

Ω1,F1,F 1,t

s ,P1,WQ,1

and

Ω2,F2,F 2,t

s ,P2,WQ,2

be two gen-

eralized reference probability spaces. Let ai : [t, T ] × Ωi → Λ, i = 1, 2 be an F i,ts -

progressively measurable process, and let ζi ∈ Lp(Ωi,Fi,tt ,Pi), i = 1, 2, p > 2. Let

LP1(a1(·),WQ,1(·), ζ1) = LP2(a2(·),WQ,1(·), ζ2) on some subset D ⊂ [0, T ] of fullmeasure. Denote by Xi(·), i = 1, 2, the unique mild solution of

dXi(s) = (AXi(s) + b(s,X(s), ai(s))) ds+ σ(s,Xi(s), ai(s))dWQ,i(s)Xi(t) = ζi

(1.53)

on [t, T ]. Then LP1(X1(·), a1(·)) = LP2(X2(·), a2(·)) on D.

Proof. It is known (see [130], proof of Theorem 7.2, pages 188-193) thatthe map Ki is a contraction in Hp,i if [t, T ] is small enough. Thus if we divide [t, T ]into such small intervals [t, T1], ...[Tk, T ], Xi(·) on [t, T1] is obtained as the limit inHp,i (restricted to [t, T1]) of the iterates (Kn

i (x))(·). Therefore, using Lemma 1.130and passing to the limit as n → +∞ we obtain

LP1(1[t,T1](·)X1(·), a1(·),WQ,1(·)) = LP2(1[t,T1](·)X2(·), a2(·),WQ,1(·)) on D.


Without loss of generality we may assume that T1 ∈ D. The solutions on [T1, T2] areobtained as the limits in Hp,i (restricted to [T1, T2]) of the iterates (Kn

i (Xi(T1)))(·),where now

Ki(Zi(·))(s) := e(s−T1)AXi(T1) +

s

T1

e(s−r)Ab(r, Zi(r), ai(r))dr

+

s

T1

e(s−r)Aσ(r, Zi(r), ai(r))dWQ,i(r).

Thus, again using Lemma 1.130 and passing to the limit as n → +∞, it followsthat

LP1(1[t,T2](·)X1(·), a1(·),WQ,1(·)) = LP2(1[t,T2](·)X2(·), a2(·),WQ,1(·)) on D.

We repeat the procedure to obtain the required claim.

1.5. Further existence and uniqueness results in special cases

Throughout this section T > 0 is a fixed constant, H,Ξ, Q, and the generalizedreference probability space µ = (Ω,F , Fss∈[0,T ],P,WQ) are as in Section 1.3(with t = 0), A is the infinitesimal generator of a C0-semigroup on H, and Λ is aPolish space. As in previous sections we will only consider equations on the interval[0, T ], however all results are the same if instead of [0, T ] we took an interval [t, T ],for 0 ≤ t < T .

1.5.1. SDE coming from boundary control problems. In this sectionwe study SDE that include equations coming from optimal control problems withboundary control and noise. To see how they arise the reader can look at theexamples in Subsections 2.6.2 and 2.6.3, and Appendix C. We consider the followingSDE in H:

dX(s) =AX(s) + b(s,X(s), a(s)) + (λI −A)βGab(s)

ds

+σ(s,X(s), a(s))dWQ(s), s ∈ (0, T ]

X(0) = ξ.

(1.54)

Hypothesis 1.132(i) A generates an analytic semigroup

etA

t≥0

and λ is a real constantsuch that (λI −A)−1 ∈ L(H).

(ii) a : [0, T ] × Ω → Λ is progressively measurable, b(·, ·, ·) satisfies (1.32)and (1.34).

(iii) Λb is a Hilbert space and ab(·) : [0, T ]×Ω → Λb is progressively measur-able.

(iv) G ∈ L(Λb;H).(v) β ∈ [0, 1).(vi) γ is a constant belonging to the interval

0, 1

2

, σ is a mapping such that

(λI −A)−γσ : [0, T ]×H ×Λb → L2(Ξ0;H) is continuous. There existsa constant C > 0 such that

(λI −A)−γσ(s, x, a)L2(Ξ0,H) ≤ C(1 + |x|)

for all s ∈ [0, T ], x ∈ H, a ∈ Λ and

(λI −A)−γ [σ(s, x1, a)− σ(s, x2, a)]L2(Ξ0;H) ≤ C|x1 − x2|

for all s ∈ [0, T ], x1, x2 ∈ H, a ∈ Λ.

1.5. FURTHER EXISTENCE AND UNIQUENESS RESULTS IN SPECIAL CASES 59

Remark 1.133 Part (i) of Hypothesis 1.132 implies, thanks to (B.15), that forevery θ ≥ 0 there exists Mθ > 0 such that

|(λI −A)θetAx| ≤ Mθ

tθ|x|, for every t ∈ (0, T ], x ∈ H. (1.55)

Following Remark 1.114, the definition of a mild solution of (1.54) is given byDefinition 1.113 in which the term

s

0e(s−r)A(λI −A)βGab(r)dr

is interpreted as s

0(λI −A)βe(s−r)AGab(r)dr,

and the term s

0e(s−r)Aσ(r,X(r), a(r))dWQ(r)

as s

0(λI −A)γe(s−r)A(λI −A)−γσ(r,X(r), a(r))dWQ(r).

This is natural since (λI − A)βe(s−r)A is an extension of e(s−r)A(λI − A)β and(λI −A)γe(s−r)A(λI −A)−γ = e(s−r)A.

Remark 1.134 SDE of type (1.54) appear most frequently in optimal controlproblems of parabolic equations on a domain O ⊂ Rn with boundary control/noise,see Section 2.6.2. More precisely the cases β ∈

34 , 1

and β ∈

14 ,

12

are related

respectively to the Dirichlet and Neumann boundary control problems when onetakes Λb = L2(∂O) and H = L2(O). γ ∈

14 ,

12

arises when one treats problems

with boundary noise of Neumann type where again Λb = L2(∂O) and H = L2(O).γ,β ∈

12 − , 1

2

arise in some specific Dirichlet boundary control/noise problems

when one considers Λb = L2(∂O) and a suitable weighted L2 space as H.

Theorem 1.135 Assume that Hypothesis 1.132 holds, p ≥ 2, and let α := 12 −γ.

Suppose that

p >1

α(1.56)

and ab(·) ∈ Mqµ(0, T ;Λb) for some q ≥ p, q > 1

1−β . Then, for every initial conditionξ ∈ L2(Ω,F0,P) and processes a(·), ab(·), there exists a unique mild solution X(·) =X(·; 0, ξ, a(·), ab(·)) of (1.54) in H2(0, T ;H) with continuous trajectories P-a.s.. Ifthere exists a constant C > 0 such that

(λI −A)−γσ(s, x, a)L2(Ξ0,H) ≤ C (1.57)

for all s ∈ [0, T ], x ∈ H, a ∈ Λ, then the solution has continuous trajectories P-a.s.without the restriction p > 1

α . If ξ ∈ Lp(Ω,F0,P) then X(·) ∈ Hp(0, T ;H) andthere exists a constant CT,p independent of ξ such that

sups∈[0,T ]

E|X(s)|p ≤ CT,p(1 + E|ξ|p). (1.58)

Proof. Assume first that ξ ∈ Lp(Ω,F0,P) where p ≥ 2 without the restric-tion (1.56). Similarly to the proof of Theorem 1.121, we will show that for some


T0 ∈ (0, T ] the map

K : Hp(0, T0) → Hp(0, T0),

K : Y → esAξ +

s

0e(s−r)Ab(r, Y (r), a(r))dr +

s

0(λI −A)βe(s−r)AGab(r)dr

+

s

0(λI −A)γe(s−r)A(λI −A)−γσ(r, Y (r), a(r))dWQ(r)

(1.59)is well defined and is a contraction. The only difference between our case here andthat considered in Theorem 1.121 are the last two terms in (1.59).

First we prove that K maps Hp(0, T0) into Hp(0, T0). We only show how todeal with the non standard terms. For the third term in (1.59) we can argue asfollows. If Mβ is the constant from (1.55) for θ = β, using (1.55), Hölder andJensen’s inequalities, and q ≥ p, q > 1

1−β , we obtain

sups∈[0,T0]

E s

0(λI −A)βe(s−r)AGab(r)dr

p

≤ sups∈[0,T0]

MpβGpE

s

0

1

(s− r)β|ab(r)|dr

p

≤ MpβGp

T0

0

1

(T0 − r)βq

q−1

dr

p(q−1)q

E T0

0|ab(r)|qdr

p

q

≤ C1

E T0

0|ab(r)|qdr

p

q

< +∞. (1.60)

As regards the stochastic integral term, using Theorem 1.106, (1.55), and Hypoth-esis 1.132-(vi), we estimate

sups∈[0,T0]

E s

0(λI −A)γe(s−r)A(λI −A)−γσ(r, Y (r), a(r))dWQ(r)

p

≤ sups∈[0,T0]

C1E s

0

1

(s− r)2γ(λI −A)−γσ(r, Y (r), a(r))2L2(Ξ0,H)dr

p

2

≤ sups∈[0,T0]

C2

T0

0

1

(T0 − r)2γdr

p

2−1 s

0

1

(s− r)2γE[(1 + |Y (r)|)p]dr

≤ C3

1 + |Y |pHp(0,T0)

(1.61)

for some constant C3.Regarding the proof that, for T0 small enough, K is a contraction, the only non-

standard term to check is the stochastic convolution term, since the third term in(1.59) does not depend on X. Arguing as before we have that for X,Y ∈ Hp(0, T0),


thanks to Theorem 1.106, (1.55), Hypothesis 1.132-(vi), and Jensen’s inequality,

sups∈[0,T0]

E s

0(λI −A)γe(s−r)A(λI −A)−γ [σ(r,X(r), a(r))− σ(r, Y (r), a(r))] dWQ(r)

p

≤ sups∈[0,T0]

C1E s

0

1

(s− r)2γ(λI −A)−γ [σ(r,X(r), a(r))− σ(r, Y (r), a(r))]

2L2(Ξ0,H)

dr

p

2

≤ sups∈[0,T0]

C2E s

0

1

(s− r)2γ|X(r)− Y (r)|2dr

p

2

≤ sups∈[0,T0]

C2

T0

0

1

(T0 − r)2γdr

p

2−1 s

0

1

(s− r)2γE[|X(r)− Y (r)|p]dr

≤ ω(T0)|X − Y |pHp(0,T0), (1.62)

where ω(r)r→0+−−−−→ 0. So for T0 small enough (which is independent of the initial

condition) we can apply the Banach fixed point theorem in Hp(0, T0) as in the proofof Theorem 1.121 (see the proof of [130], Theorem 7.2, page 188). The process cannow be reapplied on intervals [T0, 2T0], ..., [kT0, T ], where k = [T/T0], to obtain theexistence of a a unique mild solution in Hp(0, T ) in the sense of the integral equalitybeing satisfied for a.e. s ∈ [0, T ].

Estimate (1.58) follows from similar arguments using the growth assumptionson b,σ in Hypothesis 1.132 and a version of Gronwall’s lemma, Proposition D.25.

We will now prove the continuity of the trajectories if condition (1.56) is satis-fied. We will only prove the continuity of the stochastic convolution term in (1.59)since the continuity of the other terms is easier to show. In particular, the continuityof the trajectories of the third term in (1.59) follows from Lemma 1.110-(ii).

Let now p > 1α . Hence there is 0 < α < α such that p > 1

α . Then, forr ∈ [t, T ], using (1.55), (1.58), Hypothesis 1.132-(vi), and Jensen’s inequality

E r

0(r − h)−2α

(λI −A)γe(r−h)A(λI −A)−γσ(h,X(h), a(h))2

L2(Ξ0;H)dh

p

2

≤ E r

0(r − h)−2α(λI −A)γe(r−h)A2L(H)

(λI −A)−γσ(h,X(h), a(h))2L2(Ξ0;H)

ds

p

2

≤ C1E r

0(r − h)−2α

(r − h)−2γ(1 + |X(h)|)2dh p

2

≤ C1

T

0(T − h)−2α

(T − h)−2γdh

p

2

suph∈[0,T ]

E[(1 + |X(h)|)p] =: C2 < +∞.

(1.63)

Observe that C2 does not depend on r ∈ [0, T ]. This proves (1.27) and thus theclaim follows from Proposition 1.111. When (1.57) holds, estimate (1.63) is easierand can be done for any exponent p > 1/α in place of p, and thus (1.27) is alwayssatisfied.

Finally we need to discuss the continuity of the trajectories if ξ ∈ L2(Ω,F0,P).We argue as in the proof of Theorem 7.2 of [130]. For n ≥ 1 we define the randomvariables

ξn =

ξ if |ξ| ≤ n0 if |ξ| > n.


The solutions X(·; 0, ξ, a(·), ab(·)) and X(·; 0, ξn, a(·), ab(·)) on [0, T0] are obtainedas fixed points in H2(0, T0) and Hp(0, T0), with p large enough, of the same con-traction map (1.59) with the second map having the term esAξn in place of esAξ.Therefore both solutions can be obtained as limits of successive iterations start-ing from processes esAξn and esAξn respectively. It is then easy to see that wehave X(·; 0, ξ, a(·), ab(·)) = X(·; 0, ξn, a(·), ab(·)), P-a.s. on ω : |ξ(ω)| ≤ n.However the solutions X(·; 0, ξn, a(·), ab(·)) have continuous trajectories. ThusX(·; 0, ξ, a(·), ab(·)) has continuous trajectories P-a.s. on [0, T0] and we can thencontinue the argument on intervals [T0, 2T0], .... Proposition 1.136 Let the assumptions of Theorem 1.135 be satisfied. Denotethe unique mild solution of (1.54) in Hp(0, T ;H) by X(·) = X(·; 0, ξ, a(·), ab(·)).

(i) If ξ1 = ξ2 P-a.s., a1(·) = a2(·) dt ⊗ dP-a.s. a1b(·) = a2b(·) dt ⊗ dP-a.s.,then P-a.s., X(·; 0, ξ1, a1(·), a1b(·)) = X(·; 0, ξ2, a2(·), a2b(·)) on [0, T ].

(ii) LetΩ1,F1,F 1

s ,P1,WQ,1

and

Ω2,F2,F 2

s ,P2,WQ,2

be two gen-

eralized reference probability spaces. Let ζi ∈ Lp(Ωi,F i0,Pi), i =

1, 2. Let (ai, aib) : [0, T ] × Ωi → Λ × Λb, i = 1, 2 be F is-

progressively measurable processes satisfying the assumptions ofTheorem 1.135. Suppose that LP1(a

1(·), a1b(·),WQ,1(·), ζ1) =LP2(a

2(·), a2b(·),WQ,1(·), ζ2) on some subset D ⊂ [t, T ] offull measure. Then LP1(X(·; 0, ζ1, a1(·), a1b(·)), a1(·), a1b(·)) =LP2(X(·; 0, ζ2, a2(·), a2b(·)), a2(·), a2b(·)) on D.

(iii) The solution of (1.54) is unique in M2µ(0, T ;H) as well.

Proof. (i): We argue as in the proof of Corollary 1.122. If Xi(·) :=X

·; 0, ξi, ai(·), aib(·)

, using Theorem 1.98, Jensen’s inequality and Hypothesis

1.132, we obtain

E|X1(s)−X2(s)|2

≤ CT

s

0

1

(s− r)2γE|X1(r)−X2(r)|2dr,

and the claim follows using a version of Gronwall’s lemma (Proposition D.25), andthe continuity of the trajectories.

(ii): The argument is the same as the one used to prove Lemma 1.130 andProposition 1.131, since in the current case the solution is also found iterating themap K.

(iii): Using the same steps as these in the prof of Theorem 1.135 it is notdifficult to see that K is also a contraction in the space M2

µ(0, T0;H) for some T0,and this gives the required uniqueness.

1.5.2. Semilinear SDE with additive noise. In this section we give moreprecise results for some semilinear SDE with additive noise, i.e. for equation (1.29)when the coefficient σ is constant and we have possible unboundedness in the drift.

Consider first the stochastic convolution process

WA(s) =

s

0e(s−r)AσdWQ(r). (1.64)

Hypothesis 1.137(i) The linear operator A is the generator of a strongly continuous semi-

groupetA, t ≥ 0

in H and, for suitable M ≥ 1 and ω ∈ R,

|etAx| ≤ Meωt|x|, ∀t ≥ 0, x ∈ H. (1.65)

(ii) σ ∈ L(Ξ, H) and the symmetric positive operator

Qt : H → H, Qt :=

t

0esAσQσ∗esA

∗

ds,


is of trace class for every t ≥ 0, i.e.Tr [Qs] < +∞. (1.66)

Proposition 1.138 Suppose that Hypothesis 1.137 is satisfied. Then the processWA defined in (1.64) is a Gaussian process with mean 0 and covariance operatorQs, and is mean square continuous. The trajectories of WA(·) are P-a.s. squareintegrable, and WA(·) ∈ M2

µ(0, T ;H). Moreover, if there exists γ > 0 such that T

0s−γTr

esAσQσ∗esA

∗ds < ∞, (1.67)

then WA(·) has continuous trajectories4 and, for p ≥ 2,

E

sup0≤s≤T

|WA(s)|p< +∞.

Proof. See [130] Chapter 5, Theorems 5.2 and 5.11. The last estimate canbe found e.g. as a particular case of [211, Proposition 3.2].

A completely analogous result also holds for the stochastic convolution startingat a point t ≥ 0 for WA(t, s) :=

st e(s−r)AσdWQ(r).

Let T > 0. We consider the SDEdX(s) = [AX(s) + b(s,X(s))] ds+ σdWQ(s), s > 0

X(0) = ξ.(1.68)

Hypothesis 1.139 p ≥ 2 and b(s, x) = b0(s, x, a1(s)) + a2(s), where:(i) The process a1(·) : [0, T ] × Ω → Λ (where Λ is a given Polish space) is

Fs-progressively measurable. The map b0 : [0, T ]×H×Λ → H is Borelmeasurable and there exists a non-negative function f ∈ L1(0, T ;R)such that

|b0(s, x, a1)| ≤ f(s)(1 + |x|) ∀s ∈ [0, T ], x ∈ H and a1 ∈ Λ.

|b0(s, x1, a1)−b0(s, x2, a1)| ≤ f(s)|x1−x2| ∀s ∈ [0, T ], x1, x2 ∈ H and a1 ∈ Λ.

(ii) The process a2(·) is such that for all t > 0, the process (s,ω) →e(t−s)Aa2(s,ω), when interpreted properly, is Fs-progressively measur-able on [0, t]× Ω with values in H, and

|etAa2(s,ω)| ≤ t−βg(s,ω) ∀(s,ω) ∈ [0, T ]× Ω, x ∈ H,

for some β ∈ (0, 1) and g ∈ Mqµ(0, T ;R), where q ≥ p, q > 1

1−β .

Hypothesis 1.139 covers some cases which are not standard and for which aseparate proof of existence and uniqueness of mild solutions of (1.68) is required.

Remark 1.140 Hypothesis 1.139-(ii) is satisfied for example when A is the gen-erator of an analytic C0-semigroup and the process a2(·) is of the form a2(s) =(λI − A)βa3(s), where λ ∈ R is such that (λI − A) is invertible, β ∈ (0, 1),a3(·) ∈ Mq

µ(0, T ;H), q ≥ p, q > 11−β . In such cases the definition of a mild so-

lution of (1.68) is given by Definition 1.113 in which the formal term s

0e(s−r)Aa2(r)dr =

s

0e(s−r)A(λI −A)βa3(r)dr

appearing in the definition of mild solution is interpreted as s

0(λI −A)βe(s−r)Aa3(r)dr.

4Without assuming (1.67) such continuity of trajectories may fail to hold, see e.g. [268].


This is natural since (λI −A)βe(s−r)A is an extension of e(s−r)A(λI −A)β .

Proposition 1.141 Let ξ ∈ Lp(Ω,F0,P) and Hypotheses 1.137 and 1.139 besatisfied. Then equation (1.68) has a unique mild solution X(·; 0, ξ) ∈ Hµ

p (0, T ;H).The solution satisfies, for some Cp(T ) > 0 independent of ξ,

sups∈[0,T ]

E [|X(s; 0, ξ)|p] ≤ Cp(T )(1 + E|ξ|p). (1.69)

Moreover, if ξ1, ξ2 ∈ Lp(Ω,F0,P), we have, P-a.s.,

|X(s; 0, ξ1)−X(s; 0, ξ2)| ≤ MeαT |ξ1 − ξ2|eMeαT

s

0 f(r)dr. (1.70)

Finally, if also (1.67) holds for some γ > 0, then the solution X(·; 0, ξ) has P-a.s.continuous trajectories, and if ξ = x ∈ H is deterministic we then have

E( sups∈[0,T ]

|X(s)|p) ≤ Cp(T )(1 + |x|p) (1.71)

for a suitable constant Cp(T ) > 0 independent of x. In particular if g in Hypothesis1.139-(ii) is in Mq

µ(0, T ;R) for any q ≥ 1, the previous estimate holds for anyp > 0. The same is true for (1.69) if ξ = x ∈ H.

Proof. The proof of existence and uniqueness uses the same techniquesemployed in the Lipschitz case (Theorem 1.121) but contains a small additionaldifficulty due the presence of the term a2(·) and possible singularities in s of theLipschitz norm of b0(s, ·). We will write Hp(0, T ) for Hµ

p (0, T ;H). For Y ∈ Hp(0, T )we set

K(Y )(s) = e(s−t)Aξ +

s

0e(s−r)Ab0(r, Y (r), a1(r))dr+

s

0e(s−r)Aa2(r)dr+WA(s).

(1.72)WA belongs to Hp(0, T ) thanks to Proposition 1.138. Hypotheses 1.139-(i) and1.139-(ii) ensure respectively that the second and third term in the definition of themap K belong to Hp(0, T ) as well (see (1.60)). So K maps Hp(0, T ) into itself. ForY1, Y2 ∈ Hp(0, T ), s ∈ [0, T ],

|K(Y1)(s)−K(Y2)(s)| ≤ MeωT

s

0f(r)|Y1(r)− Y2(r)|dr,

which yields, for T0 ∈ (0, T ],

|K(Y1)−K(Y2)|pHp(0,T0)≤ MeαT sup

s∈[0,T0]E s

0f(r)|Y1(r)− Y2(r)|dr

p

≤ MeαT T0

0f(r)dr

p

sups∈[0,T0]

E|Y1(s)− Y2(s)|p

= MeαT T0

0f(r)dr

p

|Y1 − Y2|pHp(0,T0). (1.73)

Therefore, if T0 sufficiently small, we can apply the contraction mapping principle tofind the unique mild solution of (1.68) in Hp(0, T0). The existence and uniqueness ofa solution on the whole interval [0, T ] follows, as usual, by repeating the procedurea finite number of steps, since the estimate (1.73) does not depend on the initialdata, and the number of steps does not blow up since f is integrable. Estimate(1.69) follows from (1.72) applied to the solution X if we perform estimates similarto these above and use Gronwall’s Lemma.


To show (1.70) we notice that if Z(s) = X(s; 0, ξ1) −X(s; 0, ξ2), then for s ∈[0, T ]

Z(s) = esA(ξ1−ξ2)+

s

0e(s−r)A[b0(r,X(r; 0, ξ1), a1(r))−b0(r,X(r; 0, ξ2), a1(r))]dr.

By Hypothesis 1.139 we thus have

|Z(s)| ≤ MeαT |ξ1 − ξ2|+MeαT s

0f(r)|Z(r)|dr, s ∈ [0, T ]

so that, by Gronwall’s inequality (see Proposition D.24),

|Z(s)| ≤ MeαT |ξ1 − ξ2|eMeαT

s

0 f(r)dr

which gives the claim. The continuity of trajectories follows from Proposition 1.138,Hypothesis 1.139 and Lemma 1.110 for the second and fourth terms in (1.72), andfrom direct computations, together with the dominated convergence theorem, forthe

s0 e(s−r)Aa2(r)dr term.

The last estimate (1.71) follows by standard arguments (see the proof of (1.38)in Theorem 1.124) if we use Proposition 1.138. This implies that if g ∈ Mq

µ(0, T ;R)for any q > 0, (1.71) holds for any p ≥ 2. For p ∈ (0, 2), denoting Zr(s) :=sups∈[0,T ] |X(s)|r, we have

E(Zp(s)) ≤ [E(Zp(s)2/p)]p/2 ≤ (C(1 + |x|2))p/2 ≤ C1(1 + |x|p).

Proposition 1.142 Assume that Hypotheses 1.137, 1.139 hold, and let a2(·) beas in Remark 1.140. Then:

(i) Let ξ1, ξ2 ∈ L2(Ω,F0,P), ξ1 = ξ2 P-a.s.. Let (a11(·), a13(·)), (a21(·), a23(·))be two processes satisfying Hypothesis 1.139, together with Remark1.140, such that (a11(·), a13(·)) = (a21(·), a23(·)), dt ⊗ dP-a.s. Then, de-noting by Xi(·; 0, ξi) the solution of (1.68) for b(s, x) = (λ−A)βai3(s)+b0(s, x, ai1(s)), we have X1(·; 0, ξ1) = X2(·; 0, ξ2), P-a.s. on [0, T ].

(ii) LetΩ1,F1,F 1

s ,P1,WQ,1

and

Ω2,F2,F 2

s ,P2,WQ,2

be two gener-

alized reference probability spaces. Let ξi ∈ L2(Ωi,F i0,Pi), i =

1, 2. Let ai1(·), ai3(·), i = 1, 2, be processes on [0, T ] × Ωi sat-isfying Hypothesis 1.139, together with Remark 1.140. Supposethat LP1(a

11(·), a13(·),WQ,1(·), ξ1) = LP2(a

21(·), a23(·),WQ,2(·), ξ2). Then

LP1(X1(·; 0, ξ1), a11(·), a13(·)) = LP2(X(·; 0, ξ2), a21(·), a23(·)).

(iii) If f ∈ L2(0, T ;R) then the solution of (1.68) ensured by Proposition1.141 is unique in M2

µ(0, T ;H) as well.

Proof. Parts (i) and (ii) are proved similarly as Proposition 1.136 (i)-(ii).Part (iii) follows from (1.70) which is also true in this case. We also point outthat if f ∈ L2(0, T ;R) then K maps M2

µ(0, T ;H) into itself and is a contraction inM2

µ(0, T0;H) for small T0.

1.5.3. Semilinear SDE with multiplicative noise. This section containsa result for a class of semilinear SDE with multiplicative noise. Let T > 0, and letH, Ξ, Λ and a generalized reference probability space

Ω,F , Fss∈[0,T ] ,P,W

be as in Section 1.3, where W (t), t ∈ [0, T ], is a cylindrical Wiener process withcovariance Q = I (so here Ξ0 = Ξ). We consider the following SDE in H:

dX(s) = AX(s) ds+ b(s,X(s), a(s)) ds+ σ(s,X(s), a(s)) dW (s), s ∈ [0, T ],X(0) = x ∈ H.

(1.74)


Hypothesis 1.143(i) The operator A generates a strongly continuous semigroup

etA

t≥0

inH.

(ii) a(·) is a Λ-valued progressively measurable process.(iii) b is a function such that, for all s ∈ (0, T ], esAb : [0, T ] ×H × Λ → H

is measurable and there exist a non-negative function f1 ∈ L1(0, T ;R)such that

|esAb(t, x, a)| ≤ f1(s)(1 + |x|), (1.75)

|esA(b(t, x, a)− b(t, y, a))| ≤ f1(s)|x− y|, (1.76)

for any s ∈ (0, T ], t ∈ [0, T ], x, y ∈ H, a ∈ Λ.(iv) The function σ : [0, T ]×H×Λ → L(Ξ, H) is such that, for every v ∈ Ξ,

the map σ(·, ·, ·)v : [0, T ] × H × Λ → H is measurable and, for everys > 0, t ∈ [0, T ], a ∈ Λ and x ∈ H, esAσ(t, x, a) belongs to L2(Ξ, H).Moreover there exists a non-negative function f2 ∈ L2(0, T ;R) such that

|esAσ(t, x, a)|L2(Ξ,H) ≤ f2(s)(1 + |x|), (1.77)|esAσ(t, x, a)− esAσ(t, y, a)|L2(Ξ,H) ≤ f2(s)|x− y|, (1.78)

for every s ∈ (0, T ], t ∈ [0, T ], x, y ∈ H, a ∈ Λ.

The solution of equation (1.74) is defined in the mild sense of Definition 1.113,where the convolution term

s

0e(s−r)Aσ(r,X(r), a(r)) dW (r), s ∈ [0, T ],

makes sense thanks to (1.77) and Remark 1.117. Moreover, since s → esAb(t, x, a)is continuous on (0, T ] for every t ∈ [0, T ], x ∈ H, a ∈ Λ, we can consider e·Ab to beB([0, T ])⊗ B([0, T ])⊗ B(H)⊗ B(Λ)/B(H) measurable.

Theorem 1.144 Let Hypothesis 1.143 hold and let a(·) be a Λ-valued, progres-sively measurable process. Then, the SDE (1.74) has a unique mild solution X(·)in Hp(0, T ;H) for all p ≥ 2. The solution satisfies

sups∈[0,T ]

E [|X(s)|p] ≤ C(1 + |x|p), for all p > 0, (1.79)

for some constant C depending only on p, f1, f2, T, L and M := sups∈[0,T ] |esA|.Moreover if there exists α ∈ (0, 1/2) such that

1

0s−2αf2

2 (s)ds < +∞ (1.80)

then the mild solution X(·) has continuous trajectories, and we have

E

sups∈[0,T ]

|X(s)|p≤ C(1 + |x|p), for all p > 0, (1.81)

for some C depending on the same quantities as the constant in (1.79).

Proof. Let p ≥ 2. The existence of a unique solution is proved usingthe Banach contraction mapping theorem in Hp(0, T0) for some T0 ∈ (0, T ) smallenough. We define K : Hp(0, T ) → Hp(0, T ) by

K(Y )(s) := esAx+

s

0e(s−r)Ab(r, Y (r), a(r))dr+

s

0e(s−r)Aσ(r, Y (r), a(r))dW (r).

(1.82)


We observe first that this expression belongs to Hp(0, T ). Thanks to (1.75), (1.77)and Theorem 1.106, we have

E s

0e(s−r)Ab(r, Y (r), a(r))dr +

s

0e(s−r)Aσ(r, Y (r), a(r))dW (r)

p

≤ Cp

E s

0[f1(s− r)(1 + |Y (r)|)] dr

p

+ E s

0e(s−r)Aσ(r, Y (r), a(r))dW (r)

p

≤ Cp

T

0f1(r)dr

p

supr∈[0,T ]

E(1 + |Y (r)|)p

+ Cp

T

0f22 (r)dr

p

2

supr∈[0,T ]

E(1 + |Y (r)|)p, (1.83)

where the constant Cp depends only on p. Therefore, for any Y ∈ Hp(0, T ), K(Y ) ∈Hp(0, T ). The estimates showing that K is a contraction on Hp(0, T0) for T0 ∈ (0, T ]small enough are essentially the same. Using (1.76) and (1.78) instead of (1.75) and(1.77) we obtain, for all Y1, Y2 ∈ Hp(0, T0),

|K(Y1)−K(Y2)|pHp(0,T0)≤ Cp

T0

0f1(r)dr

p

+

T0

0f22 (r)dr

p

2

supr∈[0,T0]

E(|Y1(r)− Y2(r)|p), (1.84)

and thus K is a contraction in Hp(0, T0) if T0 ∈ (0, T ] is small enough. The existenceand uniqueness of solution in Hp(0, T ) follows, as usual, by repeating the procedurea finite number of steps, since the estimate does not depend on the initial data,and the number of steps does not blow up since the f1 and f2

2 are integrable.Estimate (1.79) follows in a standard way by applying estimates like these in (1.83)to the fixed point of the map K and using Gronwall’s Lemma (see also the proofof Theorem 7.5 in [130]). We can then extend it to 0 < p < 2 as in the proof ofProposition 1.141.

Finally the continuity of the trajectories and (1.81) is proved using the factor-ization method as in the proof of Proposition 6.9 for p > 2. We extend (1.81) to0 < p ≤ 2 in the same way as for (1.79). Proposition 1.145 Assume that Hypothesis 1.143 holds with f1(s) = Ls−γ1

and f2(s) = Ls−γ2 for some L > 0, γ1 ∈ (0, 1) and γ2 ∈ (0, 1/2). Let (t1, x1),(t2, x2) ∈ [0, T ]×H with t1 ≤ t2. Denote by X(·; t1, x1), X(·; t2, x2) the correspond-ing mild solutions of (1.74) with the same progressively measurable process a(·) andinitial conditions X(ti) = xi, i = 1, 2. Then, for all s ∈ [t2, T ] we have, settingγ3 := [2(1− γ1)] ∧ [1− 2γ2],

E[|X(s; t1, x1)−X(s; t2, x2)|2] ≤

≤ C2

|x1 − x2|2 + (1 + |x1|2)|t2 − t1|γ3 + |e(t2−t1)Ax1 − x1|2

(1.85)

for some constant C2 depending only on γ1, γ2, T, L and M := sups∈[0,T ] |esA|.Moreover the term |e(t2−t1)Ax1 − x1|2 can be replaced by |e(t2−t1)Ax2 − x2|2.

Proof. To simplify notation we denote Xi(s) := X(s; ti, xi), b(r,Xi(r)) :=b(r,Xi(r), a(r)),σ(r,Xi(r)) := σ(r,Xi(r), a(r)), i = 1, 2. By the definition of a mild


solution we have

Xi(s) = e(s−ti)Axi +

s

ti

e(s−r)Ab(r,Xi(r))dr +

s

ti

e(s−r)Aσ(r,Xi(r))dW (r),

hence|X1(s)−X2(s)| ≤ |e(s−t1)Ax1 − e(s−t2)Ax2|+

+

t2

t1

e(s−r)Ab(r,X1(r))dr

+ s

t2

e(s−r)A (b(r,X1(r))− b(r,X2(r))) dr

+

+

t2

t1

e(s−r)Aσ(r,X1(r))dW (r)

+ s

t2

e(s−r)A (σ(r,X1(r))− σ(r,X2(r))) dW (r)

.

Therefore

E|X1(s)−X2(s)|2 ≤ 5|e(s−t1)Ax1 − e(s−t2)Ax2|2

+ 5E t2

t1


2

+ 5E s

t2


2

+ 5E t2

t1


2

+ 5E s

t2


2

. (1.86)

To estimate the second and the third terms we use Jensen’s inequality applied tothe inner integral. Using Hypothesis 1.143-(ii) and (1.81) we then obtain

E t2

t1


2

≤ L2E t2

t1

(s− r)−γ1(1 + |X1(r)|)dr2

≤ L2

t2

t1

(s− r)−γ1dr

t2

t1

(s− r)−γ1E(1 + |X1(r)|)2dr

≤ 2L2[1 + C(1 + |x1|2)] t2

t1

(s− r)−γ1dr

2

≤ 2L2[1 + C(1 + |x1|)2)]1

1− γ1(t1 − t2)

2(1−γ1).

In the same way we estimate the third term getting, by Hypothesis 1.143-(ii),

E s

t2


2

≤ L2

s

t2

(s− r)−γ1dr

s

t2

(s− r)−γ1E|X1(r)−X2(r)|2dr

≤ L2(s− t2)1−γ1

1− γ1

s

t2

(s− r)−γ1E|X1(r)−X2(r)|2dr. (1.87)

The fourth and the fifth term of (1.86) are estimated using the isometry formula.We have

E t2

t1


2

=

t2

t1

E|e(s−r)Aσ(r,X1(r))|2dr

≤ L2

t2

t1

(s− r)−2γ2E(1 + |X1(r)|)2dr ≤ 2L2[1 + C(1 + |x1|2)] t2

t1

(s− r)−2γ2dr

≤ 2L2[1 + C(1 + |x1|2)]1

1− 2γ2(t1 − t2)

1−2γ2


and

E s

t2


2

=

s

t2

E|e(s−r)A (σ(r,X1(r))− σ(r,X2(r))|2 dr

≤ L2

s

t2

(s− r)−2γ2E|X1(r)−X2(r)|2dr.

Using all these estimates in (1.86) we obtain, for a suitable constant C1 > 0, forγ3 := [2(1− γ1)] ∧ [1− 2γ2] and γ4 := γ1 ∨ [2γ2],

E|X1(s)−X2(s)|2 ≤ 5|e(s−t1)Ax1 − e(s−t2)Ax2|2 + C1(1 + |x1|2)|t2 − t1|γ3+

+C1

s

t2

(s− r)−γ4E|X1(r)−X2(r)|2dr.

Observing that

|e(s−t1)Ax1 − e(s−t2)Ax2| ≤ MeωT |x1 − x2|+ |e(s−t2)A(e(t2−t1)Ax1 − x1)|,

we can thus apply a version of Gronwall’s Lemma (see Proposition D.25). It givesus

E|X1(s)−X2(s)|2 ≤ C2

|x1 − x2|2 + (1 + |x1|2)|t2 − t1|γ3 + |e(t2−t1)Ax1 − x1|2

for some C2 > 0 with the required properties.

Lemma 1.146 Assume that Hypothesis 1.143 holds with f1(s) = Ls−γ1 andf2(s) = Ls−γ2 for some L > 0, γ1 ∈ [0, 1) and γ2 ∈ [0, 1/2). Fix a Λ-valuedprogressively measurable process a(·). Let X be the unique mild solution of (1.74)described in Theorem 1.144. Let eii∈N an orthonormal basis of Ξ and, for anyk ∈ N, let P k : Ξ → Ξ be the orthogonal projection onto spane1, ..., ek. Let Xk bethe unique mild solution of

dXk(s) = (AXk(s) + e1kAb(s,Xk(s), a(s)))ds+ e

1kAσ(s,Xk(s), a(s))P kdW (s)

Xk(0) = x.(1.88)

Then, for any p > 0, there exists Mp > 0 such that

supk∈N

E

sups∈[0,T ]

|Xk(s)|p≤ Mp. (1.89)

Moreover, for every s ∈ [0, T ],

limk→∞

E|Xk(s)−X(s)|2

= 0 (1.90)

and, for every ϕ ∈ Cm(H) (m ≥ 0),

limk→∞

Eϕ(Xk(s))

= E [ϕ(X(s))] . (1.91)

Proof. To simplify the notation we drop the dependence on the variable ain the expressions for b and σ.

It is easy to see that, thanks to Hypothesis 1.143, for any k ∈ N, the functionse

1kAb(s, x) and e

1kAσ(s, x)Pk satisfy 1.143 with fi(s) = L1fi(s), i = 1, 2, where

L1 =supr∈[0,1]

erA. Thus we obtain (1.81) for Xk uniformly in k. This gives

(1.89).


We now prove (1.90). We have

EX(s)−Xk(s)

2 ≤ 2E s

0e(s−r)A

b(r,X(r))− e

1kAb(r,Xk(r))

dr

2

+ 2E s

0e(s−r)A

σ(r,X(r))− e

1kAσ(r,Xk(r))P k

dW (r)

2

.

Regarding the first term,

E s

0e(s−r)A

b(s,X(s))− e

1kAb(r,Xk(r))

dr

2

≤ I1 + I2 := 2E s

0e

1kAe(s−r)A

b(s,X(s))− b(r,Xk(r))

dr

2

+ 2E s

0e(s−r)A

b(r,X(r))− e

1kAb(r,X(r))

dr

2

.

We denote Cs := s0 f1(s−r)dr and we set µs(dr) =

f1(s−r)drCs

= LL1(s−r)−γ1

Cs

. UsingJensen inequality we get

I1 ≤ 2E s

0LL1(s− r)−γ1

X(r)−Xk(r) dr

2

≤ 2EC2s

s

0

X(r)−Xk(r)µs(dr)

2

≤ 2EC2s

s

0

X(r)−Xk(r)2 µs(dr) = 2LL1Cs

s

0(s−r)−γ1E

X(r)−Xk(r)2 dr.

We also have

I2 ≤ 2E s

0

I − e

1kAe(s−r)Ab(r,X(r))

dr2

=: ω1(k, s).

By (1.81) and the dominated convergence theorem we have that for every k, ω1(k, ·)is a bounded measurable function, and for every s, limk→∞ ω1(k, s) = 0.

We now estimate

2E s

0e(s−r)A

σ(r,X(r))− e

1kAσ(r,Xk(r))P k

dW (r)

2

≤ J1 + J2 + J3

:= 4E s

0e(s−r)Aσ(r,X(r))

I − P k

dW (r)

2

+ 4E s

0e(s−r)A

σ(r,X(r))− e

1kAσ(r,X(r))

P kdW (r)

2

+ 4E s

0e

1kAe(s−r)A

σ(r,X(r))− σ(r,Xk(r))

P kdW (r)

2

.


Recall that Ξ0 = Ξ. To estimate J1, we set Qk := I − P k. We have

J1 = 4E s

0e(s−r)Aσ(r,X(r))

I − P k

dW (r)

2

= 4

s

0Ee(s−r)Aσ(r,X(r))Qk

2

L2(Ξ,H)dr

= 4

s

0E

i∈N

e(s−r)Aσ(r,X(r))Qkei, e

(s−r)Aσ(r,X(r))Qkeids =: ω2 (k, s) .

Observe that

i∈N

e(s−r)Aσ(r,X(r))Qkei, e

(s−r)Aσ(r,X(r))Qkei

=+∞

i=k+1

e(s−r)Aσ(r,X(r))ei, e

(s−r)Aσ(r,X(r))ei

≤

i∈N

e(s−r)Aσ(r,X(r))ei, e

(s−r)Aσ(r,X(r))ei=

e(s−r)Aσ(r,X(r))2

L2(Ξ,H).

Since the series above has nonnegative terms, we obtain

limk→∞

e(s−r)Aσ(r,X(r))Qk2

L2(Ξ,H)= 0 ds⊗ dP a.s.

Therefore, thanks to (1.81), Hypothesis 1.143 and the dominated convergence the-orem, we obtain that for every s ∈ [0, T ]

limk→∞

J1 ≤ limk→∞

ω2 (k, s) = 0.

The term J2 is estimated similarly.For J3 we estimate

J3 = 4E s

0e

1kAe(s−r)A


P kdW (r)

2

= 4E s

0

e1kAe(s−r)A


P k

2

L2(Ξ,H)dr

≤ 4L2L21

s

0(s− r)−2γ2E

X(r)−Xk(r)2 dr.

Combining the above estimates we have

EX(s)−Xk(s)

2 ≤ ω3 (k, s) +

s

0L2(s− r)−γE

X(r)−Xk(r)2 dr,

for some L2 > 0, γ ∈ [0, 1) and some bounded measurable functions ω3(k, ·) suchthat for every s ∈ [0, T ] limk→∞ ω3 (k, s) = 0. The claim (1.90) now follows from aversion of Gronwall’s Lemma (see Proposition D.25).

Thanks to (1.90), for any subsequence of Xk(s) we can extract a sub-subsequence converging to X(s) almost everywhere and then, thanks to (1.89),(1.81) and the dominated convergence theorem, we obtain (1.91) along the sub-subsequence. This implies (1.91) for the whole sequence Xk(s).

Remark 1.147 Observe that, thanks to the specific form of the functions f1 andf2 considered in Lemma 1.146 and the fact that b and σ satisfy Hypothesis 1.143,the functions e

1kAb(s, x, a) and e

1kAσ(s, x, a)Pk satisfy Hypothesis 1.119.


Remark 1.148 If in Hypothesis 1.119 we set WQ = Q1/2W for a suitable cylin-drical Wiener process W in Ξ, and we substitute σ with σ := σQ1/2, it is easy tosee that Hypothesis 1.143 is more general. Also Hypothesis 1.143 is more generalthan Hypotheses 1.137 and 1.139 if we take f bounded and a2(·) ≡ 0 there.

1.6. Transition semigroups

Let T > 0 and t ∈ [0, T ]. Let H,Ξ, Q, and the generalized reference probabilityspace µ = (Ω,F , F t

ss∈[t,T ],P,WQ) be the same as in Section 1.3. Consider nowthe following SDE with non-random coefficients

dX(s) = (AX(s) + b(s,X(s))) ds+ σ(s,X(s))dWQ(s)

X(t) = x ∈ H,(1.92)

where b : [0, T ]×H → H and σ : [0, T ]×H → L2(Ξ0, H). If Hypothesis 1.119, wherewe drop the dependence on a in all conditions, (respectively, Hypotheses 1.137 and1.139 with a2(·) ≡ 0 and with no dependence on a1, respectively, Hypothesis 1.143with no dependence on a) is satisfied, then Theorem 1.121 (respectively, Proposition1.141, respectively, Theorem 1.144) ensures that (1.92) has a unique mild solutionX(·; t, x).Notation 1.149 (Transition semigroup) The (two parameter) transition semi-group Pt,s associated to equation (1.92) is defined as follows. For a bounded, Borelmeasurable function φ : H → R and 0 ≤ t ≤ s ≤ T ,

Pt,sφ : H → RPt,sφ : x → E[φ(X(s; t, x))].

(1.93)

Under our assumptions, by the measurability properties of the solutionX(s; t, x) with respect to the initial datum x, Pt,sφ is a bounded measurable func-tion on H for all φ ∈ Bb(H). Moreover, thanks to estimates (1.37), (1.69) and(1.79), Pt,sφ is also well defined for any φ ∈ Bm(H), m > 0.

Definition 1.150 [Feller and strong Feller property] A transition semigroupPt,s defined in (1.93) is said to possess the Feller property if

Pt,s(Cb(H)) ⊆ Cb(H)

and it is said to possess the strong Feller property if

Pt,s(Bb(H)) ⊆ Cb(H).

Theorem 1.151 [Feller property of the transition semigroup] Assume that Hy-pothesis 1.143 is satisfied with b and σ independent of a and with f1 = L1s−γ1 , forL1 > 0 and γ1 ∈ (0, 1), and f2 = L2s−γ2 , for L2 > 0 and γ2 ∈ (0, 1/2). Then forevery φ ∈ Cm(H) (m ≥ 0), the function Pt,sφ : H → R belongs to Cm(H). Thesame holds if we assume that Hypotheses 1.137 and 1.139 hold without dependenceon a1 and with a2(·) = 0.

Proof. The result is a consequence of the continuous dependence andgrowth estimates of Theorem 1.144 and Propositions 1.145, 1.141.

Theorem 1.152 (Markov property of the transition semigroup) Let the as-sumptions of Proposition 1.145 be satisfied with b and σ independent of a. Thenfor every φ ∈ Bm(H) (m ≥ 0) and 0 ≤ t ≤ s ≤ r ≤ T ,

Eφ(X(r; t, x)|Fs) = Ps,rφ(X(s; t, x)) P− almost surely,

1.7. ITÔ’S AND DYNKIN’S FORMULAE 73

andPt,rφ(x) = Pt,s(Ps,rφ)(x) for all x ∈ H.

The same result is true if Hypotheses 1.137 and 1.139 hold without dependence ona1 and with a2(·) = 0.

Proof. See [130] Theorem 9.14, page 248, and Corollary 9.15, page 249.The hypothesis in the cylindrical case is a little different from the cylindrical casecontained in [130] however the same arguments can be easily adapted using Proposi-tion 1.145 in the last part of the proof. The proof in [130] is given for φ ∈ Bb(H) butthe argument is exactly the same when φ ∈ Bm(H) (m > 0) simply recalling thatthe operator Pt,s is well defined on such functions thanks to estimate (1.81).

As a particular case we have the following corollary.

Corollary 1.153 Let the assumptions of Proposition 1.145 be satisfied with band σ independent of a and the time variable t. Equation (1.92) then reduces to

dX(s) = (AX(s) + b(X(s))) ds+ σ(X(s))dWQ(s)

X(0) = x ∈ H.(1.94)

Denote by X(·;x) the unique mild solution of this equation. In this case, for anyφ ∈ Bm(H) (m ≥ 0) and s ≥ 0, we define Psφ as follows

Psφ : H → RPsφ : x → Eφ(X(s;x)).

(1.95)

ThenPs+rφ(x) = Ps(Prφ)(x) for all x ∈ H, s, r ≥ 0.

1.7. Itô’s and Dynkin’s formulae

In this section we assume that T > 0, H,Ξ, Q, and the generalized referenceprobability space µ = (Ω,F , Fss∈[0,T ],P,WQ) are the same as in Section 1.3.The operator A is the generator of a C0-semigroup on H, and Λ is a Polish space.The various Itô’s and Dynkin’s formulae presented in this section are used in provingexistence of viscosity solutions (Chapter 3) and verification theorems (Chapters 4and 5).

Theorem 1.154 (Itô’s Formula) Assume that Φ is a process in N 2Q(0, T ;H),

f is an H-valued progressively measurable (P-a.s.) Bochner integrable process on[0, T ], and define, for s ∈ [0, T ],

X(s) := X(0) +

s

0f(r)dr +

s

0Φ(r)dWQ(r),

where X(0) is an F0-measurable H-valued random variable. Consider F : [0, T ] ×H → R and assume that F and its derivatives Ft, DF,D2F , are continuous andbounded on bounded subsets of [0, T ]×H. Then, for P a.e. ω,

F (s,X(s)) = F (0, X(0)) +

s

0Ft(r,X(r))dr +

s

0DF (r,X(r)), f(r) dr+

+

s

0DF (r,X(r)),Φ(r)dWQ(r)+

+1

2

s

0Tr

Φ(r)Q1/2

Φ(r)Q1/2

∗D2F (r,X(r))

dr on [0, T ]. (1.96)


Proof. See [220], Theorems 2.9 and 2.10. See also, under the assumptionof uniform continuity on bounded sets of F and its derivatives, [130] Theorem 4.32page 106. Proposition 1.155 Let F : [0, T ] ×H → R be such that F and its derivativesFt, DF,D2F are continuous in [0, T ]×H. Suppose that DF : [0, T ]×H → D(A∗),that A∗DF is continuous in [0, T ]×H, and that there exist C > 0 and N ≥ 1 suchthat

|F (s, x)|+|DF (s, x)|+|Ft(s, x)|+D2F (s, x)+|A∗DF (s, x)| ≤ C(1+|x|)N (1.97)

for all x ∈ H, s ∈ [0, T ]. Let f ∈ M4Nµ (0, T ;H), Φ ∈ N 4N

Q (0, T ;H) and x ∈ H. LetX(·) be the unique mild solution of (1.43) such that X(0) = x. Then, for P a.e. ω,

F (s,X(s)) = F (0, x) +

s

0Ft(r,X(r))dr

+

s

0A∗DF (r,X(r)), X(r) dr +

s

0DF (r,X(r)), f(r) dr

+1

2

s

0Tr

Φ(r)Q1/2

Φ(r)Q1/2

∗D2F (r,X(r))

dr

+

s

0DF (r,X(r)),Φ(r)dWQ(r) on [0, T ]. (1.98)

Proof. Since both sides of (1.98) are continuous processes, it is enough toprove the formula for a single s. We approximate X(·) by the sequence Xn(·)introduced in Proposition 1.126 (observe that the hypotheses of the proposition aresatisfied with p = 4N). By definition Xn(·) solves the integral equation

Xn(s) =

s

0(AnX

n(r) + f(r)) dr +

s

0Φ(r)dWQ(r).

We can apply Itô’s formula (1.96) (A∗n is the adjoint of An) obtaining

F (s,Xn(s)) = F (0, x) +

s

0Ft(r,X

n(r))dr

+

s

0A∗

nDF (r,Xn(r)), Xn(r) dr + s

0DF (r,Xn(r)), f(r) dr

+1

2

s

0Tr

Φ(r)Q1/2

Φ(r)Q1/2

∗D2F (r,Xn(r))

dr

+

s

0DF (r,Xn(r)),Φ(r)dWQ(r) . (1.99)

Estimates (1.45) and (1.46) yield,

supn≥1

E

sups∈[0,T ]

(|Xn(s)|4N + |X(s)|4N )

≤ C, (1.100)

and, up to a subsequence (still denoted by Xn),

limn→+∞

sups∈[0,T ]

|Xn(s)−X(s)| = 0 for a.e. ω. (1.101)

Therefore, (1.100), (1.101), (1.97), and the hypotheses on f and Φ guarantee that fora.e. ω, the integrands of the first four integrals of (1.99) converge to the integrandsof the corresponding integrals of (1.98), and that the convergence is dominated. (Forthe second integral we use a well known fact that if xn → x ∈ D(A∗) and A∗xn →A∗x, then A∗

nxn → A∗x.) It then follows from the Lebesgue dominated convergencetheorem that, up to a subsequence, the first four integrals of (1.99) converge to


the corresponding integrals of (1.98). Obviously F (s,Xn(s)) also converges toF (s,X(s)).

Regarding the stochastic integral, we have

E s

0DF (r,Xn(r)),Φ(r)dWQ(r) −

s

0DF (r,X(r)),Φ(r)dWQ(r)

2

≤ E s

0|DF (r,Xn(r))−DF (r,X(r))|2Φ(r)2L2(Ξ0,H)dr

≤E T

0|DF (r,Xn(r))−DF (r,X(r))|4dr

1/2

Φ2N 4Q(t,T ;H) → 0 as n → +∞

by (1.100), (1.101), (1.97), and the Lebesgue dominated convergence theorem.Therefore, up to a subsequence, we have that for a.e. ω

s

0DF (r,Xn(r)),Φ(r)dWQ(r) →

s

0DF (r,X(r)),Φ(r)dWQ(r) as n → +∞.

Thus, passing to the limit along a subsequence in (1.99) yields (1.98).

Proposition 1.156 Let b and σ satisfy Hypothesis 1.119 and let a : [t, T ] → Λbe a progressively measurable process. Let X(·) be the unique mild solution of (1.31)such that X(0) = x ∈ H. Let F satisfy the hypotheses of Proposition 1.155. Then:

(i) For P a.e. ω,

F (s,X(s)) = F (0, x) +

s

0Ft(r,X(r))dr

+

s

0A∗DF (r,X(r)), X(r) dr +

s

0DF (r,X(r)), b(r,X(r), a(r)) dr

+1

2

s

0Tr

σ(r,X(r), a(r))Q1/2

σ(r,X(r), a(r))Q1/2

∗D2F (r,X(r))

dr

+

s

0DF (r,X(r)),σ(r,X(r), a(r))dWQ(r) on [0, T ]. (1.102)

(ii) Let η be a real process solving

dη(s) = b(s)dsη(0) = η0 ∈ R,

where b : [0, T ] → R is bounded and progressively measurable. Then, forP a.e. ω,

F (s,X(s))η(s) = F (0, x)η0 +

s

0(Ft(r,X(r))η(r) + F (r,X(r))b(r))dr

+

s

0A∗DF (r,X(r)), X(r) η(r)dr +

s

0DF (r,X(r)), b(r,X(r), a(r)) η(r)dr

+1

2

s

0Tr

σ(r,X(r), a(r))Q

12

σ(r,X(r), a(r))Q

12

∗D2F (r,X(r))

η(r)dr

+

s

0DF (r,X(r))η(r),σ(r,X(r), a(r))dWQ(r) on [0, T ]. (1.103)


In particular

E [F (s,X(s))η(s)] = F (0, x)η0 + E s


+E s

0A∗DF (r,X(r)), X(r) η(r)dr+E

s


+1

2E s

0Tr

σ(r,X(r), a(r))Q

12

σ(r,X(r), a(r))Q

12

∗D2F (r,X(r))

η(r)dr.

(1.104)

Proof. Part (i) follows directly from Proposition 1.155 applied with f(s) :=b(s, a(s), X(s)) and Φ(s) := σ(s, a(s), X(s)), noticing that, thanks to (1.34), (1.35)and (1.38), we have f ∈ Mp

µ(0, T ;H) and Φ ∈ N pQ(0, T ;H) for every p ≥ 1.

Part (ii) is a corollary of (i). We introduce the Hilbert space H := H×R (withthe usual inner product), and set

A =

A0

, b =

bb

, σ =

σ 00 0

.

Then the process

X(s) =

X(s)η(s)

is the mild solution of the SDE

dX(s) =AX(s) + b(s, X(s), a(s))

ds+ σ(s, X(s), a(s))dWQ(s)

X(0) =

xη0

.

Therefore, (1.103) follows from (1.102) applied to the function F (s, x) = F (s, x)η0,where x = (x, η0). Taking expectation in (1.103) we obtain (1.104).

Proposition 1.157 Let b and σ satisfy Hypothesis 1.119 and let a : [t, T ] →Λ be a progressively measurable process. Assume in addition that A is maximaldissipative. Let X(·) be the unique mild solution of (1.31) such that X(0) = x ∈ H.Let F ∈ C1,2([0, T ] × H) be of the form F (t, x) = ϕ(t, |x|) for some ϕ(t, r) ∈C1,2([0, T ] × R), where ϕ(t, ·) is even and non-decreasing on [0,+∞). Moreoversuppose that there exist C > 0 N ≥ 1 such that

|F (s, x)|+ |DF (s, x)|+ |Ft(s, x)|+ D2F (s, x) ≤ C(1 + |x|)N (1.105)

for all x ∈ H, s ∈ [0, T ]. Then:

(i) For P a.e. ω,

F (s,X(s)) ≤ F (0, x) +

s

0

Ft(r,X(r)) + b(r,X(r), a(r)), DF (r,X(r))

+1

2Tr

σ(r,X(r), a(r))Q

12

σ(r,X(r), a(r))Q

12

∗D2F (r,X(r))

dr

+

s

0DF (r,X(r)), b(r,X(r), a(r))dWQ(r) on [0, T ]. (1.106)


(ii) If η is as in part (ii) of Proposition 1.156 and η is positive then, for Pa.e. ω,

F (s,X(s))η(s) ≤ F (0, x)η0

+

s


+

s


+1

2

s

0Tr

σ(r,X(r), a(r))Q

12

σ(r,X(r), a(r))Q

12

∗D2F (r,X(r))

η(r)dr

+

s

0DF (r,X(r))η(r),σ(r,X(r), a(r))dWQ(r) on [0, T ]. (1.107)

In particular

E [F (s,X(s))η(s)] ≤ F (0, x)η0

+ E s


+ E s


+1

2E s

0Tr

σ(r,X(r), a(r))Q

12

σ(r,X(r), a(r))Q

12

∗D2F (r,X(r))

η(r)dr.

(1.108)

Proof. (i): We set f(s) := b(s, a(s), X(s)) and Φ(s) := σ(s, a(s), X(s)) andconsider the approximation Xn(·) of X(·) as in Proposition 1.126. Observe that,thanks to (1.34), (1.35) and (1.38) we have f ∈ Mp

µ(0, T ;H) and Φ ∈ N pQ(0, T ;H)

for every p ≥ 1 so the assumptions of Proposition 1.126 are satisfied.We observe that DF (s, x) = ∂ϕ

∂r (s, |x|)x|x| and, since ϕ(s, ·) is non-decreasing

on [0,+∞), ∂ϕ∂r (s, r) ≥ 0. Therefore, since A, and thus An, is dissipative,

AnXn(s), DF (r,Xn(s)) = ∂ϕ

∂r(s, |Xn(s)|) 1

|Xn(s)| AnXn(s), Xn(s) ≤ 0

(1.109)for every s ≥ 0.

Hence, applying Itô formula for Xn(·) and using (1.109), we obtain

F (s,Xn(s)) = F (0, x) +

s

0

Ft(r,X

n(r)) + AnXn(r), DF (r,Xn(r))

+ f(r), DF (r,Xn(r))+ 1

2Tr

Φ(r)Q

12

Φ(r)Q

12

∗D2F (r,Xn(r))

dr

+

s

0DF (r,Xn(r)), b(r,Xn(r), a(r))dWQ(r) .

≤ F (0, x) +

s

0

Ft(r,X

n(r)) + f(r), DF (r,Xn(r))

+1

2Tr

Φ(r)Q

12

Φ(r)Q

12

∗D2F (r,Xn(r))

dr

+

s

0DF (r,Xn(r)), b(r,Xn(r), a(r))dWQ(r) . (1.110)

It remains to pass to the limit as n → +∞ in (1.110). This can be done followingthe same arguments as these used in the proof of Proposition 1.155.


(ii): The proof combines the proof of (i) with the arguments used in the proofof Proposition 1.156-(ii).

Remark 1.158 Propositions 1.156 and 1.157 are used to work with test functionsin Chapter 3. In particular parts (ii) of them are useful when discount factorsare present (see e.g. Lemma 3.65). Finally we remark that Theorem 1.154 andPropositions 1.155-1.157 are still true if s there is replaced by a stopping time τ on[0, T ].

The next two non-standard versions of Dynkin’s formula will be used to proveverification theorems in Chapters 4 and 5.

Proposition 1.159 Let Q = I. Assume that Hypothesis 1.143 holds withf1(s) = Ls−γ1 and f2(s) = Ls−γ2 for some L > 0, γ1 ∈ [0, 1) and γ2 ∈ [0, 1/2).Assume that there exists λ ∈ R such that (λI−A)−1b : [0, T ]×H×Λ → H is measur-able and is continuous in the x variable. Assume that σ : [0, T ]×H×Λ → L(Ξ, H)and is continuous in the x variable. Suppose moreover that there exists C > 0 suchthat, for all (t, x, a) ∈ [0, T ]×H × Λ,

|(λI −A)−1b(t, x, a)| ≤ C(1 + |x|)σ(t, x, a)L(Ξ,H) ≤ C(1 + |x|).

(1.111)

Fix a Λ-valued progressively measurable process a(·). Let X be the unique mildsolution of (1.74) described in Theorem 1.144. Let F : [0, T ] × H → R be suchthat F and its derivatives Ft, DF,D2F are continuous in [0, T ]×H. Suppose thatDF : [0, T ] × H → D(A∗), that A∗DF is continuous in [0, T ] × H, that D2F :[0, T ]×H → L1(H) is continuous, and that there exist C > 0 and N ≥ 1 such that

|F (s, x)|+ |DF (s, x)|+ |Ft(s, x)|+D2F (s, x)L1(H)+ |A∗DF (s, x)| ≤ C(1+ |x|)N .(1.112)

Then, for any s ∈ [0, T ],

E [F (s,X(s))] = F (0, x) + E s

0Ft(r,X(r))dr

+ E s

0A∗DF (r,X(r)), X(r) dr

+ E s

0

(λI −A∗)DF (r,X(r)), (λI −A)−1b(r,X(r), a(r))

dr

+1

2E s

0Tr

σ(r,X(r), a(r))σ(r,X(r), a(r))∗D2F (r,X(r))

dr. (1.113)

Proof. We will use the notation b(r, x) := b(r, x, a(r)), σ(r, x) :=σ(r, x, a(r)). We approximate the process X(·) by processes Xk(·) from Lemma1.146. As observed in Remark 1.147, the approximating problems satisfy the hy-potheses of Proposition 1.156 so we have

EF (s,Xk(s))

= F (0, x) +

s

0EFt(r,X

k(r))dr

+

s

0EA∗DF (r,Xk(r)), Xk(r)

dr

+

s

0EDF (r,Xk(r)), e

1kAb(r,Xk(r))

dr

+1

2

s

0ETr

e

1kAσ(r,Xk(r))Pk(e

1kAσ(r,Xk(r))Pk)

∗D2F (r,Xk(r))dr. (1.114)


The claim will follow if we can pass to the limit as k → +∞ in each term of (1.114).Using (1.90), (1.89), (1.81) and the the dominated converge theorem it is easy

to see thatlimk→∞

|X(·)−Xk(·)|M2µ(0,T ;H) = 0.

Therefore we can find a subsequence, still denoted by Xk(·), that converges to X(·)dt⊗ dP a.e..

We will only show how to prove the convergence of the last two terms in (1.114)since the arguments for other terms are similar. Using the assumptions it is obviousthat

DF (r,Xk(r)), e

1kAb(r,Xk(r))

=(λI −A∗)DF (r,Xk(r)), e

1kA(λI −A)−1b(r,Xk(r))

→(λI −A∗)DF (r,X(r)), (λI −A)−1b(r,X(r))

as k → +∞. Moreover, thanks to (1.111), (1.112) and (1.89), s

0E(λI −A∗)DF (r,Xk(r)), e

1kA(λI −A)−1b(r,Xk(r))

2dr

≤ C1

s

0E(1 + |Xk(r)|)2(N+1)dr ≤ C2

for some C2 independent of k. Similarly we obtain s

0E(λI −A∗)DF (r,X(r)), (λI −A)−1b(r,X(r))

2 dr ≤ C3

for some C3. Therefore it follows from Lemma 1.50 that

limk→+∞

s

0EDF (r,Xk(r)), e

1kAb(r,Xk(r))

dr

=

s

0E(λI −A∗)DF (r,X(r)), (λI −A)−1b(r,X(r))

dr.

Regarding the last term in (1.114),

Tre

1kAσ(r,Xk(r))Pk(e

1kAσ(r,Xk(r))Pk)

∗D2F (r,Xk(r))

− Trσ(r,X(r))σ(r,X(r))∗D2F (r,X(r))

= I1 + I2 :=

Tre

1kAσ(r,Xk(r))Pk(e

1kAσ(r,Xk(r))Pk)

∗(D2F (r,Xk(r))−D2F (r,X(r)))

+Tre

1kAσ(r,Xk(r))Pk(e

1kAσ(r,Xk(r))Pk)

∗ − σ(r,X(r))σ(r,X(r))∗D2F (r,X(r)))

.

By Proposition B.26 and (1.111) we have

|I1| ≤ C1(1 + |Xk(r)|)2D2F (r,Xk(r))−D2F (r,X(r))L1(H) → 0 as k → +∞

dt⊗dP a.e.. Let e1, e2, ... be an orthonormal basis of eigenvectors of D2F (r,X(r))and λ1,λ2, ... be the corresponding eigenvalues. Then

Tre

1kAσ(r,Xk(r))Pk(e

1kAσ(r,Xk(r))Pk)

∗D2F (r,X(r)))

=∞

n=1

λn

Pkσ(r,Xk(r))∗e

1kA∗

en2

Ξ→

∞

n=1

λn |σ(r,X(r))∗en|2Ξ

= Trσ(r,X(r))σ(r,X(r))∗D2F (r,X(r)))

as k → +∞


dt⊗ dP a.e. Therefore limk→+∞(I1 + I2) = 0 dt⊗ dP a.e. Since we also have s

0E|I1 + I2|2dr ≤ C2

for some constant C2 independent of k, the convergence of the last term in (1.114)now follows from Lemma 1.50.

Remark 1.160 It is obvious from the proof of Proposition 1.159 that the as-sumption σ : [0, T ] × H × Λ → L(Ξ, H) and is continuous in the x variable canbe weakened by the requirement that σ : [0, T ] × H × Λ → L(Ξ, H) and σ∗v iscontinuous in the x variable for every v ∈ H.

Proposition 1.161 Let Hypotheses 1.137 and 1.139 be satisfied with f(s) =Ls−γ1 for some L > 0, γ1 ∈ [0, 1). Assume there exists γ > 0 such that (1.67) issatisfied. Suppose moreover that there exist λ ∈ R such that (λI−A)−1a2(·) : [0, T ]×Ω → H is in M2

µ(0, T ;H). Let X be the unique mild solution of (1.68) describedin Proposition 1.141. Let F : [0, T ] × H → R be such that F and its derivativesFt, DF,D2F are continuous in [0, T ]×H. Suppose that DF : [0, T ]×H → D(A∗),that A∗DF is continuous in [0, T ]×H, that D2F : [0, T ]×H → L1(H) is continuous,and that there exists C > 0 such that (1.112) holds with N = 0. Then, for anys ∈ [0, T ],

E [F (s,X(s))] = F (0, x) + E s

0Ft(r,X(r))dr

+ E s

0A∗DF (r,X(r)), X(r) dr + E

s

0DF (r,X(r)), b0(r,X(r), a1(r)) dr

+ E s

0

(λI −A∗)DF (r,X(r)), (λI −A)−1a2(r)

dr

+1

2E s

0Tr

σQσ∗D2F (r,X(r))

dr.

Proof. The proof is similar to the proof of Proposition 1.159 so we simplycomment how to adapt the latter. Without loss of generality we can assume thatQ = I. First, as we did in Lemma 1.146 for the solutions of (1.74), we need toapproximate the solution X. For example we can use the sequence Xk of thesolutions of the problems

dXk(s) =

AXk(s) + bk(s,Xk(s))

ds+ σP kdW (s), s > 0,

X(0) = x,(1.115)

where, for sufficiently big k ∈ N,- P k : Ξ → Ξ is the projection on the span of the first k vectors of an

orthonormal basis eii∈N of Ξ.- bk(s, x) = bk0(s, x, a1(s)) + (λI − Ak)(λI − A)−1a2(s)1Bk

(s,ω), wherebk0(s, x, a1(s)) = b0(s, x, a1(s))φ(ks), where φ : [0,+∞) → [0, 1] is asmooth, increasing function equal to 0 on the interval [0, 1] and equalto 1 on the interval [2,+∞), Ak is the Yosida approximation of A, andBk = (s,ω) : |(λI −A)−1a2(s,ω)| ≤ k.

Using arguments similar to these in the proof of Lemma 1.146 for the solutions of(1.74) one can prove analogues of (1.89) for p = q, p ≥ 2 (see Hypothesis 1.139),(1.90) and (1.91) and then argue as in the proof Proposition 1.159. We leave thedetails to the reader.

1.8. BIBLIOGRAPHICAL NOTES 81

Remark 1.162 Proposition 1.161 is also true if (1.112) holds with some N ≥1 however then one has to assume that q in Hypothesis 1.139 is large enough(depending on N and β).

1.8. Bibliographical notes

Section 1.1 contains elements of basic probability and measure theory. Classicalreferences are for example [12, 42, 44, 199, 281, 374, 407]. We refer in particularto [42, 44, 199, 281] for general theory of measure and probability (Section 1.1.1)and to [42, 44, 199, 407] for results on measurability (Section 1.1.2 and Section1.1.3). For the Bochner integral and the integration of Banach-valued functions(Section 1.1.3) the reader can consult [137, 138, 140, 306]; some results, useful fromthe stochastic calculus perspective are contained in [130]. For Sections 1.1.4 and1.1.5 conditional expectation for Banach-valued random variables the reader cansee [130, 267, 281, 374, 446]. Gaussian measure in Hilbert spaces (Section 1.1.6)and Fourier transform are nicely introduced in [130, 110], a more extended studyof the subject is contained in [300].

Generalities about stochastic processes, martingales and stopping times in Sec-tion 1.2 can be found in many different books, e.g. [267, 283, 294, 344, 345, 346,395, 398, 446], while for Hilbert-valued martingales (Section 1.2.2) the reader mayconsult [130, 220, 380]. For standard Wiener and Q-Wiener processes and relatedresults we refer to [93, 130, 220, 283, 345, 344, 349]. The material of Section 1.2.4is based on [130]. Definition 1.87 is not contained in the standard literature, itis introduced here because it is useful to study stochastic control problems. Thepresentation of Lemma 1.89 is based on [402] and [283]. The material of Section1.2.5 is loosely based on [130, 220, 283].

The material of Section 1.3 is based on [130, 127, 220] (see also [93, 385]). Thesebooks present the theory in Hilbert spaces while [345, 344] (see also [139]) presentthe Banach space case.

The presentation of Section 1.4 about solutions of stochastic differential equa-tions in Hilbert spaces is also based on [130, 220]. In particular [130] is a stan-dard reference in the theory. Other references on strong and mild solutions arefor example in [93, 127] while good introduction to variational solutions is in[93, 297, 385, 406]. The reader is also referred to [130] for more on weak mildsolutions. Section 1.4.4 containing some results about uniqueness in law uses theapproach of [368]. For a different approach to weak uniqueness based on the theo-rem of Yamada-Watanabe we refer the reader to [385], Appendix E.

Section 1.5 contains existence and uniqueness results for stochastic differentialequations with special unbounded terms and cylindrical additive noise. They aremore or less common knowledge however we presented proofs since no completereferences seem to be available in the literature.

Classical results on transition semigroups (Section 1.6) can be found in [130].The statements here they are a little modified and extended in order to be used inour applications to optimal control, mainly in Chapter 4.

Section 1.7 contains various versions of Itô’s and Dynkin’s formulae (Proposi-tions 1.155, 1.156, 1.157) in connection with mild solutions for functions that haveproperties of test functions used in the definition of viscosity solution (Definition3.32). Such results have been known and used in the viscosity solution literature,however complete proofs are available only in [286]. The statements here are slightlymore general than these in [286] and we presented proofs for the reader’s conve-nience. The last two results of Section 1.7 (Propositions 1.159 and 1.161) are usedto prove the verification theorems of Sections 4.8 and 5.5. They have been used inthe literature (e.g in [231]) but without complete proofs, hence we provide them for


completeness. We finally recall that Itô’s formula related to variational solutions oflinear stochastic parabolic equations is proved in [364].

CHAPTER 2

Optimal control problems and examples

In this chapter we discuss the connection between the study of infinite di-mensional stochastic optimal control problems and that of second order Hamilton-Jacobi-Bellman (HJB) equations in Hilbert spaces. This so called “dynamic pro-gramming approach” to optimal control problems is based on two main results:

• The dynamic programming principle, which is a functional equation forthe value function of the control problem, and whose differential formis the HJB equation. It is the core result in the dynamic programmingapproach.

• The verification theorem, which gives a sufficient (and sometimes nec-essary) condition for optimality. Verification theorems rely on the HJBequation and open the way to the so-called optimal synthesis i.e. theexpression of the optimal control strategy as function of the currentstate trajectory (the feedback form).

To carry out this dynamic programming approach one needs suitable existence,uniqueness and regularity results for the solutions of the HJB equation. With thisin mind we organize the chapter as follows.

In Section 2.1 we describe a general stochastic infinite dimensional optimalcontrol problem (in both strong and weak formulations).

Sections 2.2 and 2.3 contain the dynamic programming principle (with a com-plete proof) and the equivalence between weak and strong formulations when theunderlying “information structure" of the problem is based on a less general notionof a reference probability space, see Definition 2.7. The problem and the statementof the dynamic programming principle are formulated in an abstract form so thatthey can be used in many cases when the solutions of the state equation (which isan infinite dimensional SDE) are interpreted in various ways (strong, mild, varia-tional, etc.). Since this increases the level of technicalities, we recommend that thereaders assume on first reading that the state equation in the control problem is theone described in Section 2.2.3 with solutions defined in the mild sense, as this is themost common case in this book and the theory then applies more straightforwardly.Section 2.4 is devoted to the infinite horizon problem.

In Section 2.5 we present classical verification theorems and the optimal syn-thesis when the value function is regular, in both the finite and the infinite horizoncases.

Finally, in Section 2.6 we discuss various examples of stochastic infinite di-mensional optimal control problems, which arise in applications, and which can bestudied in the framework of the theory presented in this book.

The material on the dynamic programming principle and the examples are pre-sented for optimal control problems defined on the whole space. We do not discussin the book problems in bounded domains with more general cost functionals in-cluding cost of exiting through the boundary, problems with optimal stopping, state

83

84 2. OPTIMAL CONTROL PROBLEMS AND EXAMPLES

constraints problems, singular control problems, risk sensitive control problems, er-godic control problems, stochastic differential games. Some references to results forsuch problems are scattered throughout other sections.

2.1. Stochastic optimal control problem: general formulation

2.1.1. Strong formulation. We start with a description of a general stochas-tic optimal control problem in an infinite dimensional Hilbert space. We will beusing the convention of Notation 1.65.

We make the following assumptions:

Hypothesis 2.1(i) The state space H and the noise space Ξ are real separable Hilbert

spaces.(ii) The control space Λ is a Polish space.(iii) The horizon of the problem is T ∈ (0,+∞)∪+∞, and the initial time

is t ∈ [0, T ).(iv) µ :=

Ωµ,Fµ,

F t

µ,s

s∈[t,T ]

,Pµ,WµQ

is a generalized reference proba-

bility space from Definition 1.95 with WQ(t) = 0, P a.s..

We introduce the set of admissible controlsUµt :=

a(·) : [t, T ]× Ω → Λ : a(·) is F

tµ,s − progressively measurable

. (2.1)

The notation Uµt emphasizes the dependence on the generalized reference probabil-

ity space. Sometimes additional conditions (e.g. state constraints) are imposed onthe admissible controls.

In a general infinite dimensional stochastic optimal control problem, we con-sider, for every aµ(·) ∈ Uµ

t , a system driven by an abstract stochastic differentialequation in H

dX(s) = β(s,X(s), aµ(s))ds+ σ(s,X(s), aµ(s))dWµ

Q(s), s ∈ [t, T ]

X(t) = x ∈ H,(2.2)

where β,σ are appropriate functions for which the above equation is well posed(in a sense to be made precise, see Remark 2.2) for every admissible control. Suchequation is called the state equation and we denote by X(·; t, x, aµ(·)) : [t, T ] → H(or simply by X(·) when its meaning is clear) its unique solution. This is thestate trajectory of the system. The couple (aµ(·), X(·; t, x, aµ(·))) will be called anadmissible couple (or admissible pair).

The goal is to minimize, over all aµ(·) ∈ Uµt , the cost functional

Jµ(t, x; aµ(·)) = Eµ

T

te−

s

tc(X(τ ;t,x,aµ(·)))dτ l(s,X(s; t, x, aµ(·)), aµ(s))ds

+e−

T

tc(X(τ ;t,x,aµ(·)))dτg(X(T ; t, x, aµ(·)))

, (2.3)

where l : [t, T ] × H × Λ → R, c, g : H → R are Borel measurable functions, andc is bounded from below. The function l is the so called running cost, g is theterminal cost, and c is a function responsible for discounting. When T = +∞the standing convention will be that g = 0, i.e. the cost functional only dependson the running cost and discounting. When T is finite the problem is called afinite horizon problem, and when T = +∞ it is called an infinite horizon problem.The expectation Eµ is computed with respect to the probability measure Pµ soit depends on the generalized reference probability space. When the generalizedreference probability space is clear we will often drop the superscript µ in ournotation. We will refer to the above problem as the strong formulation of the

2.1. STOCHASTIC OPTIMAL CONTROL PROBLEM: GENERAL FORMULATION 85

stochastic optimal control problem (2.2)-(2.3) on [t, T ]. Strong here means that thegeneralized reference probability space is fixed.

The discounting function c may also depend on the control variable. The resultsof this chapter can be easily extended to cover such case. However we chose not toinclude this dependence in order not to overcomplicate the presentation which isalready very technical.

Remark 2.2 In the infinite dimensional case the state equation (2.2) can havedifferent forms which may call for various definitions of solutions (strong solutions,mild solutions, variational solutions, etc.) and various approaches to solve them.For this reason in our general formulation we do not specify the concept of solutionof (2.2) and we do not specify the required assumptions. Later, in Section 2.2, wewill formulate and prove the dynamic programming principle (DPP) in a generalform so that it can be applied in these different situations. Thus we will make aseries of rather abstract assumptions (see Hypotheses 2.11-2.12) about (2.2) that areverified in various cases for different concepts of solutions and which are sufficientto prove the DPP.

However the reader should keep in mind that our primary guiding examples arethe control problems of the type (2.2)-(2.3) where the state equation is a stochasticevolution equation with solutions interpreted in the mild sense. In such cases wehave β = A + b, where A is the generator of a C0 semigroup on H, and b,σ arefunctions satisfying suitable Lipschitz conditions. This case requires a less generalformulation to prove the DPP and will be discussed separately in Subsection 2.2.3.The cases which do not use mild solutions include optimal control problems for theDuncan-Mortensen-Zakai, Burgers, Navier-Stokes, and reaction diffusion equations.

The value function for problem (2.2)-(2.3) in the strong formulation with initialtime t is defined as

V µt (x) = inf

a(·)∈Uµ

t

Jµ(t, x; aµ(·)). (2.4)

Notice however, that in this strong formulation the generalized reference probabilityspace changes when we change t and so does the control set Uµ

t .

Definition 2.3 (Optimal control/couple) If, for given initial data (t, x),a∗(·) ∈ Uµ

t minimizes (2.3), i.e. if Jµ(t, x; a∗(·)) = V µt (x), we say that a∗(·) is an

optimal control at (t, x). The associated state trajectory X∗(·) := X(·; t, x, a∗(·))(i.e. the solution of (2.2) with control a∗(·)) is an optimal state at (t, x). The pair(a∗(·), X∗(·)) is called an optimal couple (or optimal pair) at (t, x).

To perform the dynamic programming approach in the strong formulation weneed to consider a family of problems (2.2)-(2.3) parameterized by the initial timet which are defined on a common generalized reference probability space, and in-troduce a value function defined on [0, T ] × H. To do this we take a generalizedreference probability space µ =

Ωµ,Fµ,

F 0

µ,s

s∈[0,T ]

,Pµ,WµQ

on [0, T ] with

WQ(0) = 0 (i.e. µ satisfies Hypothesis 2.1 with initial time t = 0). We then definethe value function

V µ(t, x) = infa(·)∈Uµ

0

Jµ(t, x; aµ(·)), (2.5)

where Jµ(t, x; aµ(·)) is defined by (2.3) with X(·; t, x, aµ(·))) solving (2.2). Wenotice that for µ as above, the generalized reference probability spaces µt :=Ωµ,Fµ,

F 0

µ,s

s∈[t,T ]

,Pµ,WµQ(·)−Wµ

Q(t)

satisfy Hypothesis 2.1 with initialtime t. Thus it is reasonable to expect that V µ(t, x) should be equal to V µt

t (x)for (t, x) ∈ [0, T ] × H. This is indeed the case for control problems considered in


this book and it is a simple consequence of the properties of the stochastic integral(see e.g. (2.16)). Thus the requirement that WQ(t) = 0 in Hypothesis 2.1-(iv) canbe dropped for all practical purposes.

2.1.2. Weak formulation. In the strong formulation of the optimal con-trol problem (2.2)-(2.3), the generalized reference probability space µ :=(Ω,F ,F t

s ,P,WQ) is fixed. However it is often more convenient or necessary toinclude the generalized reference probability space as part of the control. Thisapproach is used to prove the dynamic programming principle and to constructoptimal feedback controls (see Sections 2.2 and 2.5). This leads us to the weakformulation of the stochastic optimal control problem.

In the weak formulation the state equation and the cost functional are thesame, however, for each fixed t ∈ [0, T ], any generalized reference probability spaceµ is allowed and so the class of admissible controls is enlarged. We define

U t :=

µ

Uµt , (2.6)

where the union is taken over all generalized reference probability spaces µ satisfyingHypothesis 2.1-(iv). We say that a(·) is an admissible control if a(·) ∈ U t, i.e. ifthere exist a generalized reference probability space µ =

Ωµ,Fµ,Fµ,t

s ,Pµ,WµQ

satisfying Hypothesis 2.1-(iv) such that a(·) : [t, T ]×Ωµ → Λ is Fµ,ts -progressively

measurable. We will often write aµ(·) to indicate the dependence of a(·) on thegeneralized reference probability space. This way, choosing an admissible controlalso means choosing a generalized reference space µ, so with a slight abuse ofnotation, we will often write, a(·) = (Ω,F ,F t

s ,P,WQ, a(·)).Given a control aµ(·) ∈ U t and the related trajectory X(·; t, x, aµ(·))1, we call

the couple (aµ(·), X(s; t, x, aµ(·))) an admissible couple (or admissible pair) in theweak sense.

Remark 2.4 To avoid misunderstandings we clarify that the use of the term“weak" in the “weak formulation" of our stochastic control problem, is referred onlyto the fact that the generalized reference probability spaces vary with the controlsand not to the concept of solution of the state equation. Indeed, in our framework,once the control aµ(·) is fixed (and with it also the generalized reference space), thesolution is taken in the same generalized reference space (i.e. in the so-called strongprobabilistic sense). Solutions of the state equation in the weak probabilistic sensewill be used only to treat the closed loop equations in some cases, see Remark 2.6

In the weak formulation the goal is to minimize the same cost functional (2.3),however now over all controls aµ(·) ∈ U t. Consequently, the value function in theweak formulation is now defined by

V (t, x) = infaµ(·)∈Ut

Jµ(t, x; aµ(·)), (t, x) ∈ [0, T )×H, (2.7)

and we set V (T, x) := g(x) for x ∈ H if T < +∞. From the above definition weclearly have

V (t, x) = infµ

infa(·)∈Uµ

t

Jµ(t, x; a(·)) = infµ

V µt (x). (2.8)

1To know that such trajectory exists and is unique we need to assume that the state equation(2.2) is well posed for every admissible control aµ(·), so in particular for every generalized referencespace.

2.2. DYNAMIC PROGRAMMING PRINCIPLE: SETUP AND ASSUMPTIONS 87

Remark 2.5 For the optimal control problem we could as well have requiredthat controls in Uµ

t be measurable and adapted instead of progressively measur-able, since, by Lemma 1.69, every adapted a(·) has a progressively measurablemodification a(·). We chose to deal with progressively measurable controls to avoidunnecessary technical issues. In light of Lemma 1.94, we could have chosen to workwith predictable controls as well.

Moreover, in the definition of Uµt we did not specify possible further restrictions

on the control and on the state (state constraints, integrability conditions on thecontrols, etc.) which commonly arise in examples, see Section 2.6. Such kinds ofrestrictions usually lead to more complicated problems, however in principle theycan be treated in the same framework.

Remark 2.6 To study problems where neither existence, nor uniqueness of so-lutions of the state equation is guaranteed for arbitrary control process a(·) (inparticular to study the existence of optimal feedback controls) it is useful to extendthe formulation of an optimal control problem. In such cases the extended formula-tion of the control problem can be given as follows. Given a generalized referenceprobability space µ, we call (a(·), X(·)) an admissible control pair if a(·) is an F t

s -progressively measurable process with values in Λ and X(·) is a (not necessarilyunique) solution of (2.2) corresponding to a(·). To every admissible control pairwe associate the cost (2.3). The optimal control problem in the extended strongformulation consists in minimizing the functional Jµ(t, x; a(·), X(·)) over all admis-sible control pairs (a(·), X(·)), and in characterizing the value function (where weuse the same notation for simplicity)

V µt (x) = inf

(a(·),X(·))Jµ(t, x; a(·), X(·)).

The optimal control problem in the extended weak formulation consists in furtherminimizing with respect to all generalized reference probability spaces, i.e. in char-acterizing the value function (where again we use the same notation for simplicity)

V (t, x) = infµ

V µt (x).

Such formulation is often much more suitable for construction of optimal feedbackcontrols, see Corollary 2.37, and is employed for instance in Chapter 6 (and partlyalso in Chapters 4 and 5).

2.2. Dynamic Programming Principle: setup and assumptions

In this section we introduce the Dynamic Programming Principle (DPP). Itis one of the fundamental results of stochastic optimal control, whose formulationand proof are very technical, here even more so since we deal with the infinitedimensional case. We first present the stochastic setup and the main assumptions,and then follow with the statement of the DPP and the proof.

2.2.1. The setup.

Definition 2.7 A reference probability space is a generalized reference prob-ability space ν := (Ω,F ,F t

s ,P,WQ) (see Definition 1.95), where WQ(t) = 0, Pa.s., and F t

s = σ(F t,0s ,N ), where F t,0

s = σ(WQ(τ) : t ≤ τ ≤ s) is the filtrationgenerated by WQ, and N is the collection of the P-null sets in F .

Definition 2.8 We will say that a reference probability space ν is standard ifthere exists a σ-field F such that F

t,0T ⊂ F ⊂ F , F is the completion of F ,

and (Ω,F ) is a standard measurable space (see Section 1.1).


We will consider control problem (2.2)-(2.3) in the weak formulation in whichgeneralized reference probability spaces are replaced by reference probability spacesThis means that we are restricting the set of admissible controls. The set of alladmissible controls is now defined by

Ut :=

ν

Uνt , (2.9)

where the union is taken over all reference probability spaces ν. Obviously Ut ⊆ U t,where U t is defined by (2.6). Thus a(·) is an admissible control now if there exist areference probability space ν =

Ων ,F ν ,F ν,t

s ,Pν ,W νQ

such that a(·) : [t, T ]×Ων →

Λ is F ν,ts -progressively measurable. As before we will often write aν(·) to indicate

the dependence of a(·) on the reference probability space.With this definition the value function is now defined by

V (t, x) = infaν(·)∈Ut

Jν(t, x; aν(·)). (2.10)

(with the same convention that V (T, x) := g(x) if T < +∞) and, clearly, if V isthe value function defined in (2.7),

V (t, x) ≤ V (t, x) ≤ V νt (x), for every reference probability space ν. (2.11)

We will later see (Theorem 2.22) that the last inequality is indeed an equality underour assumptions. When solutions of the HJB equations are regular enough to allowconstruction of optimal feedbacks we will also see in Chapters 4, 5 and 6 thatboth inequalities become equalities (see e.g. Theorem 4.140). We do not studythis issue here but the reader may check [364] for an argument that for controlproblems considered in [364] the first inequality is an equality. It is possible thatthe approach from [364] can be applied to the control problems in this book.

2.2.2. The general assumptions. We make the following general assump-tion.

Hypothesis 2.9(i) The state space H and the noise space Ξ are real separable Hilbert

spaces.(ii) The control space Λ is a Polish space.(iii) The horizon of the problem is T ∈ (0,+∞)∪+∞ and the initial time

is t ∈ [0, T ).(iv) Q ∈ L+

1 (Ξ)

Remark 2.10 We assume here that Q ∈ L+1 (Ξ) (i.e. Tr (Q) < +∞), which

implies that the processes WQ in the reference probability spaces are Ξ-valuedQ-Wiener processes. We do this for technical reasons, because in our proof of theDPP it is important that the Q-Wiener processes always have values in some (fixed)space Ξ. However in many examples discussed in this chapter we will encounterQ-Wiener processes for which Tr (Q) = +∞. Recalling the definition of a Q-Wiener process (see Definition 1.83) we then have to choose and fix a space Ξ1

such that WQ is a Ξ1-valued Q1-Wiener process. (Since the space Ξ1 is often notimportant, abusing notation, we will still call such process a WQ-Wiener process,see Remark 1.84.) It puts us in the framework developed in this chapter andthis is how the reader should understand such control problems, as it will not berepeated in the future when we discuss the examples in this chapter unless it isessential. However, as mentioned in Remark 1.86, if WQ(s) =

∞k=1 gkβk(s) for

some orthonormal basis gkk∈N of Ξ0 (see Definition 1.83), then regardless ofthe choice of Ξ1, F t,0

s = σβk(r) : t ≤ r ≤ s, k ∈ N. Thus the filtration does


not depend on the choice of Ξ1, and then also the class of integrable processes isindependent of Ξ1. Therefore the control problems discussed in the examples areindependent of the choice of Ξ1 and the theory can be applied to optimal controlproblems for which Q ∈ L+(Ξ).

The following comment is important. The Q-Wiener processes in the referenceprobability spaces in general have trajectories which are only P a.e. continuous.However we can always modify them on a set of measure zero so that the trajecto-ries are continuous everywhere. Moreover it is obvious that such modified Q-Wienerprocess generates the same filtration F t

s as the original one so the set of admissiblecontrols does not change. Moreover the solutions of the stochastic differential equa-tions for control problems considered in this book are indistinguishable after thismodification of the Q-Wiener processes so the cost functional will be the same (seeassumption (A1) of Hypothesis 2.12). Therefore, unless specified otherwise, withoutloss of generality, we will always assume that the Q-Wiener processes in thereference probability spaces have everywhere continuous paths howeverwe will point it out explicitly if it is important to avoid any misunderstandings.

The assumptions about existence and uniqueness of solutions of the state equa-tion are the following.

Hypothesis 2.11 Assume that, for every 0 ≤ t < T , reference probability spaceν := (Ω,F ,F t

s ,P,WQ), a(·) ∈ Uνt and an H-valued F t

t -measurable random vari-able ζ (i.e, ζ = x, P a.s. for some x ∈ H), we have a unique, up to a modification,solution (in a certain sense) X(·) = X(·; t, ζ, a(·)) on [t, T ] of the abstract stochasticdifferential equation

dX(s) = β(s,X(s), a(s))ds+ σ(s,X(s), a(s))dWQ(s),

X(t1) = ζ.(2.12)

The solution X(·; t, ζ, a(·)) is F ts -progressively measurable, has continuous trajec-

tories in H and X(t; t, ζ, a(·)) = ζ, P a.e..

The above hypothesis in particular implies that any modification of a solutionis still a solution of the same equation as long as it has continuous trajectories. Toemphasize the dependence of the solution on the reference probability space we willsometimes use the notation Xν(·; t, ζ, a(·)).

Hypothesis 2.12 collects assumptions about the properties of the family of so-lutions of the state equation. To simplify notation we will write W instead of WQ

in Hypothesis 2.12 and in other places when notation becomes cumbersome andwhen the meaning of it is clear.

Hypothesis 2.12 Assume that Hypothesis 2.11 holds. For every 0 ≤ t ≤ η < T ,x ∈ H, reference probability space ν = (Ω,F ,F t

s ,P,W ), a(·) ∈ Uµt , and an H-

valued F tt -measurable random variable ζ such that ζ = x, P a.s., we have the

following properties:

(A0)

X (·; t, ζ, a(·)) = X (·; t, x, a(·)) on [t, T ] P a.e..

(A1) If ν1 =Ω1,F1,F t

1,s,P1,W1

, ν2 =

Ω2,F2,F t

2,s,P2,W2

are

two reference probability spaces, a1 (·) ∈ Uν1t , a2 (·) ∈ Uν2

t , andLP1(a1(·),W1(·)) = LP2(a2(·),W2(·)) (see Definition 1.127), then

LP1(X(·; t, x, a1(·)), a1(·)) = LP2(X(·; t, x, a2(·)), a2(·)).


(A2) If a1 (·) , a2 (·) ∈ Uνt are such that a1(·) = a2(·), dt⊗P a.e. on [t, η]×Ω,

then

X (·; t, x, a1(·)) = X (·; t, x, a2(·)) on [t, η] , P a.e..

(A3) Let ν = (Ω,F ,F ts ,P,W ) be a standard reference probability space (Def-

inition 2.8) with W having everywhere continuous trajectories. Letνω0 =

Ω,Fω0 ,F

ηω0,s,Pω0 ,Wη

, where Pω0 = P(·|F t,0

η )(ω0) is the regu-lar conditional probability, Fω0 is the augmentation of F by the Pω0

null sets, and F ηω0,s is the augmented filtration generated by Wη

2. Leta (·) ∈ Uν

t and a|[η,T ] (·) ∈ Uνω0η for P a.e. ω0. Then the process

Xν (s; t, x, a(·)) has an indistinguishable version such that, for P a.e.ω0, Xνω0 (·; η, Xν(η), a(·)) = Xν (·; t, x, a(·)) on [η, T ] Pω0 a.s..

Remark 2.13 It is possible to relax and slightly simplify Hypothesis 2.12 bycombining conditions (A0)− (A1) into one condition

(A1’) If ν1 =Ω1,F1,F t

1,s,P1,W1

, ν2 =

Ω2,F2,F t

2,s,P2,W2

are two ref-

erence probability spaces, x ∈ H, ζ is an H-valued F t1,t-measurable

random variable such that ζ = x, P1 a.s., a1 (·) ∈ Uν1t , a2 (·) ∈ Uν2

t , andLP1(a1(·),W1(·)) = LP2(a2(·),W2(·)), then

LP1(X(·; t, ζ, a1(·)), a1(·)) = LP2(X(·; t, x, a2(·)), a2(·)).With this change the proof of the dynamic programming principle is virtually un-changed. However since condition (A0) is standard and is satisfied by control prob-lems considered in this book, we opted to keep it in the formulation of Hypothesis2.12 hoping that this way the proof of the dynamic programming principle will beslightly easier to follow.

Remark 2.14 We point out that since the trajectories of the solutions are con-tinuous, (A1) implies in particular that

LP1(X(·; t, x, a1(·))) = LP2(X(·; t, x, a2(·))) on [t, T ].

Let us briefly explain the nature of the abstract assumptions of Hypothesis2.12. Condition (A0) guarantees pathwise uniqueness for our solutions with al-most deterministic initial conditions, while condition (A1) is a statement aboutuniqueness in law which guarantees that the joint law of (X(·; t, x, a(·)), a(·)) onlydepends on the joint law of (a(·),W (·)). Condition (A2) is a requirement that ifthe controls are “almost the same” then the solutions do not change. Finally, themost complicated condition (A3) is a technical assumption which is needed sincewe do not define precisely what we mean by a solution. It guarantees, roughlyspeaking, that if we have a solution X in one reference probability space, then, forP a.e. ω0, X is still a solution in reference probability spaces equipped with mea-sures Pω0 provided certain conditions are satisfied. We remark that for P a.e. ω0,Xν(η) is Pω0 a.e. constant and is equal to Xν(η)(ω0). We remark that condition(A3) in particular implies that the version of Xν (s; t, x, a(·)) is F η

ω0,s-progressivelymeasurable on [η, T ] for P a.e. ω0, and has continuous trajectories Pω0 a.e. for Pa.e. ω0. These two properties can be proved for every solution Xν satisfying ourHypothesis 2.11. We required that ν is a standard reference probability space toguarantee the existence of the regular conditional probability Pω0 . The requirementthat a|[η,T ] (·) ∈ Uνω0

η can be always assumed since we will see in Lemma 2.26 that

2We remark that νω0 in (A3) is a reference probability space for P a.e. ω0 by Lemma 2.25.


for every a (·) ∈ Uνt there is a1 (·) ∈ Uν

t such that a (·) = a1 (·), P ⊗ dt a.e. anda1|[η,T ] (·) ∈ Uνω0

η for P a.e. ω0.

Remark 2.15 This is a very important remark regarding optimal control prob-lems with additional conditions on the set of admissible controls. In the proof ofthe dynamic programming principle we will use the following property of admissiblecontrols.

(A4) If ν is a standard reference probability space as in (A3) and a (·) ∈ Uνt ,

then there exists a1 (·) ∈ Uνt such that a(·) = a1(·) P ⊗ dt a.e. and

a|[η,T ] (·) ∈ Uνω0η for P a.e. ω0, where νω0 is as in (A3).

This property is always true for our abstract optimal control problem and it isshown in Lemma 2.26 whose proof is only based on considerations of measurability.However, if the set of admissible controls is characterized by additional conditions,for instance some integrability conditions, property (A4) must be established ineach particular case, see for instance Remark 2.27. Therefore the reader should bevery careful adapting the abstract proof of the dynamic programming principle tosuch cases.

We close with important remarks about standard reference probability spacesand regular conditional probabilities. Let ν = (Ω,F ,F t

s ,P,WQ) be a standardreference probability space. Regular conditional probabilities will be denoted byPω, and to indicate that Pω0 is the regular conditional probability given a sigmafield F t,0

s we will write Pω0 = P(·|F t,0s )(ω0) even though this is a slight abuse of

notation. The expectation with respect to Pω0 will be denoted by Eω0 .For every Ω1 ∈ F such that P(Ω1) = 1 there exists Ω2 ⊂ Ω1,Ω2 ∈ F (from

Definition 2.8) such that P(Ω2) = 1. Therefore

1 = P(Ω2) = EE1Ω2 |F t,0

s

(ω0)

= E [Pω0(Ω2)] .

Thus we obtain that Ω1 ∈ Fω0 for P a.e. ω0 and

1 = Pω0(Ω2) = Pω0(Ω1).

Now suppose that Y ∈ L1(Ω,F ,P). Let Y ∈ L1(Ω,F ,P) be such that Y = Y ,P a.s. Hence also Y = Y Pω0 a.s. for P a.s. ω0, which implies that Y is Fω0

measurable for P a.s. ω0. Thus for P a.s. ω0

EY |F t

s

(ω0) = E

Y |F t,0

s

(ω0) =

Y (ω)dPω0(ω) =

Y (ω)dPω0(ω) = Eω0 [Y ].

Therefore Eω0 [Y ] (as a function of ω0) is in L1(Ω,F ,P) and

E[Y ] = EEY |F t

s

= E [Eω0 [Y ]] .

This fact will be used frequently in the following chapters without repeating thetechnical details.

2.2.3. The assumptions in the case of control problem for mild solu-tions. In this subsection we briefly illustrate the abstract setup for the case whichis the most frequent among problems treated in the book, namely optimal controlproblem driven by general stochastic evolution equation (with Lipschitz coefficients)with solutions interpreted in the mild sense, and explain how Hypotheses 2.11 and2.12 are satisfied. In this case the state equation (2.12) is of type (1.31) i.e.

dX(s) = (AX(s) + b(s,X(s), a(s))) ds+ σ(s,X(s), a(s))dWQ(s)

X(t) = ζ.(2.13)


where A, b and σ satisfy Hypothesis 1.119 and its solution is understood in themild sense of Definition 1.113, i.e. we have

X(s) = e(s−t)Aζ+

s

te(s−r)Ab(r,X(r), a(r))dr+

s

te(s−r)Aσ(r,X(r), a(r))dWQ(r)

(2.14)on [t, T ], P a.e..

Proposition 2.16 Consider equation (2.13) under Hypotheses 1.119 and 2.9.Then Hypotheses 2.11 and 2.12 are satisfied for its mild solutions.

Proof. Hypothesis 2.11 follows from Theorem 1.121. Regarding Hypothesis2.12, condition (A0) follows from the fact that esAζ = esAx P-a.e. for all s ∈ [t, T ]and from (1.40). Condition (A1) follows from Proposition 1.131 and (A2) followsfrom Corollary 1.122.

To show (A3), we will first show that Xν(s) := Xν(s; t, x, a(·)) has a modifica-tion Xν

1 , which is everywhere continuous on and for P a.e. ω0 is F ηω0,s-progressively

measurable on [η, T ]. In general Xν is only F ts -progressively measurable. Let Ω0

be such that P(Ω0) = 1, and for ω ∈ Ω0, Xν(·,ω) is continuous on [t, T ]. Letsk, k ≥ 1, s1 = t, be a countable dense set in [t, T ], and let Ak ∈ F t,0

sk , k ≥ 1be sets such that P(Ak) = 1, and Xν(sk) = ξk on Ak for some F t,0

sk -measurablerandom variable ξk. Set Ω1 = Ω0 ∩

∞k+1 Ak. Then P(Ω1) = 1 and thus for P a.e.

ω0, Pω0(Ω1) = 1, which implies that Ω1 ∈ F ηω0,s for P a.e. ω0. We now define

Xν1 (s) = Xν(s) for s ∈ [t, T ],ω ∈ Ω1 and Xν

1 (s) = 0 for s ∈ [t, T ],ω ∈ Ω \ Ω1. Theprocess Xν has continuous trajectories. Since for ω ∈ Ω1, Xν

1 (s) = limsk→s,sk≤s ξk,Xν

1 is σ(F t,0s ,Ω1)-adapted. However, thanks to Lemma 2.26, F t,0

s ⊂ F ηω0,s for

P a.s. ω0, and so it follows that Xν1 is F η

ω0,s-adapted, which, noticing that ithas continuous trajectories implies by Lemma 1.69 that it is F η

ω0,s-progressivelymeasurable for P a.s. ω0. From now on we will write Xν for Xν

1 .We notice that Xν(·) ∈ Mp

ν (t, T ;H), p > 2, so in particular

EE T

η|Xν(s)|pds|F t,0

η

= E

T

η|Xν(s)|pds

< +∞, (2.15)

so for P a.e. ω0, Xν(·) ∈ Mpνω0

(η, T ;H). Thus by uniqueness of mild solutions givenby Theorem 1.121 we will be done provided we know that, for P a.e. ω0, Xν(·) isa mild solution, in the interval [η, T ] in the reference probability space νω0 .

We have the flow property

Xν(s) = e(s−η)A

ζ +

η

te(η−r)Ab(r,Xν(r), a(r))dr

+

η

te(η−r)Aσ(r,Xν(r), a(r))dW (r)

+

s

ηe(s−r)Ab(r,Xν(r), a(r))dr

+

s

ηe(s−r)Aσ(r,Xν(r), a(r))dW (r) = e(s−η)AXν(η)

+

s

ηe(s−r)Ab(r,Xν(r), a(r))dr +

s

ηe(s−r)Aσ(r,Xν(r), a(r))dW (r)

= Xν(s; η, X(η; t, ζ, a(·)), a(·))

Since P a.s. s

ηe(s−r)Aσ(r,Xν(r), a(r))dW (r) =

s

ηe(s−r)Aσ(r,Xν(r), a(r))dWη(r) (2.16)


on [η, s], and since for every set Ω1 such that P(Ω1) = 1 we have Pω0(Ω1) = 1 forP a.e. ω0, the equality

Xν(s) = e(s−η)AXν(η) +

s

ηe(s−r)Ab(r,Xν(r), a(r))dr

+

s

ηe(s−r)Aσ(r,Xν(r), a(r))dWη(r).

is satisfied Pω0 a.s. for P a.e. ω0. This equality is exactly the same for the mildsolution in the interval [η, T ] in the reference probability space νω0 except for thefact that there the stochastic integral is taken in the reference probability space νω0

instead of ν. So to conclude it is enough to show that for P a.e. ω0

Iν(s) :=

s

ηe(s−r)Aσ(r,Xν(r), a(r))dWη(r)

is Pω0 a.e. equal on [η, T ] to the stochastic integral in the reference probabilityspace νω0 , which we denote by Iνω0

(s).3 By continuity of the paths of the stochasticconvolution (see Proposition 1.107) it is enough to show it for a single s. DenoteΦ(r) = e(s−r)Aσ(r,Xν(r), a(r)). By Lemma 1.93 and the proof of Lemma 1.94there exist a sequence of elementary and F t,0

s -progressively measurable processesΦn with values in L(Ξ;H), such that Φ−ΦnN 2

Q,ν(η,T ;H) → 0, where we indicated

the dependence on the reference probability space in the notation for the norm.By Lemma 2.26, Φn are also F η

ω0,s-progressively measurable. Since Eν [ sη [Φn(r)−

Φ(r)]dWη(r)]2 → 0, passing to a subsequence if necessary, we can assume that s

ηΦn(r)dWη(r) → Iν(s), P a.e., (2.17)

say on a set Ω2, where P(Ω2) = 1 and we can assume that Pω0(Ω2) = 1 for P a.e.ω0.

On the other hand, again by using conditional expectation as in (2.15), weknow that, up to a subsequence, for P a.e. ω0 we have Φ−ΦnN 2

Q,νω0(η,T ;H) → 0.

So, for P a.e. ω0, we have that there exists a subsequence of Φn such that s

ηΦn(r)dWη(r) → Iνω0

(s), Pω0 a.e.. (2.18)

Since Pω0(Ω2) = 1 for P a.e. ω0, (2.17) and (2.18), imply that, for P a.e. ω0,Iν(s) = Iνω0

(s), Pω0 a.e..A different approach to proving (A3) can be found in [427].

Remark 2.17 Another two examples of systems satisfying Hypotheses 2.11 and2.12 are given by the boundary control system described in Section 1.5.1 (Theorem1.135) and by the semilinear system with non-nuclear covariance described in Sec-tion 1.5.2 (Proposition 1.141). We briefly sketch how one can show that Hypotheses2.11 and 2.12 hold in these two cases. However we point out that these cases donot fully conform to our general abstract control problem as additional integra-bility conditions on the controls must be assumed to guarantee the existence anduniqueness of a unique mild solution. Thus the formulation of the control problemmust be slightly adjusted in an obvious way.

Concerning the case of Section 1.5.1, suppose that the assumptions of Theorem1.135 including (1.57) are satisfied. Then Hypothesis 2.11 follows from Theorem1.135. Regarding Hypothesis 2.12, (A0) and (A2) follow from part (i) of Proposition

3Observe that we can compute the integral Iνω0(s) since the control and the trajectory X

ν(·)are F

ηω0,s-progressively measurable on [η, T ] for P a.e. ω0.


1.136, (A1) follows from part (ii) of Proposition 1.136 while for (A3) one can argueas in Proposition 2.16, using (A4) which holds by Remark 2.27.

As regards the case of Section 1.5.2, suppose that the assumptions of Proposi-tions 1.141 and 1.142 are satisfied and a2(·) is as in Remark 1.140. Hypothesis 2.11follows from Propositions 1.141 For Hypothesis 2.12, (A0) and (A2) follow frompart (i) of Proposition 1.142, (A1) follows from part (ii) of Proposition 1.142, whilefor (A3) one can again argue as in Proposition 2.16 using (A4).

2.3. Dynamic Programming Principle: statement and proof

This section is devoted to the formulation and the proof of the Dynamic Pro-gramming Principle. Throughout the whole section we always assume that Hy-pothesis 2.9 is satisfied. We begin with a technical subsection.

2.3.1. Pullback to the canonical reference probability space. Fixt ∈ [0, T ]. The canonical reference probability space is the 5-tuple νW :=(W,F∗,P∗,Bt

s,W), where W := ω ∈ C([t, T ];Ξ) : ω(t) = 0 equipped withthe usual sup-norm, P∗ is the Wiener measure on (W,B(W)) (where B(W) is theBorel σ-field), i.e. the unique probability measure on W that makes the mapping

W : [t, T ]×W → Ξ

W(s,ω) = ω(s)(2.19)

a Q-Wiener process in Ξ (see [300]), F∗ is the completion of B(W), and for s ∈ [t, T ],Bt,0s = σ(W(τ) : t ≤ τ ≤ s), Bt

s = σBt,0s ,N ∗, where N ∗ are the P∗-null sets. W

is a Polish space.It is easy to see that B(W) is generated by the one dimensional cylinder sets

C = ω : ω(s) ∈ A, where s ∈ [t, T ], A is open in Ξ, and that Bt,0T = B(W) (Lemma

2.18). Theorem 1.12 thus guarantee that νW is a standard reference probabilityspace.

The canonical reference probability space on [t,+∞) is defined the same exceptthat now W := ω ∈ C([t,+∞);Ξ) : ω(t) = 0 is equipped with the metric

ρ(w1, w2) =∞

n=1

2−n(w1 − w2C([t,t+n];Ξ) ∧ 1),

which makes it a Polish space.

Lemma 2.18 Let for s ∈ [t, T ] the map ϕs : W → W be defined by ϕs(ω)(τ) =ω(τ ∧ s). Then

Bt,0s = ϕ−1

s (B(W)).

In particular Bt,0T = B(W).

Proof. Observe that for a one dimensional cylinder C = ω : ω(r) ∈ A,where r ∈ [t, T ], A is open in Ξ, we have

ϕ−1s (C) = ω ∈ W : ϕs(ω)(r) ∈ C = ω ∈ W : ω(r ∧ s) ∈ C ∈ Bt,0

s .

Since the cylinder sets C generate B(W), we thus obtain ϕ−1s (B(W)) ⊂ Bt,0

s .For the opposite inclusion, since Bt,0

s is generated by sets of the form B =W−1(r, ·) (V ), where r ∈ [t, s], V is open in Ξ, we have

B = ω ∈ W : ω(r) ∈ V = ω ∈ W : ω(r ∧ s) ∈ V = ϕ−1s (ω ∈ W : ω(r) ∈ V ).

Thus Bt,0s ⊂ ϕ−1

s (B(W)).

2.3. DYNAMIC PROGRAMMING PRINCIPLE: STATEMENT AND PROOF 95

Lemma 2.19 Let (Ω,F ,F ts ,P,W ) be a reference probability space (i.e. it satisfies

Definition 2.7), and let the paths of the Q-Wiener process W (·,ω) be continuousfor every ω ∈ Ω. Then, for s ∈ [t, T ],

Ft,0s = W (· ∧ s)−1(B(W)).

Proof. The proof is similar to that of Lemma 2.18.

We denote by PW[t,T ] to be the sigma field of Bt,0

s -predictable sets, i.e. the sigmafield generated by the sets of the form (s, r] × A, t ≤ s < r ≤ T,A ∈ Bt,0

s andt × A,A ∈ Bt,0

t . For a reference probability space (Ω,F ,F ts ,P,WQ) we denote

by PΩ[t,T ] to be the sigma field of F t,0

s -predictable sets.We will use the following simple representation lemma from [427].

Lemma 2.20 Let a(·) = (Ω,F ,F ts ,P,W, a (·)) ∈ Ut (defined by (2.9)) be F t,0

s -predictable, and let the paths of the Q-Wiener process W (·,ω) be continuous forevery ω ∈ Ω. Then there exists a PW

[t,T ]/B(Λ)-measurable function f : [t, T ]×W →Λ such that

a(s,ω) = f(s,W (·,ω)), for ω ∈ Ω, s ∈ [t, T ]. (2.20)

Proof. Define the process

β : [t, T ]× Ω → [t, T ]×W

β(τ,ω) = (τ,W (·,ω)).

The sets of the form A1 = (s, r]× ω ∈ Ω : W (η,ω) ∈ B, t ≤ η ≤ s < r ≤ T,B ∈B(Ξ), and A2 = t × ω ∈ Ω : W (t,ω) ∈ B, B ∈ B(Ξ), generate PΩ

[t,T ]. But(τ,ω) ∈ A1 if and only if τ ∈ (s, r] and W (·,ω) ∈ B1 = ξ ∈ W : ξ(η) ∈ B ∈ Bt,0

s ,and (t,ω) ∈ A2 if and only if W (·,ω) ∈ B2 = ξ ∈ W : ξ(t) ∈ B ∈ Bt,0

t .Therefore, A1 = β−1((s, r] × B1), A2 = β−1(t × B2). Since the sets of the form(s, r] × ξ ∈ W : ξ(η) ∈ B, t ≤ η ≤ s < r ≤ T,B ∈ B(Ξ), and t × ξ ∈ W :ξ(t) ∈ B, B ∈ B(Ξ), generate PW

[t,T ], we have PΩ[t,T ] = β−1(PW

[t,T ]). Therefore, byTheorem 1.9, there exists a PW

[t,T ]/B(Λ)-measurable function f : [t, T ] × W → Λ

such that (2.20) is satisfied.

Corollary 2.21 Let a(·) = (Ω,F ,F ts ,P,W, a (·)) ∈ Ut be F t,0

s -predictable. LetΩ1,F1,F t

1,s,P1,W1

be another reference probability space. Suppose that W and

W1 have everywhere continuous trajectories. Let f : [t, T ]×W → Λ be the functionfrom Lemma 2.20 satisfying (2.20). Then the process

a (s,ω1) = f (s,W1 (·,ω1)) (2.21)

is Ft,01,s -predictable and hence F

t,01,s -progressively measurable on [t, T ]× Ω1, and

LP(a(·),W (·)) = LP1(a(·),W1(·)). (2.22)

2.3.2. Independence of reference probability spaces. To prove the Dy-namic Programming Principle we have formulated our optimal control problem in aspecial weak form in which we only use reference probability spaces. Here we showthat the control problem does not depend on the choice of a reference probabilityspace ν and thus the strong and weak formulations are equivalent.

We will formulate the result only for the case T < +∞ however the reader caneasily modify the assumptions so that the result also holds for T = +∞.

Theorem 2.22 (Independence of the reference probability space) Let T ∈(0,+∞). Let Hypotheses 2.9, 2.11, 2.12 be satisfied. Let the functions l : [0, T ] ×H × Λ → R, g : H → R, c : H → R be Borel measurable, c be bounded from


below, and let, for every 0 ≤ t < T, x ∈ H, every reference probability space ν =(Ω,F ,F t

s ,P,W ) and a(·) ∈ Uνt ,

l(·, X(·; t, x, a(·)), a(·)) ∈ M1ν (t, T ;R), g(X(T ; t, x, a(·))) ∈ L1(Ω,F ,P),

Then, for every 0 ≤ t < T, x ∈ H, every two reference probability spaces ν1 =Ω1,F1,F t

1,s,P1,W1

, ν2 =

Ω2,F2,F t

2,s,P2,W2

, and a(·) ∈ Uν1

t , there existsa2(·) ∈ Uν2

t such that

LP1(Xν1(·; t, x, a(·)), a(·)) = LP2(X

ν2(·; t, x, a2(·)), a2(·)).In particular, for every reference probability space ν,

V νt (x) = V (t, x).

Proof. Let a(·) ∈ Uν1t . Let a1(·) be the F t,0

s -predictable processesfrom Lemma 1.94 such that a1(·) = a(·), P1 ⊗ dt a.e.. Let a1(·) ∈ Uν2

t bethe process from Corollary 2.21. (Without loss of generality we can assumethat W1,W2 have everywhere continuous trajectories.) Since LP1(a1(·),W1(·)) =LP2(a1(·),W2(·)), it follows from (A1), (A2), and Theorem 1.128 that, denotingXν1(·) = Xν1(·; t, x, a1(·)), Xν2(·) = Xν2(·; t, x, a1(·)),

f1(s) = e−

s

ηc(Xν1 (τ))dτ , f2(s) = e−

s

ηc(Xν2 (τ))dτ ,

we haveLP1(f1(·), Xν1(·), a(·)) = LP2(f2(·), Xν2(·), a1(·)). (2.23)

This shows the first claim.Using (2.23) we thus obtain

Jν1 (t, x; a(·)) = Jν1 (t, x; a1(·)) = Jν2 (t, x; a1(·)) ,which implies

infa(·)∈Uν1

t

Jν1 (t, x; a(·)) ≥ infa(·)∈Uν2

t

Jν2 (t, x; a(·)) .

The opposite inequality is obtained in the same way and thus it follows that

infa(·)∈Uν1

t

Jν1 (t, x; a(·)) = infa(·)∈Uν2

t

Jν2 (t, x; a(·)) . (2.24)

This completes the proof.

2.3.3. The proof of the abstract principle of optimality. We now stateand prove the Dynamic Programming Principle (DPP) in an abstract formulation.We will do it only for the finite horizon problem. However the same proof appliesto the infinite horizon case if Hypothesis 2.23 is slightly changed to accommodatefor T = +∞. A special infinite horizon case is discussed in more details in Section2.4.

Hypothesis 2.23 Let T ∈ (0,+∞). The functions l : [0, T ] × H × Λ → R,g : H → R, c : H → R are Borel measurable, and c is bounded from below. Forevery 0 ≤ t ≤ η < T, x ∈ H, every reference probability space ν = (Ω,F ,F t

s ,P,W )and a(·) ∈ Uν

t

l(·, X(·; t, x, a(·)), a(·)) ∈ M1ν (t, T ;R), g(X(T ; t, x, a(·))) ∈ L1(Ω,F ,P),

V (η, X(η; t, x, a(·))) ∈ L1(Ω,F ,P).Moreover the functional J (t, y; a(·)) is uniformly continuous in the variable y onbounded sets of H, uniformly for a(·) ∈ Ut.

Hypothesis 2.23 in particular ensures that the value function V is finite.


Theorem 2.24 (Dynamic Programming Principle) Assume that Hypotheses2.9, 2.11, 2.12, and 2.23 are satisfied. Let 0 ≤ t < η < T, x ∈ H. Then

V (t, x) = infa(·)∈Ut

E η

te−

s

tc(X(τ))dτ l (s,X (s) , a (s)) ds+ e−

η

tc(X(τ))dτV (η, X (η))

.

(2.25)

The proof is very technical so we will proceed slowly. We begin with two simplelemmas.

Lemma 2.25 Let 0 ≤ t ≤ η < T . Let (Ω,F ,F ts ,P,W ) be a standard reference

probability space, and let W have everywhere continuous trajectories. Define, fors ∈ [η, T ], Wη(s) := W (s) − W (η). Then for P a.e. ω0 ∈ Ω, Wη is a Q-Wienerprocess on

Ω,Fω0 ,F

ηω0,s,Pω0

, where Pω0 = P(·|F t,0

η )(ω0) is the regular condi-tional probability, Fω0 is the augmentation of F (see Definition 2.8) by the Pω0

null sets, and F ηω0,s is the augmented filtration generated by Wη.

Proof. We notice that for η ≤ s ≤ T

Fη,0ω0,s = σ (Wη(τ) : η ≤ τ ≤ s) ⊂ F

t,0s

(observe that Fη,0ω0,t1 is independent of ω0) and, by Lemma 2.26-(i), for P a.e. ω0,

F t,0s ⊂ F η

ω0,s for every η ≤ s ≤ T . Thus for P a.e. ω0, F ηω0,s is the augmentation

of F t,0s by the Pω0 null sets for every η ≤ s ≤ T .We fix η ≤ t1 < t2, y ∈ Ξ. We want to apply Lemma 1.88 (with Ξ1 = Ξ and

Q1 = Q) so we need to compute for P a.s ω0,

h := Eω0

eiy,Wη(t2)−Wη(t1)Ξ |F η

ω0,t1

= Eω0

eiy,Wη(t2)−Wη(t1)Ξ |F t,0

t1

Pω0 a.s.

Thus we can assume that h is Ft,0t1 -measurable. We have

Ah(ω)dPω0(ω) =

Aeiy,Wη(t2)−Wη(t1)Ξ(ω)dPω0(ω) ∀A ∈ F

t,0t1 ,

which (by the definition of Pω0) means that for P a.s ω0

Ah(ω)dPω0(ω) = E

eiy,Wη(t2)−Wη(t1)Ξ1A|F t,0

η

(ω0)

= E1AE

eiy,W (t2)−W (t1)Ξ |F t

t1

|F t,0

η

(ω0)

= Ee−

Qy,yΞ2 (t2−t1)1A|F t,0

η

(ω0) = e−

Qy,yΞ2 (t2−t1)Pω0(A). (2.26)

Therefore, since Ft,0t1 is countably generated, it follows that h = e−

Qy,yΞ2 (t2−t1) for

P a.s. ω0. Thus by the separability of Ξ, for P a.s. ω0 we have h = e−Qy,yΞ

2 (t2−t1)

for all y ∈ Ξ. Consider now all pairs (tk1 , tk2), k = 1, 2, ..., where tk1 = η or tk1 is

rational, tk2 is rational and η ≤ tk1 < tk2 ≤ T . We can conclude from the above thatthere is a set Ω0 such that P(Ω0) = 1 and such that for every ω0 ∈ Ω0, y ∈ Ξ andk = 1, 2, ...

Eω0

eiy,Wη(t

k

2 )−Wη(tk

1 )Ξ |F t,0tk1

= e−

Qy,yΞ2 (tk2−tk1 ). (2.27)

It remains to prove that if ω0 ∈ Ω0, y ∈ Ξ and η ≤ t1 < t2 ≤ T , then

Eω0


t1

= e−

Qy,yΞ2 (t2−t1). (2.28)

So let ω0 ∈ Ω0, y ∈ Ξ and η ≤ t1 < t2 ≤ T . We will assume that t1 = η andt1, t2 are not rational since in such cases the argument is similar and easier. Then


for some subsequence of our sequence of pairs, which we will still denote by (tk1 , tk2),

we have tk1 → t1, tk2 → t2 and tk1 < t1, tk2 < t2. We claim that Pω0 a.s.

limk→+∞

Eω0

eiy,Wη(t

k

2 )−Wη(tk

1 )Ξ |F t,0tk1

= Eω0


t1

(2.29)

which, together with (2.27), will establish (2.28). First we notice that, since thefiltration F t,0

s is left continuous, by Proposition 1.40-(viii)

limk→+∞

Eω0


tk1

= Eω0


t1

Pω0 a.s.

Then we observe that by Proposition 1.40-(vi)

Eω0

Eω0

eiy,Wη(t

k

2 )−Wη(tk

1 )Ξ |F t,0tk1

− Eω0


tk1

≤√2Eω0

eiy,Wη(tk

2 )−Wη(tk

1 )Ξ − eiy,Wη(t2)−Wη(t1)Ξ → 0 as k → +∞.

Thus for some subsequence, Pω0 a.s.

limk→+∞

Eω0

eiy,Wη(t

k

2 )−Wη(tk

1 )Ξ |F t,0tk1

− Eω0


tk1

= 0.

These two convergences show (2.29).

The reader can consult [427] for a different argument to prove Lemma 2.25.

Lemma 2.26 Let 0 ≤ t ≤ η < T . Let ν = (Ω,F ,F ts ,P,W ) be a standard

reference probability space and let W have everywhere continuous trajectories. Leta(·) ∈ Uν

t , and let a1(·) be from Lemma 1.94. Then(i) For P a.e. ω0 ∈ Ω, F t,0

s ⊂ F ηω0,s for every η ≤ s ≤ T .

(ii) For P a.e. ω0 ∈ Ω, aω0(·) :=Ω,Fω0 ,F

ηω0,s,Pω0 ,Wη, a1|[η,T ](·)

∈ Uη.

Proof. To show Part (i), we take a countable generating family Ak ofB(Ξ) and a countable dense subset sm in [t, T ]. We will show that for a.e.ω0 ∈ Ω, W (sm)−1(Ak) ∈ F η

ω0,s for all k ≥ 1, sm ≤ s. If sm ≤ η, since F t,0η is

countably generated, we obtain by Theorem 1.44 that W (sm)(ω) = W (sm)(ω0),Pω0 a.e., for P a.e. ω0 ∈ Ω. Thus, up to a set of Pω0 measure 0, W (sm)−1(Ak) iseither empty or is equal to Ω and so it is in F η

ω0,s. If sm > η then again up to a setof Pω0 measure 0, W (sm)−1(Ak) = Wη(sm)−1(Ak −Wη(ω0)) and so it is in F η

ω0,s

for P a.e. ω0 ∈ Ω. This implies that σW (sm) : sm ≤ s ⊂ F ηω0,s for P a.e. ω0 ∈ Ω.

It remains to notice that F t,0s = σW (sm) : sm ≤ s.

Part (ii): In view of Lemma 2.25 it is enough to show that for P a.e. ω0 ∈ Ω,a1(·) is F η

ω0,s-progressively measurable on [η, T ]. This fact follows from Part (i).

Remark 2.27 If a(·) in Lemma 2.26 is such that a(·) ∈ Mpν (t, T ;E), p ≥ 1, for

some Banach space E, it is easy to see that we also have a1|[η,T ](·) ∈ Mpνω0

(η, T ;E),for P a.e. ω0, where νω0 =

Ω,Fω0 ,F

ηω0,s,Pω0 ,Wη

is as in Lemma 2.26.

Proof of Theorem 2.24. We first do the following reduction. Denote(similarly to (2.9))

Ut =

ν

Uνt : ν is a standard reference probability space

.

The set Ut is nonempty since for instance the canonical reference probability spaceνW is in it. It is clear from Theorem 2.22 (and the same argument used to justify


(2.23) there) that (2.25) will follow if we can prove it with Ut replaced by Ut.Therefore it remains to show that


E η

te−

s

tc(X(τ))dτ l (s,X (s) , a (s)) ds

+ e−

η


. (2.30)

(In fact, using Theorem 2.22 it would be enough to replace Ut by UνW

t and doeverything on canonical reference probability spaces, however we will do the proofin the more general setup since the arguments and technicalities are similar.) Weremind that we assume that all Q-Wiener processes in the reference probabilityspaces have everywhere continuous trajectories.

Part 1. (inequality ≥ in (2.30)): Let a(·) ∈ Ut. Denoting X(s) =X(s; t, x, a(·))) we have

J (t, x; a(·)) = E η

te−

s


+E T

ηe−

s


T

tc(X(τ))dτg (X (T ))

.(2.31)

Let a1(·) be from Lemma 1.94, and for P a.e. ω0 ∈ Ω, aω0(·) be the control fromLemma 2.26. Let Pω0 = P

·|F t,0

η

(ω0) and Eω0 be the expectation with respect to

Pω0 .Let X1(s) = X(s; t, x, a1(·)). By (A2), X1 and X are indistinguishable.

Thus we obtain by (A3) that (up to an indistinguishable modification) X1(s) =X(s; η, X(η), aω0(·)) in

Ω,Fω0 ,F

ηω0,s,Pω0 ,Wη

for P a.s. ω0.

Therefore, using this, (A0) and the fact that for P a.s. ω0, Pω0(ω : X1(η,ω) =X1(η,ω0)) = 1, we have

E T

ηe−

s


T


= E T

ηe−

s

tc(X1(τ))dτ l (s,X1 (s) , a1 (s)) ds+ e−

T

tc(X1(τ))dτg (X1 (T ))

= Ee−

η

tc(X(τ))dτE

T

ηe−

s

ηc(X1(τ))dτ l (s,X1 (s) , a1 (s)) ds

+ e−

T

ηc(X1(τ))dτg (X1 (T )) |F t

η

= Ee−

η

tc(X(τ))dτJ (η, X1(η,ω0); a

ω0(·))≥ E

e−

η

tc(X(τ))dτV (η, X1(η,ω0))

= Ee−

η

tc(X(τ))dτV (η, X(η))

, (2.32)

where we used the remarks at the end of Section 2.2.2. Thus, using (2.31), weobtain

J (t, x; a(·)) ≥ E η

te−

s


η


and the claim follows by taking the infimum over all a(·) ∈ Ut.Part 2. (inequality ≤ in (2.30)): Let t ≤ η ≤ T . We fix a(·) =

(Ω,F ,F ts ,P,W, a (·)) ∈ Ut.


We can choose δ1 > 0 be such that, for |x − x| < δ1 and |x|, |x| ≤ 1 we have,for each a(·) ∈ Uη

|J (η, x; a(·))− J (η, x; a(·))|+ |V (η, x)− V (η, x)| <

Since H is separable we can choose a partitionDR

j

j∈N of BH(0, 1) into countable

disjoint Borel subsets with diam(D1j ) < δ1. Similarly we can choose a (possibly

smaller) δ2 > 0 such that, for |x − x| < δ2 and |x|, |x| ≤ 2 we have, for eacha(·) ∈ Uη

|J (η, x; a(·))− J (η, x; a(·))|+ |V (η, x)− V (η, x)| <

and we can choose a partitionD2

j

j∈N of BH(0, 2)\BH(0, 1) into countable disjoint

Borel subsets with diam(D2j ) < δ2.

Iterating the argument we can find a partition Djj∈N of H into countabledisjoint Borel subsets with the following property: for all Dj and all x, x ∈ Dj wehave, for each a(·) ∈ Uη,

|J (η, x; a(·))− J (η, x; a(·))|+ |V (η, x)− V (η, x)| <

For each j ∈ N we choose xj ∈ Dj and aj(·) =Ωj ,Fj ,F

ηj,s,Pj ,Wj , aj(·)

∈

Uνj

η such thatJ (η, xj ; aj(·)) < V (η, xj) + . (2.33)

We define a new control aη(·) ∈ Ut on the probability space (Ω,F ,F ts ,P,W )

as follows. Let aj,1(·) be the Fη,0j,s -predictable processes from Lemma 1.94 such

that aj,1(·) = aj(·), Pj ⊗ dt a.e.. Let fj : [η, T ]×C ([η, T ] ;Ξ) → Λ be the functionsfrom Lemma 2.20 such that

fj (s,Wj(·,ω)) = aj,1 (s,ω) , for ω ∈ Ωj , s ∈ [η, T ].

We now set aj (s,ω) = fj (s,Wη (·,ω)). By Corollary 2.21 and Lemma 2.26 the pro-cess aj(·) is F t,0

s -progressively measurable and, for P-a.e. ω0, is F ηω0,s-progressively

measurable in the reference probability spaces νω0 :=Ω,Fω0 ,F

ηω0,sPω0 ,Wη

de-

fined in Lemma 2.25. Moreover LPω0(aj(·),Wη(·)) = LPj

(aj,1(·),Wj(·)). We define

aη (s,ω) = a (s,ω)1t≤s<η + 1s≥η

j∈Naj (s,ω)1X(η;t,x,a(·))∈Dj. (2.34)

Obviously (Ω,F ,F ts ,P,W, aη(·)) ∈ Ut.

Let X(s) = X(s; t, x, aη(·)). Notice that X(s; t, x, aη(·)) = X(s; t, x, a(·)) on[t, η], P a.e..

Denote Oj := ω : X(η; t, x, a(·)) ∈ Dj. Since for P-a.s. ω0, Pω0(ω :X(η,ω) = X(η,ω0)) = 1, if ω0 ∈ Oj , then Pω0(Ω \Oj) = 0, which implies that inthis case aj(·) = aη(·) on [η, T ], Pω0 a.s., and thus, for P a.s. ω0, aη|[η,T ] ∈ Uνω0

η ,and

LPω0(aη(·),Wη(·)) = LPj

(aj,1(·),Wj(·)), j ∈ N. (2.35)Moreover we can assume, by (A3), that for P a.e. ω0, X(s) = Xνω0 (·; η, X(η), aη(·))on [η, T ] Pω0 a.s..

By the definition of V ,

V (t, x) ≤ E T

te−

s

tc(X(τ))dτ l (s,X (s) , aη (s)) ds+ e−

T


= E η

te−

s


+E T

ηe−

s


T


. (2.36)


We have

E T

ηe−

s


T


= Ee−

η

tc(X(τ))dτE

T

ηe−

s

ηc(X(τ))dτ l (s,X (s) , aη (s)) ds

+e−

T

ηc(X(τ))dτg (X (T )) |F t

η

=

j∈N

Oj

e−

η

tc(X(τ))dτEω0

T

ηe−

s

ηc(X(τ))dτ l (s,X (s) , aη (s)) ds

+e−

T

ηc(X(τ))dτg (X (T ))

dP(ω0)

By (2.35) and (A0)− (A1) we obtain

LPω0(X(·), aη(·)) = LPj

(Xνj (·), aj,1(·)), j ∈ N,

where Xνj (s) = Xνj (s; η, X(η; t, x, a(·))(ω0), aj,1(·)). Thus, it follows from Theo-rem 1.128 that, denoting

f(s) = e−

s

ηc(X(τ))dτ , fj(s) = e−

s

ηc(Xνj (τ))dτ ,

LPω0(f(·), X(·), aη(·)) = LPj

(fj(·), Xνj (·), aj,1(·)), j ∈ N.Therefore,

E T

ηe−

s


T


=

j∈N

Oj

e−

η

tc(X(τ))dτJPω0

(η, X(η; t, x, a(·))(ω0); aη(·)) dP(ω0)

=

j∈N

Oj

e−

η

tc(X(τ))dτJPj

(η, X(η; t, x, a(·))(ω0); aj,1(·)) dP(ω0).

Moreover, using (2.33), we get for a.s. ω0 ∈ Oj

JPj(η, X (η; t, x, a(·)) (ω0) ; aj(·)) ≤ JPj

(η, xj ; aj(·)) +

≤ V (η, xj) + 2 ≤ V (η, X (η; t, x, a(·)) (ω0)) + 3,

so we finally obtain

E T

ηe−

s


T


≤ Ee−

η

tc(X(τ))dτV (η, X (η; t, x, a(·)))

+ C.

Therefore, by (2.36) and the arbitrariness of a(·),

V (t, x) ≤ infa(·)∈Ut

E η

te−

s


+e−

η


+ C

and the claim follows by letting → 0.


If we know more information about the value function and the control problem,in particular that the value function is continuous in both variables, the dynamicprogramming principle can be strengthened to include stopping times. We do notdo it here in the abstract case. We explain how to obtain such formulation of thedynamic programming principle for a control problem for mild solutions in Section3.6, Theorem 3.70.

2.4. Infinite horizon problems

In this section we consider a special infinite horizon problem described by anevolution equation

dX(s) = β(X(s), a(s))ds+ σ(X(s), a(s))dWQ(s)

X(t) = x,(2.37)

with a cost functional of the form

J (t, x; a(·)) = E +∞

te−

s

tc(X(τ ;t,x,a(·)))dτ l(X (s; t, x, a(·)) , a(s))ds

, (2.38)

where c ≥ λ > 0. We are really only interested in the case t = 0 but we will keepthe dependence on t for a while.

This is a very important class of problems which are semi-“autonomous” in asense that the coefficients β,σ and the cost l do not depend explicitly on time.In this case the value function does not depend on time and the DPP takes on asimpler form.

We define the value function for t ≥ 0 as


E +∞

te−

s


(2.39)

and setJ (x; a(·)) := J (0, x; a(·)) , V (x) := V (0, x). (2.40)

We assume now that Hypotheses 2.9, 2.11 and 2.12 are satisfied with T = +∞(i.e. reference probability spaces and solutions are defined on [t,+∞)). We alsoreplace Hypothesis 2.23 by the following one.

Hypothesis 2.28 The functions l : H×Λ → R, c : H → R are Borel measurableand there exists λ > 0 such that c(x) ≥ λ for every x ∈ H. Moreover for every0 ≤ η < +∞, x ∈ H, reference probability space ν, a(·) ∈ Uν

0

e−

·

0 c(X(·;0,x,a(·)))dτ l(X(·; 0, x, a(·)), a(·)) ∈ M1ν (0,+∞;R),

V (X(η; 0, x, a(·))) ∈ L1(Ω,F ,P).Finally J (·; a(·)) is uniformly continuous on bounded sets of H, uniformly for a(·) ∈U0.

Since we are dealing with an abstract state equation we have to add anotherhypothesis which reflects the “autonomous” nature of the system. First we noticethat

Ω,F ,F

ts

s≥t

,P,WQ(·), a (·)∈ Ut

⇐⇒Ω,F ,

F

ts+t

s≥0

,P,WQ(t+ ·), a (t+ ·)∈ U0.

Hypothesis 2.29 Assume that the family of the solutions X(·; t, x, a(·)) of(2.37) satisfies the following property. For every (Ω,F ,F t

s ,P,WQ(·), a (·)) ∈ Ut

(A5) LP(X(t+·; t, x, a(·)), a(t+·)) = LP(X(·; 0, x, a(t+·)), a(t+·)) on [0,+∞),where X(·; 0, x, a(t+ ·)) is the solution of (2.37) with WQ(·) replaced byWQ(t+ ·).

2.5. HJB EQUATION AND OPTIMAL SYNTHESIS IN THE SMOOTH CASE 103

Remark 2.30 An example of a state equation satisfying Hypothesis 2.29 is givenby the mild solution of an SDE

dX(s) = AX(s)ds+ b(X(s), a(s))ds+ σ(X(s), a(s))dWQ(s)

X(t) = x,(2.41)

where A, b and σ satisfy the assumptions described in Hypothesis 1.119.

Using Hypothesis 2.29, by a change of variable and (A5), we observe that


E +∞

te−

s


= infa(·)∈Ut

E +∞

0e−

s

0 c(X(t+τ ;t,x,a(·)))dτ l(X (t+ s; t, x, a(·)) , a(t+ s))ds

= infa(·)∈Ut

E +∞

0e−

s

0 c(X(τ ;0,x,a(t+·)))dτ l(X (s; 0, x, a(t+ ·)) , a(t+ s))ds

= infa(·)∈U0

E +∞

0e−

s

0 c(X(τ ;0,x,a(·)))dτ l(X (s; 0, x, a(·)) a(s))ds= V (x).

We thus have the following theorem whose proof is obtained by a simple mod-ification of the proofs of Theorems 2.22 and 2.24.

Theorem 2.31 (DPP, infinite horizon case) Assume that Hypotheses 2.9, 2.11and 2.12 for T = +∞ hold, and that Hypotheses 2.28 and 2.29 are satisfied. Thenthe value function V satisfies the Dynamic Programming Principle: For every η >0, x ∈ H,

V (x) = infa(·)∈U0

E η

0e−

s

0 c(X(τ))dτ l (X (s) , a (s)) ds+ e−

η

0 c(X(τ))dτV (X (η))

.

(2.42)Moreover

V (x) = V ν(x) := infa(·)∈Uν

0

E +∞

0e−

s

0 c(X(τ))dτ l (X (s) , a (s)) ds

for every reference probability space ν.

2.5. HJB equation and optimal synthesis in the smooth case

Once we know that the dynamic programming principle (DPP) holds, we wantto use it to solve the control problem i.e. to find optimal couples and, possibly, tostudy their properties. In the dynamic programming approach the path to do thisconsists in:

• Writing a differential form of the DPP (the HJB equation);• Finding a solution v of the HJB equation (which we do not know ex

ante to be the value function);• Using such solution v to prove a verification theorem i.e. sufficient,

and possibly necessary, conditions for optimality which express optimalcontrols as functions of the current state (feedback controls);

• Performing the optimal synthesis, i.e. using the optimality conditionsof the previous step to find optimal feedback controls: this will alsoimply that v is indeed the value function.

Such a program can be performed if we know in advance that the HJB equationhas a smooth solution or if we know that the value function is sufficiently regular,both of which may not be true even in finite dimensions. However it is still useful


to present how the program works in the smooth case to understand the machineryof the dynamic programming approach. We do it for our model problem, when thestate equation admits a solution in the mild sense, assuming that the value functionis smooth. We prove the following three results (in both finite and infinite horizoncases):

• The value function solves the HJB equation.• The verification theorem (necessary and sufficient conditions for opti-

mality).• The existence of optimal couples in feedback form.

One of the main goals of the theory presented in this book is to obtain some ofthese results under more realistic assumptions.

It is important to note that, if one finds a sufficiently smooth solution of theHJB equation, then the verification theorem and the existence of optimal feedbackscan be done without using the DPP and this is done in Chapters 4-6.

We will present everything for a control problem in the weak formulation ofSection 2.1.2, i.e. when the set of admissible controls is equal to U t, as this setupis more convenient when discussing optimal feedback controls. However the sameresults are also true for control problems in the weak formulation of Section 2.2used to prove the DPP, with U t replaced by Ut, or in the strong formulation ofSection 2.1.1.

2.5.1. Finite horizon problem: Parabolic HJB equation. Let Hypothe-sis 2.1 hold. Consider an optimal control problem of minimizing the cost functional(2.3) for the system governed by (2.13), where for simplicity we do not have dis-counting in (2.3), i.e. we set c = 0. We rewrite it here for the reader’s convenience.The state equation is

dX(s) = (AX(s) + b(s,X(s), aµ(s))) ds+ σ(s,X(s), aµ(s))dWQ(s)

gX(t) = x,(2.43)

where aµ(·) ∈ Uµt for some generalized reference probability space µ satisfying

Hypothesis 2.1, and the cost functional

J(t, x; aµ(·)) = Eµ

T

tl(s,X(s; t, x; aµ(·)), aµ(s))ds+g(X(T ; t, x, aµ(·)))

. (2.44)

We consider the control problem in the weak formulation of Section 2.1.2, andassume that Hypothesis 1.119 is satisfied. The HJB equation associated with thisproblem is

vt + Dv,Ax+ infa∈Λ

12Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗D2v

+ Dv, b(t, x, a)+ l(t, x, a)= 0,

v(T, x) = g(x).

(2.45)

In the above equation Dv,D2v are the Fréchet derivatives of v with respect to x,which are identified respectively with elements of H and S(H), the set of bounded,self-adjoint operators in the Hilbert space H. For (t, x, p, S, a) ∈ [0, T ]×H ×H ×S(H)× Λ, the term

FCV (t, x, p, S, a) :=1

2Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗S+p, b(t, x, a)+l(t, x, a)

(2.46)


will be called the current value Hamiltonian of the system and its infimum overa ∈ Λ

F (t, x, p, S) := infa∈Λ

1

2Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗S

+ p, b(t, x, a)+ l(t, x, a)

(2.47)

will be called the Hamiltonian.4 Using this notation, the HJB equation (2.45) canbe rewritten as

vt + Dv,Ax+ F (t, x,Dv,D2v) = 0,

v(T, x) = g(x).(2.48)

The HJB equation (2.45) can be viewed as a differential form of the DPP.

Definition 2.32 (Classical solution, parabolic case) A function v : (0, T ] ×H → R is a classical solution of (2.45) if v ∈ C1,2((0, T ) × H) ∩ C((0, T ] × H),Dv : (0, T )×H → D(A∗), A∗Dv ∈ C((0, T )×H;H) and v satisfies

vt + A∗Dv, x+ infa∈Λ

12Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗D2v

+ Dv, b(t, x, a)+ l(t, x, a)= 0, (t, x) ∈ (0, T )×H,

v(T, x) = g(x), x ∈ H.

pointwise.

We will use the following assumption.

Hypothesis 2.33(i) The functions σ(t, x, a), b(t, x, a) and l(t, x, a) are uniformly continuous

in t on [0, T ], uniformly for (x, a) ∈ B(0, R)× Λ for every R > 0.(ii) There exist C,N > 0 such that

|l(t, x, a)| ≤ C(1 + |x|)N (2.49)

for all (t, x, a) ∈ [0, T ]×H × Λ.(iii) The function v : [0, T ] × H → R is uniformly continuous on bounded

subsets of [0, T ] × H, and its derivatives Dv, D2v, vt are uniformlycontinuous on bounded subsets of (0, T ) × H. Moreover Dv : (0, T ) ×H → D(A∗) and A∗Dv is uniformly continuous on bounded subsets of(0, T )×H. Finally there exist C,N > 0 such that

|v(t, x)|+ |Dv(t, x)|+ |vt(t, x)|+ D2v(t, x)+ |A∗Dv(t, x)| ≤ C(1 + |x|)N (2.50)

for all (t, x) ∈ (0, T )×H.

Theorem 2.34 Let Hypotheses 1.119, 2.1 and 2.33 be satisfied, v(T, x) = g(x)for every x ∈ H, and let the function v satisfy the DPP, i.e. for every 0 < t < η <T, x ∈ H,

v(t, x) = infa(·)∈Ut

E η

tl(s,X(s), a(s))ds+ v(η, X(η))

. (2.51)

Then v is a classical solution of (2.45).

Proof. To prove that equation (2.45) is satisfied we show separately thetwo inequalities. We will not present all the details here as the proof follows thelines of the proof of Theorem 3.66 where it is showed that the value function is aviscosity solution applying Dynkin’s formula to a suitable family of test functions.

4Sometimes it is called the minimum value Hamiltonian.


Part 1. (Supersolution inequality). We fix (t, x) ∈ (0, T ) × H. By (2.51), forevery ∈ (0, T − t) we can choose a control aµ(·) ∈ Uµ

t , such that,

v(t, x) + 2 ≥ Eµ

t+

tl(s,Xµ(s), aµ(s))ds+ v(t+ , Xµ(t+ ))

,

where Xµ(·) is the trajectory starting at (t, x) driven by aµ(·). Dividing the aboveby we have

≥ Eµv(t+ , Xµ(t+ ))− vµ(t, x)

+

1

Eµ

t+

tl(s,Xµ(s), aµ(s))ds

and, using Dynkin’s formula from Proposition 1.156,

≥ 1

Eµ

t+

t

vt(s,X

µ(s)) + A∗Dv(s,Xµ(s)), Xµ(s)

+ Dv(s,Xµ(s)), b(s,Xµ(s), aµ(s))

+1

2Tr

σ(s,Xµ(s), aµ(s))Q1/2

σ(s,Xµ(s), aµ(s))Q1/2

∗D2v(s,Xµ(s))

+ l (s,Xµ (s) , aµ(s))

ds

=

1

Eµ

t+

tΨ(s,Xµ(s), aµ(s))ds, (2.52)

where

Ψ(s, y, a) := vt(s, y) + A∗Dv(s, y), y+ Dv(s, y), b(s, y, a)

+1

2Tr

σ(s, y, a)Q1/2

σ(s, y, a)Q1/2

∗D2v(s, y)

+ l (s, y, a) .

By our assumptions we have, for some h > 0 and modulus ρ, depending on t, x,

|Ψ(s, y, a)−Ψ(t, x, a)| ≤ ρ(|s− t|+ |y− x|) for all (s, y) ∈ [t, t+ h]×B1(x), a ∈ Λ(2.53)

and, for some C and M ≥ 0,

|Ψ(s,Xµ (s) , aµ(s))| ≤ C(1 + |Xµ(s)|M ). (2.54)

Moreover it follows from (1.39) that there is r > 0, independent of aµ , such thatr → 0 as → 0, and denoting

Ω1 = ω ∈ Ωµ : sup

s∈[t,t+]|Xµ(s)− x| ≤ r,

thenPµ(Ω

1) ≥ γ() → 1 as → 0. (2.55)Thus, using (1.38), (2.52), (2.53), (2.54), and (2.55) we obtain (see the proof ofTheorem 3.66 for more details) that there exists a modulus ρ1(), depending on tand x, such that

ρ1() ≥1

Eµ

t+

tΨ(t, x, aµ(s))ds

≥ vt(t, x) + A∗Dv(t, x), x+ 1

Eµ

t+

tinfa∈Λ

1

2Tr

σ(t, x, a)Q1/2

×σ(t, x, a)Q1/2

∗D2v(t, x)

+ Dv(t, x), b(t, x, a)+ l (t, x, a)

ds

= vt(t, x) + A∗Dv(t, x), x+ infa∈Λ

1

2Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗D2v(t, x)

+ Dv(t, x), b(t, x, a)+ l (t, x, a)

. (2.56)


The inequality follows letting → 0.

Part 2. (Subsolution inequality). Choose a ∈ Λ and consider the constantcontrol a(·) ≡ a ∈ Λ for some generalized reference probability space µ. Denote byX (s) the trajectory starting from (t, x) driven by the control a(·). From (2.51) wehave for ∈ (0, T − t)

v(t, x) ≤ Eµ

t+

tl (s,X (s) , a) ds+ v (t+ , X (t+ ))

.

Using again Dynkin’s formula from Proposition 1.156, we thus obtain

0 ≤ Eµ[v(t+ , X(t+ ))− v(t, x)]

+

1

Eµ

t+

tl (s,X (s) , a) ds

=1

Eµ

t+h

t

vt(s,X(s)) + A∗Dv(s,X(s)), X(s)

+ Dv(s,X(s)), b(s,X(s), a)

+1

2Tr

σ(s,X(s), a)Q1/2

σ(s,X(s), a)Q1/2

∗D2v(s,X(s))

+ l (s,X (s) , a)

ds

. (2.57)

We can now pass to the limit as → 0 above like in Part 1 to obtain that for everya ∈ Λ

0 ≤ vt(t, x) + A∗Dv(t, x), x+ 1

2Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗D2v(t, x)

+ Dv(t, x), b(t, x, a)+ l(t, x, a), t ∈ (0, T ), x ∈ H. (2.58)

The inequality follows by taking the infimum over a ∈ Λ above.

We now show how to use the HJB equation to characterize optimal controls.First we prove the so-called verification theorem.

Theorem 2.35 (Smooth Verification Theorem, Sufficient Condition) Letv : [0, T ] × H → R be a classical solution of (2.45) as defined in Definition 2.32.Let Hypotheses 1.119, 2.1 and 2.33-(ii)(iii) be satisfied. Then:

(i) We havev(t, x) ≤ V (t, x) for all (t, x) ∈ [0, T ]×H. (2.59)

(ii) Let (a∗(·), X∗(·)) be an admissible pair at (t, x) such that

a∗(s) ∈ argmina∈Λ

FCV (s,X∗(s), Dv(s,X∗(s)), D2v(s,X∗(s)), a),

(2.60)for almost every s ∈ [t, T ] and P-almost surely. Then the pair(a∗(·), X∗(·)) is optimal at (t, x), and v(t, x) = V (t, x).

Proof. We prove first the following identity5: for every a(·) ∈ U t

v(t, x) = J(t, x; a(·))

− E T

t

FCV

r,X(r), Dv(r,X(r)), D2v(r,X(r)), a(r)

− Fr,X(r), Dv(r,X(r)), D2v(r,X(r))

dr. (2.61)

5This is often called the fundamental identity for the optimal control problem.


Indeed, consider a(·) ∈ U t and the corresponding trajectory X(·) starting at x attime t. We apply Proposition 1.156, to the process v(s,X(s)), s ∈ [t, T ] obtaining

Ev(T,X(T )) = v(t, x) + E T

tvt(r,X(r))dr

+ E T

tA∗Dv(r,X(r)), X(r) dr + E

T

tDv(r,X(r)), b(r,X(r), a(r)) dr

+1

2E T

tTr

σ(r,X(r), a(r))Q1/2

σ(r,X(r), a(r))Q1/2

∗D2v(r,X(r))

dr.

(2.62)

We now use that v(T, ·) = g, rearrange the terms, and we add and subtractE Tt l(r,X(r), a(r))dr obtaining, by the definition of the current value Hamiltonian

FCV in (2.46),

v(t, x) = Eg(X(T )) + E T

tl(r,X(r), a(r))dr

− E T

tvt(r,X(r)) + A∗Dv(r,X(r)), X(r) dr

− E T

tFCV (r,X(r), Dv(r,X(r)), D2v(r,X(r)), a(r))dr. (2.63)

Equality (2.61) is now a consequence of the definition of the functional J and thefact that v is a classical solution of the HJB equation (2.45).

Therefore (i) follows by observing that, by definition, FCV −F ≥ 0 everywhere,and by taking the infimum over a(·) ∈ U t in the right hand side of (2.61).

Regarding (ii), let (a∗(·), X∗(·)) be an admissible pair at (t, x) satisfying (2.60)for almost every s ∈ [t, T ] and P-almost surely. We then have

E T

t

FCV

r,X(r), Dv(r,X(r)), D2v(r,X(r)), a(r)

− Fr,X(r), Dv(r,X(r)), D2v(r,X(r))

dr = 0. (2.64)

Thus, by (2.61), we getv(t, x) = J(t, x, a∗(·)), (2.65)

which, together with (i), implies that (a∗(·), X∗(·) is optimal at (t, x) and v(t, x) =V (t, x).

Note that part (i) of above theorem remains true if v is any classical subsolutionof the HJB equation (2.45)6 with the required regularity.

If we know from the beginning that the solution v in Theorem 2.35 is the valuefunction V then (2.60) becomes also a necessary condition for optimality.

Corollary 2.36 (Smooth Verification Theorem, Necessary Condition) Letthe assumptions of Theorem 2.35 hold for v = V . Let (a∗(·), X∗(·)) be an optimalpair at (t, x). Then we must have


FCV (s,X∗(s), DV (s,X∗(s)), D2V (s,X∗(s)), a), (2.66)

for almost every s ∈ [t, T ] and P-almost surely.

6in the sense that v(T, x) ≤ g(x) and it satisfies (2.45) with the inequality ≥.


Proof. Now the function v = V satisfies (2.61). Since (a∗(·), X∗(·)) is anoptimal pair at (t, x), we have V (t, x) = J(t, x; a∗(·)). Therefore, (2.61) for Vimplies that the integrand of the last term of (2.61) is zero dt ⊗ P-a.e. and theclaim follows.

Assume now that we have a classical solution v of the HJB equation (2.45).Define the multivalued function

Φ : (0, T )×H → P(Λ)

Φ : (t, x) → argmina∈Λ FCV (t, x,Dv(t, x), D2v(t, x), a).(2.67)

The Closed Loop Equation (CLE) associated with our problem and v is then formallydefined as

dX(s) ∈ AX(s)dt+ b(s,X(s),Φ(s,X(s)))ds+ σ(s,X(s),Φ(s,X(s)))dWQ(s)

X(t) = x.(2.68)

If we can find a solution (in a suitable sense) XΦ(·) of such stochastic differentialinclusion, we expect that, if aΦ(·) is a suitable measurable selection of Φ(·, X(·)),then the couple (aΦ(·), XΦ(·)) is optimal at (t, x). This is indeed the statement ofthe next corollary.

Corollary 2.37 (Optimal Feedback Controls) Let the assumptions of Theo-rem 2.35 hold. Assume moreover that the feedback map Φ defined in (2.67) admitsa measurable selection φ : (0, T )×H → Λ such that the Closed Loop Equation

dX(s) = AX(s)ds+ b(s,X(s),φ(s,X(s)))ds+ σ(s,X(s),φ(s,X(s)))dWQ(s)

X(t) = x,(2.69)

has a mild weak solution (see Definition 1.115) Xφ(·) in some generalized referenceprobability space µ satisfying Hypothesis 2.1-(iv). Then the couple (aφ(·), Xφ(·)),where the control aφ(·) is defined by the feedback law aφ(s) = φ(s,Xφ(s)), is admis-sible and it is optimal at (t, x).

Proof. By construction the couple (aφ(·), Xφ(·)) satisfies (2.60). Then, byTheorem 2.35-(ii) we obtain that such couple is optimal. Observe that since theassumptions of Theorem 2.35 are satisfied, Xφ(·) is the unique mild solution (in thestrong probabilistic sense) of the state equation associated to the control aφ(·) ingeneralized reference probability space µ.

In the above corollary we assumed that the closed loop equation has a weak mildsolution to obtain existence of an optimal feedback control in the weak formulation.If we consider the control problem (2.43)-(2.44) in the strong formulation with thevalue function defined by (2.5), all the results above remain true except for Corollary2.37. Indeed the weak mild solution Xφ(·), and so the control aφ(·) may be definedin a different reference probability space than the starting one. To get existence ofan optimal feedback control in the strong formulation one needs to have existenceof solutions of the closed loop equation (2.69) in the strong probabilistic sense (i.e.in the mild or strong sense of Definitions 1.112 and 1.113).

To conclude let us reiterate the three step process to do the so-called synthesisof optimal control for the problem (2.43)-(2.44) once we have a classical solution vof our HJB equation.

(1) Define the function Φ as in (2.67) .(2) Look for a solution X∗(·) of the closed loop equation (2.68).(3) Define a∗(s) := φ(s,X∗(s)). It is optimal thanks to Theorem 2.35.


Of course, given an optimal control problem like (2.43)-(2.44), it may not be possibleto perform the above steps as they are. However, even if the HJB equation does nothave a classical solution, we may still be able to synthesize optimal controls. Thiswill be explained in later chapters for some special cases. The general synthesis ofoptimal controls is still a largely open problem.

As it was explained in Remark 2.6, the extended weak formulation may bemore suitable for Corollary 2.37 (and also Corollary 2.43 in the infinite horizoncase). This is done in Chapter 6 (see also Chapters 4 and 5).

2.5.2. Infinite horizon problem: Elliptic HJB equation. Consider theoptimal control problem of minimizing the infinite horizon functional (2.38) for thesystem governed by the state equation (2.41) with t = 0 and the set of admissiblecontrols equal to U0. For simplicity we will assume that c(·) ≡ λ > 0. This is atypical infinite horizon problem with constant discounting.

The current value Hamiltonian is now defined by

FCV (x, p, S, a) :=1

2Tr

σ(x, a)Q

12

σ(x, a)Q

12

∗S+p, b(x, a)+l(x, a), (2.70)

the Hamiltonian is given by

F (x, p, S) := infa∈Λ

1

2Tr

σ(x, a)Q

12

σ(x, a)Q

12

∗S+ p, b(x, a)+ l(x, a)

,

(2.71)and the HJB equation associated to our infinite horizon optimal control problem is

λv − Dv,Ax − F (x,Dv,D2v) = 0 (2.72)

for the unknown function v : H → R.We present here the infinite horizon versions of the results of the previous

subsection, adapted to the infinite horizon case.

Definition 2.38 (Classical solution, elliptic case) A function v : H → R is aclassical solution of (2.72) if v ∈ C2(H), A∗Dv ∈ C(H;H) and v satisfies

λv − A∗Dv, x − F (x,Dv,D2v) = 0.

pointwise.

Similarly to the previous section we will need the following assumption.

Hypothesis 2.39(i) There exist C,N > 0 such that

|l(x, a)| ≤ C(1 + |x|)N (2.73)

for all (x, a) ∈ H × Λ.(ii) The function v : H → R and its derivatives Dv, D2v, vt are uniformly

continuous on bounded subsets of H. Moreover Dv : H → D(A∗) andA∗Dv are uniformly continuous on bounded subsets of H.

|v(x)|+ |Dv(x)|+ D2v(x)+ |A∗Dv(x)| ≤ C(1 + |x|)N (2.74)

for all x ∈ H.

Theorem 2.40 Let Hypotheses 1.119, 2.1 for T = +∞, and 2.39 be satisfied.Assume that the function v satisfy the DPP, i.e. for every 0 < η < +∞, x ∈ H,

v(x) = infa(·)∈U0

E η

0e−λsl(s,X(s), a(s))ds+ e−ληv(X(η))

. (2.75)

Then, the function v is a classical solution of (2.72).

Proof. The proof follows the lines of the proof of Theorem 2.34.


We now pass to the verification theorem, the necessary conditions and theclosed loop equation. In these results we may encounter integrability problems. Toavoid technical complications, here we consider the case where the discount factorλ is sufficiently big.

Theorem 2.41 (Smooth Verification, Sufficient Condition, Infinite Horizon)Let v : H → R be a classical solution of (2.72) as defined in Definition 2.38.

Let Hypotheses 1.119, 2.1 for T = +∞, and 2.39 be satisfied, and let λ > λ =2(N + 2)(C + (N + 1)C2), where C is the constant from (1.34) and (1.35) (seeProposition 3.24 for m = N + 2). Then:

(i) For all x ∈ H

v(x) ≤ V (x) for all x ∈ H. (2.76)

(ii) Let (a∗(·), X∗(·)) be an admissible pair at x such that


FCV (X∗(s), Dv(s,X∗(s)), D2v(s,X∗(s)), a), (2.77)

for almost every s ∈ [0,+∞) and P-almost surely. Then the pair(a∗(·), X∗(·)) is optimal at x, and v(x) = V (x).

Proof. The proof is similar to that of Theorem 2.35 except for the fact thatwe now have to take the limit as T → +∞ , in (2.61). Indeed, arguing as in theproof of Theorem 2.35, we obtain that for every a(·) ∈ U0, and every T > 0,

v(x) = e−λTEv(X(T )) +

T

0e−λrl(X(r), a(r))dr

− E T

0e−λr

FCV

X(r), Dv(r,X(r)), D2v(r,X(r)); a(r)

− FX(r), Dv(r,X(r)), D2v(r,X(r))

dr. (2.78)

The condition λ > λ guarantees, due to estimate (3.34), that we can pass to thelimit as T → +∞ above, obtaining the fundamental identity:

v(x) =

+∞

0e−λrl(X(r), a(r))dr

− E +∞

0e−λr

FCV

X(r), Dv(r,X(r)), D2v(r,X(r)); a(r)

− FX(r), Dv(r,X(r)), D2v(r,X(r))

dr. (2.79)

The claims now follow as in the proof of Theorem 2.35. Corollary 2.42 (Smooth Verification, Necessary Cond., Infinite Horizon)Let the assumptions of Theorem 2.41 hold for v = V . Let (a∗(·), X∗(·)) be an

optimal pair at x. Then we must have


FCV (X∗(s), DV (s,X∗(s)), D2V (s,X∗(s)), a), (2.80)

for almost every s ∈ [0,+∞) and P-almost surely.

Proof. The same as Corollary 2.36 using (2.79). As in the finite horizon case we assume that we have a classical solution v of

the HJB equation (2.72). We define the multivalued function

Φ : H → P(Λ)

Φ : x → argmina∈Λ FCV (x,Dv(t, x), D2v(t, x); a)(2.81)


The Closed Loop Equation (CLE) associated with our problem and v is then formallydefined as

dX(s) ∈ AX(s)dt+ b(X(s),Φ(X(s)))ds+ σ(X(s),Φ(X(s)))dWQ(s)

X(0) = x.(2.82)

Again, if a solution XΦ(·) of this stochastic differential inclusion can be found, andwe can find aΦ(·), a suitable measurable selection of Φ(·, X(·)), we would expectthe couple (aΦ(·), XΦ(·)) to be optimal at x.

Corollary 2.43 (Optimal Feedback Controls, Infinite Horizon) Let theassumptions of Theorem 2.41 hold. Assume moreover that the feedback map Φdefined in (2.81) admits a measurable selection φ : H → Λ such that the ClosedLoop Equation

dX(s) = AX(s)dt+ b(X(s),φ(X(s)))ds+ σ(X(s),φ(X(s)))dW (s)

X(0) = x,(2.83)

has a mild weak solution (see Definition 1.115) Xφ(·) in some generalized refer-ence probability space satisfying Hypothesis 2.1-(iv). Then the couple (aφ(·), Xφ(·)),where the control aφ(·) is defined by the feedback law aφ(s) = φ(s,Xφ(s)), is admis-sible and it is optimal at (t, x).

Proof. The proof is the same as that of Corollary 2.37. The optimal synthesis is performed in the same way as in the finite horizon

case.

Remark 2.44 In Section 2.4 and in this subsection we have considered infinitehorizon problems satisfying Hypothesis 2.29 which substantially means that thedata b, σ, l and c must be time independent. We did this restriction partly forsimplicity of exposition and partly because such cases are very common in appliedmodels. However with little effort, it is possible to apply the dynamic programmingapproach to infinite horizon problems with “non-autonomous" data (so withoutHypothesis 2.29). In such cases the value function would be a function of (t, x),the DPP would have the form (2.25), and the HJB equation would be a parabolicequation on (0,+∞)×H like (2.48) but with a zeroth order term coming from thediscount factor, and without a terminal condition:

vt − λv + Dv,Ax+ F (t, x,Dv,D2v) = 0. (2.84)Such problems are more difficult but are still interesting e.g. in some financialapplications (see [107, 174, 218]).

2.6. Some motivating examples

In this section we describe several examples that motivate the study of sto-chastic optimal control problems in infinite dimensions. Our goal here is to showhow various applied problems, arising in different areas of science and engineering,are naturally modeled within the framework of the infinite dimensional stochasticanalysis. The first five examples are concerned with the control of various kindsof stochastic PDE, while the last deals with the control of stochastic delay equa-tions. Despite similarities, these examples are very different from each other andit is difficult to find a general theory that includes all of them. This is an unpleas-ant feature of the infinite dimensional optimal control which will force us to applydifferent approaches and adaptations of the main general theory.

We have chosen our examples, among many others, since they are representativeof interesting applied models and since they motivate the four different approaches

2.6. SOME MOTIVATING EXAMPLES 113

described in this book (viscosity solutions, strong solutions, L2µ solutions, solutions

via BSDE). In all examples the criteria to maximize/minimize are given as the ex-pectation of a Bolza-type functionals (in finite or infinite horizon cases). We leaveaside other types of criteria (mean-variance, risk sensitive, ergodic, etc) for whichwe will refer the reader to the existing literature. For each example we provide ashort motivation, the main mathematical framework (state equation, objective func-tional and constraints), and show how to translate the problem into the abstractframework of infinite dimensional stochastic optimal control introduced before. Fi-nally we discuss the issue of the Dynamic Programming Principle, and we presentthe associated HJB equations with references to further material in the book andin the literature. For purposes of the DPP we always take the optimal controlproblems in the weak formulation of Section 2.2, however the control problems canbe studied with other formulations.

We remark that the existing theory is far from providing a satisfactory treat-ment of all problems: many challenging questions remain open and call for furtherresearch. An important issue in this respect is the one of constraints: to obtainmore realistic control problems it is often necessary to impose suitable constraintson the state and on the control variables. Such constraints strongly depend on theparticular problems. Since the addition of state constraints makes the dynamicprogramming approach much harder and not much is known at the present stage,we will mention how state constraints arise in specific problems, however we willnot deal with state constraint problems.

Finally we mention a few things about the notation.

• To be consistent with the general setting introduced before, the initialtime is always t ≥ 0.

• In all examples we start first with the finite dimensional notation. Thusthe state equation is first written as a PDE (or a functional equation) infinite dimensions with solutions defined informally, then in a subsequentsection the state equation is rewritten as an evolution equation in aninfinite dimensional space. To distinguish the two cases we write yto denote the finite dimensional state and α for the finite dimensionalcontrol, while the infinite dimensional state and control will be denoted,as before, by X and a, respectively.

• Following the usual convention, even if all variables depend on the “sce-nario” ω, we will always drop such dependence unless needed in thecontext.

WARNING: The HJB equations that appear in the examples in this sectionare formal and are all written using the convention adapted from the natural waythe equation was written in (2.45)-(2.47) in Section 2.5.1. This form is preferablefrom the PDE point of view as all the terms (when they are defined) use only thereference Hilbert space H of the independent variable x. We want to focus onthe examples and leave the details of how the equations are interpreted and solvedto later chapters. In some cases the Hamiltonians appearing here are always welldefined and will not need any interpretation. In some cases some terms may notmake sense the way they are written here and they will need special interpretationwhich would take too long to explain here. This is especially true of equationsdiscussed in Sections 2.6.2 and 2.6.3. The formal versions are enough to point outthe main difficulties posed by the equations, however the reader should be carefulas the equations may not be what they appear.


2.6.1. Stochastic controlled heat equation: distributed control. Ourfirst example concerns the problem of controlling a nonlinear stochastic heat equa-tion in a given space region O ⊂ RN . This is a very popular example and here,under “reasonable” assumptions, the theory applies quite well giving rise to resultsabout the HJB equation and the synthesis of optimal controls.

The optimal control problems of this type arise in various applied contexts. Werecall some of them.

• Optimal control of the heat distribution of a given body (the region O).The deterministic case is described for instance in the monograph [312](pages 3-5). The presence of the stochastic additive term in the equationcan be justified by the presence of (small) random perturbations in thesystem. (It may be of interest to see what happens when such termgoes to zero, see e.g. [63]).

• Optimal control of stochastic reaction diffusion equations, where thewhite noise term describes the internal fluctuation of the system due toits many-particle nature (see e.g. page 8 of [130], and [10] for the modelwithout control and [79, 81] for the model with control).

• Optimal control of the motion of an elastic string in a random viscousenvironment (see e.g. page 4 of [130] and [214] for the model withoutcontrol).

• Optimal control of the stochastic cable equation (arising also in neuro-sciences, see page 9 of [130] and [444] for the stochastic model, and e.g[53] for the optimal control problem in the deterministic case).

• Optimal advertising problems arising in economics (see [331]).

2.6.1.1. Setting of the problem. We are given an open, connected and boundedset O ⊂ RN with C1 boundary ∂O ⊂ RN and a reference probability space(Ω,F , (Fs)s∈[t,T ],P,WQ). We consider a controlled dynamical system driven bythe following stochastic PDE in the time interval [t, T ], for 0 ≤ t ≤ T < +∞

dy(s, ξ) = [∆ξy(s, ξ) + f(y(s, ξ)) + α(s, ξ)] ds+ dWQ(s)(ξ), s ∈ (t, T ], ξ ∈ O

y(s, ξ) = 0 (s, ξ) ∈ (t, T ]× ∂O

y(t, ξ) = x(ξ) ∈ L2(O), ξ ∈ O,(2.85)

where:

• the function y : [t, T ]×O×Ω → R, (s, ξ,ω) → y(s, ξ,ω) is a stochasticprocess that describes e.g. the evolution of the temperature distributionand is the state trajectory of the system;

• the function α : [t, T ]×O×Ω → R, (s, ξ,ω) → α(s, ξ,ω) is a stochasticprocess giving e.g. the dynamics of the external source of heat actingat every interior point of O and is the control strategy of the system.

We will omit the variable ω writing simply y(s, ξ) and α(s, ξ). Moreover:

• ∆ξ is the Laplace operator. We consider the Dirichlet boundary condi-tions, however the problem can be studied similarly with the Neumannboundary conditions. Conditions of mixed type are also possible, seeon this e.g. Chapters 3 and 5 of [323];

• f ∈ C0(R) is a nonlinear function of the state (which may represent a“reaction” term);


• WQ is a Q-Wiener process with Q ∈ L+(L2(O)) and (Fs)s∈[t,T ] is theaugmented filtration generated by WQ

7 (see Remark 2.10 if Tr(Q) =+∞);

• x(·) ∈ L2(O) is the initial state (e.g. temperature distribution) in theregion O.

The solution8 of (2.85) will be denoted by yα,t,x to underline the dependence ofthe state y on the control α and on the initial data t, x. Having in mind the controlof the temperature distribution, a reasonable objective of the controller here canbe the one of getting such distribution yα,t,x to be close to a required one y (foreach time s ∈ [t, T ] or only at the final time T ) while spending the fewest amountof energy doing this. In such case a reasonable cost functional may be of the form(for suitable constants c0, c1, c2 ∈ R)

I1(t, x;α) = E T

t

O

c0|yα,t,x(s, ξ)− y(s, ξ)|2 + c1|α(s, ξ)|2

dξds

+

Oc2|yα,t,x(T, ξ)− y(T, ξ)|2dξ

, (2.86)

and the objective would be to minimize the functional I1 above over all controlstrategies α, progressively measurable with respect to the filtration generated byWQ, and satisfying suitable constraints and integrability conditions (e.g. such thatthe state equation and above integrals make sense). More generally one couldconsider a cost functional

I2(t, x;α) = E T

t

Oβ(yα,t,x(s, ξ),α(s, ξ))dξds+

Oγ(yα,t,x(T, ξ))dξ

(2.87)

where β ∈ C0(R2) an γ ∈ C0(R) are given functions depending on the objective ofthe controller.

Finally, the constraints: If the state is the absolute temperature it is naturalto require the positivity of yα,t,x; moreover it is reasonable to assume bounds onthe control strategies depending on the physical device used to control the system(e.g. α(s, ξ) ∈ [m,M ] for given m < M). The constraints depend on a particularproblem.

2.6.1.2. The infinite dimensional setting and the HJB equation. Take H =Ξ = L2(O) and let Λ be a closed, bounded subset of L2(O). For instance,if α(s, ξ) ∈ [m,M ] for every (s, ξ) ∈ [t, T ] × O then we would take Λ =f ∈ L2(O) : f(ξ) ∈ [m,M ], ∀ξ ∈ O

. Consider the Laplace operator with Dirich-

let boundary conditions defined as (see e.g. [432], Section 5.2 page 180):

D(A) = H2(O) ∩H10 (O),

Ax = ∆x, for x ∈ D(A)(2.88)

that generates an analytic semigroup of compact operators etAt≥0. Moreoverdefine the Nemytskii operator b : H → H as

b(x)(ξ) = f(x(ξ)). (2.89)

7Indeed also stochastic PDE’s with more general types of noise can be treated, see e.g. [380]but this is beyond the scope of this book.

8For the concept of solution and the assumptions on the data f,Q and on the control strategyα that guarantee the existence and uniqueness of it see next section.


Defining X(s) := y(s, ·) ∈ L2(O) and a(s) := α(s, ·) ∈ Λ, the state equation (2.85)can be rewritten as an SDE in H as follows

dX(s) = [AX(s) + b(X(s)) + a(s)] ds+ dWQ(s)

X(t) = x ∈ H.(2.90)

From Proposition 1.141 we know that, when f is Lipschitz9 (and so is b) andQr :=

r0 eτAQeτA

∗

dτ is trace class for all r > 0, the above equation admits aunique mild solution denoted by X(s; t, x, a) (or simply X(s) when no confusionis possible). If also (1.67) holds, then such solution has continuous trajectories.10If Tr(Q) < +∞ then, thanks to Proposition 2.16, Hypotheses 2.11 and 2.12 aresatisfied. If Tr(Q) = +∞ the claim is still true as outlined in Remark 2.17.

Defining l : H × Λ → R

l(x, a) =

Oβ(x(ξ), a(ξ))dξ,

and g : H → R as

g(x) =

Oγ(x(ξ))dξ,

the functional I2 of (2.87) can be rewritten in the Hilbert space setting as

J2(t, x; a(·)) = E T

tl(X(s), a(s))ds+ g(x(T ))

. (2.91)

Suppose that β and γ satisfy the right conditions so that Hypothesis 2.23 holds.This is done for instance in Section 3.6, Propositions 3.61 and 3.62. Then, all theassumptions of Theorem 2.24 are satisfied and hence the dynamic programmingprinciple holds.

The Hamilton-Jacobi-Bellman equation associated with problem (2.90)-(2.91)is the following:

vt +1

2Tr [QD2v] + Ax+ b(x), Dv+ inf

a∈Λa,Dv+ l(x, a) = 0,

v(T, x) = g(x).

(2.92)

This problem falls into the classes studied for instance in [286, 421, 422]11 by theviscosity solution approach, in [23, 63, 64, 79, 136, 231, 232, 333, 335, 339]12 by thestrong solutions approach, in [94, 223] by the L2 approach, in [211, 337]13 by theBSDE approach. The theory of such HJB equations is described in Chapters 3-6.We also refer to Section 4.8.3 for a specific example and to Section 4.10.1 wherean explicit solution of the HJB equation (2.92) is found in the case of a quadraticHamiltonian.

9In the case studied in [79, 81], b is not Lipschitz, see on this Section 4.910Such assumptions are true e.g. when N = 1 and Q is the identity, or when N = 2 and

Q = A−α for some α > 0.

11These papers treat the fully nonlinear case.12[79] deals with nonlipschitz b, [333, 335, 339] also treat the case of multiplicative noise, and

[335] treats a Banach space case.13These papers also treat the case of multiplicative noise and [337] considers it in a Banach

space.


2.6.1.3. The infinite horizon case. For the infinite horizon case we again rewritethe state equation (2.85) as (2.90), starting at time 0 at the point x ∈ H. The costfunctional

I3(x; a) = E +∞

0e−ρs

Oβ(yα,0,x(s, ξ),α(s, ξ))dξds

(2.93)

is then expressed as

J3(x; a) = E +∞

0e−ρsl(X(s; 0, x, a(·)), a(s))ds

. (2.94)

Hypotheses 2.11 and 2.12 are satisfied as in the finite horizon part. Hypothesis 2.29holds thanks to Remark 2.30. Hypothesis 2.28 holds if β satisfies proper conditions.This is discussed in Section 3.6, Propositions 3.73 and 3.74. The Hamilton-Jacobi-Bellman equation associated with the problem is now

ρv − 1

2Tr [QD2v]− Ax+ b(x), Dv − inf

a∈Λa,Dv+ l(x, a) = 0. (2.95)

As regards the literature, we refer for to [286, 421, 422] for the viscosity solutionapproach in the fully nonlinear case and with multiplicative noise, to [81, 241, 334]14for the strong solutions approach, to [212] for the BSDE approach in the case ofmultiplicative noise. In [226], an ergodic control problem is studied, using theresults for the infinite horizon problem.

2.6.2. Stochastic controlled heat equation: boundary control. Thesecond example is also concerned with the control of a nonlinear stochastic heatequation in a given space region O but perhaps in a more realistic case, when thecontrol can be exercised only at the boundary of O or in a subset of O. We presentonly the case of the control at the boundary, remarking that the case of the controlon a subdomain of O (that may even reduce to a point) gives rise to very similarmathematical difficulties that are treated e.g. in [206, 336]. We consider two casesthat are the most standard and commonly used: the first when the control at theboundary enters through the Dirichlet boundary condition, and the second whenone controls the flow, i.e. the Neumann boundary condition.

2.6.2.1. Setting of the problem: Dirichlet case. As in the previous example,assume O to be an open, connected, bounded subset of RN with smooth boundary∂O. We consider the controlled dynamical system driven by the following stochasticPDE on the time interval [t, T ], for 0 ≤ t ≤ T < +∞,

dy(s, ξ) = [∆ξy(s, ξ) + f (y(s, ξ))] + dWQ(s)(ξ) in (t, T ]×O

y(t, ξ) = x(ξ) on O

y(s, ξ) = α(s, ξ) on (t, T ]× ∂O,

(2.96)

where ∆ξ, f , WQ, x, y are as in equation (2.85). The difference with respect toequation (2.85) is that here the control is no longer in the drift term of the stateequation but it influences the system through its values at the boundary (the so-called Dirichlet boundary condition). So here the control strategy of the system isthe function α : [t, T ]× ∂O ×Ω → R which may be interpreted as the dynamics ofan external source of heat acting at every boundary point of O.

Following the notation of Section 2.6.1.1, we denote the unique solution (when-ever it exists, see next subsection for more precise setting) of (2.96) by yα,t,x to

14The paper [334] also treats the case of a multiplicative noise.


underline the dependence of the state y, on the control α and on the initial datat, x.

Similarly to the distributed control case, a reasonable objective of the controllercan be the one of minimizing a functional

I(t, x;α) = E T

t

Oβ1(y

α,t,x(s, ξ))dξ+

∂Oβ2(α(s, ξ))dξds+

Oγ(yα,t,x(T, ξ))dξ

(2.97)where β1,β2, γ ∈ C0(R) are given functions depending on the objective of thecontroller. Observe that the difference with respect to the cost functional I2 in(2.87) is that here we take the integral on ∂O when the control α is involved.The goal of the controller here would be to minimize the functional I above, overall control strategies α which are progressively measurable with respect to theaugmented filtration generated by WQ, and such that the above integrals makesense. The constraints can be the same as in the distributed control case in Section2.6.1.1.

2.6.2.2. Setting of the problem: Neumann case. In this case the boundary con-dition in (2.96) is replaced by

∂y(s, ξ)

∂n= α(s, ξ) on (t, T ]× ∂O, (2.98)

where n is the outward unit normal vector to ∂O. This means that one controlsthe heat flow across the boundary. The goal again is to minimize a cost functionalof type (2.97) over all admissible controls α.

2.6.2.3. The infinite dimensional setting and the HJB equation. To rewrite thestate equation (2.96) (with either Dirichlet or Neumann boundary condition) in aninfinite dimensional setting we take H = L2(O) and Λ to be a closed subset ofL2(∂O) depending on the control constraints.

We first consider the Dirichlet case. Let A be the operator defined in (2.88)and b : H → H be the Nemytskii operator defined in (2.89). Let D be the Dirichletoperator defined in (C.2). We denote, as before, X(s) := y(s, ·) ∈ L2(O), anda(s) := α(s, ·) ∈ Λ. We assume in addition that Λ is bounded in L2(∂O). Then, asexplained in Appendix C, Section C.2 (Notation C.14), the state equation (2.96)can be formally rewritten as

dX(s) = [AX(s) + b(X(s))−ADa(s)] ds+ dWQ(s), t < s ≤ T

X(t) = x, x ∈ H.(2.99)

Now, thanks to (C.5), if we write the term −AD as (−A)βB for β ∈ (3/4, 1), whereB := A1−βD, the operator B is bounded from L2(∂O) to H. Thus, passing to theintegral form (see again Notation C.14), we can write (2.99) as follows

X(s) = e(s−t)Ax+

s

te(s−r)Ab(X(r))dr +

s

t(−A)βe(s−r)ABa(r)dr

+

s

te(s−r)AdWQ(s). (2.100)

The Neumann boundary control case is handled similarly. Here we take Λ =L2(∂O) and Uν

t = M2ν (t, T ;L

2(∂O)). Let A be the Laplace operator with Neumannboundary conditions (see e.g. [432], Section 5.2, page 180):

D(A) =

x ∈ H2(O) : ∂x

∂n = 0,

Ax = ∆x, for x ∈ D(A).(2.101)


It generates an analytic semigroup of compact operators etAt≥0 in H. We con-sider, for fixed λ > 0 the Neumann operator Nλ defined in (C.7). Similarly to theDirichlet boundary control case, as explained in Appendix C, Section C.3 (Nota-tion C.17), the state equation can be formally expressed as an evolution equationas follows:

dX(s) = [AX(s) + b(X(s)) + (λI −A)Nλa(s)] ds+ dWQ(s), t < s ≤ T

X(t) = x, x ∈ H.(2.102)

Now, thanks to (C.10), if we write the term (λI − A)Nλ as (λI − A)βBλ, forβ ∈ (1/4, 1/2) and Bλ := (λI−A)1−βNλ, the operator Bλ is bounded from L2(∂O)to H. Then, passing to the integral form (see again Notation C.17), we can rewrite(2.102) as

X(s) = e(s−t)Ax+

s

te(s−r)Ab(X(r))dr

+

s

t(λI −A)βe(s−r)ABλa(r)dr +

s

te(s−r)AdWQ(s). (2.103)

If f (and thus b) is Lipschitz15 and (1.67) holds (if Tr(Q) = +∞) both integralequations (2.100) and (2.103) have unique mild solutions (see Theorem 1.135 andProposition 1.141) with continuous trajectories, which we denote by X(s ; t, x, a(·))(or simply X(s) if its meaning is clear).

Thus it follows from the discussion in Remark 2.17 that Hypotheses 2.11 and2.12, and condition (A4) in the Neumann case, needed for the dynamic program-ming principle hold for both problems.

We now define l1 : H → R by

l1(x) =

Oβ1(x(ξ))dξ,

l2 : Λ → R by

l2(a) =

∂Oβ2(a(ξ))dξ,

and g : H → R by

g(x) =

Oγ(x(ξ))dξ.

The functional I in (2.97) can be rewritten in the Hilbert space setting as

J(t, x; a(·)) = E T

t[l1(X(s)) + l2(a(s))] ds+ g(x(T ))

. (2.104)

Thus, if β1,β2, γ satisfy proper continuity and growth conditions that guaranteeHypothesis 2.23, then the hypotheses of Theorem 2.24 are satisfied, and thus thedynamic programming principle stated there holds.

The associated HJB equation in both cases can be written as

vt +1

2Tr [QD2v] + Ax+ b(x), Dv+ inf

a∈Λ

(λI −A)βBλa,Dv

+ l2(a)

+ l1(x) = 0,

v(T, x) = g(x),

(2.105)

15More general assumptions on f could be used, like e.g. in [79, 81] in the distributed controlcase.


where in the Dirichlet case we take λ = 0 and β ∈ (3/4, 1), while in the Neumanncase we take λ > 0 and β ∈ (1/4, 1/2).

Observe that the term(λI −A)βBλa,Dv

caused by the presence of the

boundary control term in the state equation does not make sense in general.However if his term is interpreted as

Bλa, (λI −A)βDv

then (writing l(x, a) =

l1(x) + l2(a)), the Hamiltonian F0 defined by

F0(x, p) = infa∈Λ

Bλa, (λI −A)βp

+ l(x, a)

(2.106)

is well defined on H × D(λI −A)β

. Such unboundedness of F0 is difficult to

treat and typically requires better regularity properties of the solution, e.g. thatthe Dv(t, x) belongs to the narrower space D

(λI −A)β

. Since D

(λI −A)β

is

larger in the Neumann case, the regularity needed for the value function to solvethe HJB equation in the Neumann case is weaker than in the Dirichlet case. Thusthe Neumann case can be studied under weaker assumptions and/or with betterresults. In the framework of viscosity solutions, the unboundedness of F0 mayrequire additional conditions on test functions. Overall this problem is much moredifficult to study than the one of the previous section.

The theory of viscosity solutions has been developed for such equations in[242].16 It is presented in Section 3.12, where existence and uniqueness of viscositysolutions for stationary HJB equations is proved in large generality, covering Dirich-let boundary conditions and very general drift and diffusion coefficients allowing forpossibly fully nonlinear Hamiltonians. However no results about feedback controlsexist with this approach. A Cauchy problem was also studied in [439] using thetechniques of [242]. A related finite horizon problem has been studied partly with aviscosity solution approach in [451] in a case with boundary control and boundarynoise: uniqueness of solutions is not proved.

Regarding the strong solution approach, only a special one dimensional Neu-mann case has been investigated in [136, 235] when the term b is zero or regular (seealso Chapter 4 and, in particular, the examples of Section 4.8). The existence anduniqueness of a regular solution of the HJB equation, and the existence of feedbackcontrols were obtained there.17 The L2 approach presented in Chapter 5 and theBSDE approach of Chapter 6 have not yet been applied to such equations, howeverthere are results when the boundary control comes together with the boundarynoise, see Section 2.6.3.

2.6.3. Stochastic controlled heat equation: boundary control andboundary noise. The third example still concerns the control of the stochas-tic heat equation in a given space region O. In this case we assume that both thenoise and the control act only at the boundary of that region. The problem is veryhard since the presence of the noise at the boundary introduces a strong unbound-edness in the model. For this reason, up to now, only one dimensional cases havebeen studied, and so we present here a one-dimensional example with a Neumannboundary condition taken from [131]. We will also mention what happens in a moredifficult case of Dirichlet boundary condition (see [165, 338]).

2.6.3.1. Setting of the problem. We consider an optimal control problem fora state equation of parabolic type on a bounded interval, which, for convenience,we take to be [0,π]. We consider a Neumann boundary condition in which thederivative of the unknown function is equal to the sum of the control and of a white

16See also, for the deterministic case, the papers [67, 70, 71, 163, 164].17See also, for the deterministic case, [167, 168, 171].


noise in time, namely:

∂y

∂s(s, ξ) = ∆ξy(s, ξ) + f (y(s, ξ)) in (t, T ]× (0,π),

y(t, ξ) = x(ξ) on (0,π),

∂y(s, 0)

∂n= a1(s) + W1(s),

∂y(s,π)

∂n= a2(s) + W2(s) on (t, T ],

(2.107)In the above equation, Wi(t), t ≥ 0, i = 1, 2, are independent standard realWiener processes; the unknown y(s; ξ,ω), representing the state of the system, is areal-valued process; the control is modeled by the real-valued processes ai(s,ω), i =1, 2 acting, respectively, at ξ = 0 and ξ = π; x is in L2(0,π), which are progressivelymeasurable with respect to the augmented filtration generated by W = (W1,W2).The function f belongs to Cb(R) and is globally Lipschitz continuous.

The functional to minimize is

I(t, x; a1(·), a2(·)) =

E T

t

π

0β1(ξ, y(s, ξ))dξ + β2(a1(s), a2(s))

ds+

π

0γ(ξ, y(T, ξ))ds

(2.108)

2.6.3.2. The infinite dimensional setting. To rewrite the problem in an infinitedimensional setting we take H = L2(0,π), Λ = Ξ = R2, Q is the identity operatoron Ξ, WQ = W , and a(·) = (a1(·), a2(·)). As in the previous example, using theresults of Section 1.4 and Appendix C Section C.3, we get, formally, the followinginfinite dimensional state equation for the variable X(s) = y(s, ·):

dX(s) = [AX(s) + b(X(s)) + (λI −A)Nλa(s)] ds+ (λI −A)NλdWQ(s), s ∈ (t, T ]

X(t) = x, x ∈ H,(2.109)

where b and Nλ are as in the Section 2.6.2.3. This equation is interpreted in themild form as

X(s) = e(s−t)Ax+

s

te(s−r)A[b(X(s)) + (λI −A)Nλa(s)]dr

+

s

t(λI −A)βe(s−r)ABλdWQ(r), (2.110)

where β ∈ (1/4, 1/2) and Bλ := (λI − A)1−βNλ is a bounded operator. We takeUt to be the set of processes a(·) belonging to M2

ν (t, T ;R2) for a given referenceprobability space.

Theorem 1.135 guarantees the existence and uniqueness of a mild solutionX(s) := X(s; t, x, a(·)) of (2.110) with continuous trajectories. The validity of Hy-potheses 2.11 and 2.12, and (A4) needed for the dynamic programming principleare discussed in Remark 2.17.

We now define l : H → R by

l1(x) =

π

0β1(ξ, x(ξ))dξ,

l2 : Λ → R byl2(a) = β2(a1, a2),

and g : H → R by

g(x) =

π

0γ(ξ, x(ξ))dξ.


The functional I in (2.97) can thus be rewritten as

J(t, x; a(·)) = E T

t[l1(X(s)) + l2(a(s))] ds+ g(x(T ))

. (2.111)

Again, β1,β2, γ must satisfy the right continuity and growth assumptions toguarantee Hypothesis 2.23, so that we can claim that the dynamic programmingprinciple be satisfied.

2.6.3.3. The HJB equation. The HJB equation associated with the problem(2.110)-(2.111) is

vt +1

2Tr [(λI −A)Nλ [(λI −A)Nλ]

∗ D2v] + Ax,Dv+ F0(Dv) + l2(x) = 0,

v(T, x) = g(x), x ∈ H,

where the Hamiltonian F0 is given by

F0(p) = infa∈R

(λI −A)Nλa, p+ l2(a) .

Similarly to the boundary control case, here the Hamiltonian makes sense whenone rewrites the term (λI − A)Nλa, p as Bλa, (λI − A)βp, and then F0 is un-bounded with respect to the variable p as it is only defined if p ∈ D((λI − A)β).However, the extra difficulty arises due to the second order term Tr[(λI −A)Nλ [(λI −A)Nλ]

∗ D2v] which is written here in a formal way and needs to begiven special interpretation. We notice that the same “operator” (λI − A)Nλ actson the control and on the Wiener process in (2.109), and thus the control acts onthe solution in the same way as the noise. This allows to use the BSDE approachto mild solutions (see [131] and, later, [448, 464, 465]). The L2 approach is inprinciple applicable to this problem but, up to now, it has not been developed.Concerning a viscosity solution approach we mention the paper [451] where theauthors show that the value function is a viscosity solution of the HJB equationbut without proving uniqueness. At the present stage, the perturbation approachto mild solution presented in Chapter 4 does not seem applicable here.

Remark 2.45 A one dimensional control problem in the half-line [0,+∞) withboundary control and noise in the Dirichlet case (i.e. with the boundary conditionof the type y(s, 0) = a(s)+ W (s) for s ∈ (t, T ]) has been studied in [165] and [338].

However, the choice of the infinite dimensional setting in this case presents aproblem, since, choosing as the state space H = L2(0,+∞), the continuity of thetrajectories in L2(0,+∞) is not ensured (see for example [125] Proposition 3.1, page176). We can have the continuity only in some spaces of distributions extendingL2(0,+∞), or in L2 with a suitable weight (see [165] Proposition 2.2, Lemma 2.2and Theorem 2.7).

In [165] the linear quadratic case is studied while in [338] a more general caseis studied by the BSDE approach. The problem has not been studied yet by othermethods.

We remark that, similarly to the case of boundary control, the HJB equationfor the Dirichlet boundary noise case is more difficult than the one for the Neumannboundary noise (as the unbounded operators arising in the first and second orderterms contain “higher powers of A”).


2.6.4. Optimal control of the stochastic Burgers equation. Our fourthexample concerns optimal control of the stochastic Burgers equation. Determin-istic Burgers equation has been introduced by J. M. Burgers (see e.g. [61, 62])as a model in fluid mechanics and has been later used in various areas of appliedmathematics such as acoustics, dispersive water waves, gas dynamics, traffic flow,heat conduction, etc. As explained in [126] (page 255) the deterministic Burgersequation is not a good model for turbulence since it does not display any chaoticphenomena; even when a force is added to the right hand side, all solutions convergeto a unique stationary solution as time goes to infinity. The situation is differentwhen the force is random. Several authors have indeed suggested using the sto-chastic Burgers equation as a simple model for turbulence, as [83, 88, 279]. In [284]it is used to model the growth of a one-dimensional interface. Among other paperson the subject we mention [119], [146], [118].

Here we present a simple optimal control problem for the one dimensionalstochastic Burgers equation motivated by a model of the control of turbulenceformulated in [88] and studied in [112, 113].

2.6.4.1. Setting of the problem. The state equation is the following stochasticcontrolled viscous Burgers equation

dy(s, ξ) =

∂2y(s, ξ)

∂ξ2+

1

2

∂

∂ξy2(s, ξ) +

Qα(s, ·)(ξ)

ds+ dWQ(s)(ξ),

s ∈ (t, T ], ξ ∈ (0, 1),

y(t, ξ) = x(ξ), ξ ∈ [0, 1],

y(s, 0) = y(s, 1) = 0, s ∈ [t, T ].(2.112)

Here:

• the function y : [t, T ] × [0, 1] × Ω → R, (s, ξ,ω) → y(s, ξ,ω) describesthe evolution of the velocity field of the fluid;

• the control α : [t, T ] × (0, 1) × Ω → R, (s, ξ) → α(s, ξ,ω) gives thedynamics of the external force acting at every point of (0, 1);

• WQ is a Q-Wiener process with Q ∈ L+1 (L

2(0, 1)) and (F ts)s∈[t,T ] is the

augmented filtration generated by WQ;• x(·) ∈ L2(0, 1) gives the distribution of the initial velocity field.

As before, the solution of (2.112) is denoted by yα,t,x. A possible objective of thecontroller (used in [88, 112, 113]) is to minimize a functional

I(t, x;α(·)) = E T

t

1

0

∂yα,t,x(s, ξ)

∂ξ

2

+1

2|α(s, ξ)|2

dξds

+

1

0

1

2|yα,t,x(T, ξ)− y(ξ)|2dξ

. (2.113)

where y is a given “desired” velocity profile. The main idea beyond this form of thecost functional is that we try to get the final velocity field to be close to y whileminimizing the “vorticity" of the flow (measured here by the integral of the spacederivative) and the energy spent controlling the system.

We then minimize the functional I over all control strategies α which are pro-gressively measurable with respect to F t

s , and such that E Tt

10 |a(s, ξ)|2dξds <

+∞. Sometimes we may require some additional bounds on the control strategies(e.g. α(s, ξ) ∈ [m,M ] for some m < M).


Note that the operator√Q acting on the control is the square root of the

covariance operator of the Wiener process. This can be interpreted as “the noiseacting on the control.”

2.6.4.2. The infinite dimensional setting and the HJB equation. We take H =Λ = L2(0, 1). The state equation (2.112) and the functional (2.113) can be rewrittenas an abstract evolution equation in H using the operator

D(A) = H2(0, 1) ∩H1

0 (0, 1),

Ax = ∂2

∂ξ2x, for x ∈ D(A),(2.114)

and the nonlinear operator

D(B) = H1(0, 1)

B(x)(ξ) = x(ξ) ∂∂ξx(ξ), for x ∈ D(B).

(2.115)

Indeed, once we set X(s) = y(s, ·) ∈ L2(0, 1), a(s) = α(s, ·) ∈ L2(0, 1), the stateequation (2.112) becomes

dX(s) =

AX(s) +B (X(s)) +

√Qa(s)

ds+ dWQ(s)

X(t) = x,(2.116)

and (2.113) is equivalent to

J(t, x; a(·)) = E T

t

(−A)1/2x(s)2

H+

1

2|a(s)|2H

ds+

1

2|X(T )− y|2H

.

(2.117)Differently from the previous examples, the standard mild solution approach doesnot work for equation (2.116). Therefore, the existence and uniqueness resultsrequire a different framework, see [119] and [127] Chapter 14 (the result is alsostated in Section 4.9). An unpleasant consequence of this is the fact that it isnot obvious to conclude that Hypotheses 2.11 and 2.12 needed for the dynamicprogramming principle are satisfied. We will not deal explicitly with this problemin the book, but we will see in Chapter 3 how to show the DPP for a much moredifficult problem, namely the optimal control of the 2-D stochastic Navier-Stokesequations (see next section).

The HJB equation related to our control problem is (see [113] equation (2.4))

vt(t, x) +1

2Tr

QD2v(t, x)

+ Dv(t, x), Ax+B(x)

−1

2

QDv(t, x)

2+(−A)1/2x

2= 0,

v(T, x) =1

2|x− y|2.

(2.118)

This equation is difficult to investigate due to the presence of the non-linear un-bounded term Dv(t, x), B(x), coming from the state equation, and the term(−A)−1/2x

2, coming from the objective functional. It was studied in [113] bya Hopf-type change of variable and in [112, 114] using a variant the mild solutionapproach. In these papers the authors use finite dimensional approximations of thestate equation and are able to obtain existence and uniqueness of regular solutionsto the HJB equation and to find optimal control in feedback form (see Subsection4.9.1.1).

It is interesting to note that this technique has been extended to the caseof control of stochastic Navier-Stokes equations in dimensions 2 and 3 (see nextsection). Equation (2.118) can also be investigated using the viscosity solutionapproach even though there are no explicit results. However, we refer the readersto Chapter 3 and [244]).


2.6.5. Optimal control of the stochastic Navier-Stokes equations. Thestochastic Navier-Stokes equations are used to model turbulent flows. We refer thereader to the books [299, 443], the survey article [187], Chapter 15 of [126], and thepaper [39] for more on this.

The optimal control of the Navier-Stokes equations, in the deterministic andstochastic cases, is a very challenging problem, both from the theoretical and ap-plied points of view. For a survey on this subject we refer to the book [215] forthe deterministic case and the paper [419] for the stochastic case. The dynamicprogramming approach to the optimal control of stochastic Navier Stokes equationshas been investigated in the papers [115, 244, 328].

We consider a model problem for a control of turbulent flow governed by thestochastic two-dimensional Navier-Stokes equations for incompressible fluids. Wefollow mainly the paper [115] with some changes borrowed e.g. from [419] and [244].It has been partly generalized to the three-dimensional case in [328].

2.6.5.1. Setting of the problem. We are given an open domain O ⊂ R2 withlocally Lipschitz boundary: it includes e.g. the case where O is a rectangle (which isquite common in the literature, see e.g. [116, 436, 244]). Given ξ = (ξ1, . . . , ξn), η =(η1, . . . , ηn) ∈ Rn we use the notation ξ · η :=

ni=1 ξiηi.

We take any reference probability space (Ω,F ,F ts ,P,WQ) satisfying the

usual conditions, where WQ is an L2(O;R2))-valued Q-Wiener process with Q ∈L+1 (L

2(O;R2)).The control variable is an external force α(s, ξ) acting at every point ξ of O; for

models with control on subdomains one can see e.g. [262] and [419] (page 3). Thecontrols stochastic processes progressively measurable with respect to the filtrationF t

s for a given reference probability space, and such that |α(s, ·)|L2(O;R2)) ≤ R forsome fixed R > 0, for all (s,ω) ∈ [t, T ]× Ω. The unknowns are the velocity vectorfield y(s, ξ) = (y1(s, ξ), y2(s, ξ)) and the pressure p(s, ξ): they satisfy the system

dy(s) + [(y(s, ξ) ·∇)y(s, ξ) +∇p(s, ξ)] ds

= [ν∆y(s, ξ) + α(s, ξ)] ds+ dWQ(s)(ξ) in (t, T ]×O

div (y(s, ξ)) = 0 in [t, T ]×O

y(s, ξ) = 0 on [t, T ]× ∂O

y(0, ξ) = x(ξ) on O,(2.119)

(∇ denotes (∂ξ1 , ∂ξ2) and y · ∇ denotes y1∂ξ1 + y2∂ξ2). The positive constant νrepresents the kinematic viscosity. We remark that distributed control can be ap-proximately realized for electrically conducting fluids (like salt water, liquid metals,etc.) by a suitable Lorentz force distribution. The boundary control, which is notpresent in this example, is typically implemented by blowing and suction at theboundary.

Suppose, as in the previous section, that we want to achieve the desired profiley of the flow while minimizing the turbulence of the flow and the amount of energyused to control it. This is is the most common case in engineering applications. Werecall that we can measure how turbulent a flow is by evaluating the time averagedenstrophy, which is defined by

O|curl y(s, ξ)|2 dξ,


where the rotational operator curl in dimension 2 is defined as

curl (y1, y2) =∂y1∂ξ2

− ∂y2∂ξ1

. (2.120)

Thus we consider the problem of minimizing the following functional over all controlstrategies α:

I(t, x;α(·)) = E T

t

O

|curl y(s, ξ)|2 + 1

2|α(s, ξ)|2

dξds+

O|y(T, ξ)− y(ξ)|2 dξ

.

(2.121)In areas like combustion the goal may be to maximize mixing (and hence turbulence)of the flow. As remarked in [419] (page 3) in some flow control problems and indata assimilation problems in meteorology one may also minimize the functional

I1(t, x;α(·)) = E T

t

O

|curl (y(s, ξ)− yd(s, ξ))|2 +

1

2|α(s, ξ)|2

dξds

(2.122)

for a given velocity field yd(s, ξ). (See also [215], page 167, formula (1.15), for asimilar type of functional, in the deterministic case.)

2.6.5.2. The infinite dimensional setting and the HJB equation. Denote

V :=f ∈ C∞

0 (O;R2) : div(f) = 0, (2.123)

H := the closure of V in L2(O;R2), (2.124)and

V := the closure of V in H1(O;R2). (2.125)Recall that we have an orthogonal decomposition

L2(O;R2) = H ×H⊥,

where H⊥ = f = ∇p : for some p ∈ H1(O).We define the unbounded operator in H

D(A) := H2(O;R2) ∩ V ⊂ H

A := P∆

where P is the orthogonal projection in L2(O;R2) onto H. The operator A isself-adjoint and strictly negative (see [437]), generates a C0-semigroup on H, andmoreover V = D((−A)1/2). We also define the bilinear operator

B : V × V → H

B(x, y) = P (x ·∇)y

and set B(x) := B(x, x).Applying the projection P to equation (2.119) and setting X(s) = y(s, ·) ∈ H,

a(s) = Pα(s, ·) ∈ H, we obtain

dX(s) = (νAX(s)−B (X(s)) + a(s)) ds+ PdWQ(s)

X(t) = x.(2.126)

Since |a(s)|H = |a(s)|L2(O;R2) ≤ |α(s, ·)|L2(O;R2), we can obviously restrict theset of controls to these with values in H. Moreover, for x ∈ V , |curl x|L2(O) =

|∇x|L2(O;R4) = |(−A)1/2x|H . Thus the minimization of the functional I in (2.121)is equivalent to the minimization of

J(t, x; a(·)) = E T

t

(−A)1/2X(s)2

H+

1

2|a(s)|2H

ds+ |X(T )− y|2H

(2.127)

over all a(·) ∈ Ut, which is defined as in Section 2.2.1 with Λ := BH(0, R).


There are various ways to define solutions of (2.126) and for existence anduniqueness results we refer for instance to [93, 127, 342, 443] and to Chapter 3 andSection 4.9 here. Unfortunately we cannot apply the definition of mild solution,and thus showing that Hypotheses 2.11 and 2.12 are satisfied requires a differentargument. We explain it in Chapter 3.

The Hamiltonian for the control problem is

F (p) :=

− 1

2 |p|2 if |p| ≤ R

−|p|R+ 12R

2 if |p| > R,

and the Hamilton-Jacobi-Bellman equation for the system becomes

vt +1

2Tr

PQP ∗D2v

+ Dv, νAx−B(x)

+F (Dv) +(−A)1/2x

2= 0

v(T, x) = |x− y|2.

(2.128)

We note that one can also associate a different control problem with (2.128)by considering PWQ to be a Q-Wiener process in H with Q = PQP ∗ and takingcontrols to be the progressively measurable processes with respect to the augmentedfiltration generated by PWQ.

Similarly to the case of the control of the stochastic Burgers equation discussedin the previous section, the difficulty of the HJB equation (2.128) comes from thepresence of the unbounded terms Dv, b(x) and

(−A)1/2x2. The unboundedness

caused by the operator b is much worse now. However the approach used for theBurgers case by Da Prato and Debussche still allows to obtain satisfactory results onexistence and uniqueness of regular solutions [115] under some conditions on Q. In[328] existence of regular solutions is proved for the three dimensional case. Theseresults are described in Subsections 4.9.1.2, 4.9.1.3. The viscosity solution approachcan be applied to the two dimensional case with more general cost functional andnoise and it yields existence and uniqueness of viscosity solutions (see [244] andChapter 3). The viscosity solution approach for the three dimensional case is stillopen. Some results in the deterministic case are in [412].

2.6.6. Optimal control of the Duncan-Mortensen-Zakai equation.This example concerns a class of finite dimensional stochastic optimal control prob-lems with partial observation and correlated noises. We present the problem andwe briefly show its connection with the so-called “separated” problem (see e.g.[34, 156, 157, 193, 370]) which is a fully observable infinite dimensional stochasticoptimal control problem. The setting of the partially observed control system wedescribe here is the same as in [245] and is borrowed from [370, 468, 469], (see also[257, 258, 259, 320]). Duncan-Mortensen-Zakai (DMZ) equation, separated prob-lem and optimal control of the DMZ equation are also discussed in details in [364].The presentation in [364] relies on [406] which also discusses filtering problems.

2.6.6.1. An optimal control problem with partial observation. Consider, in theinterval [t, T ] a random state process y in Rd and a random observation process y1in Rm. The state-observation equation is

dy(s) = b1(y(s), a(s))ds+ σ1(y(s), a(s))dW 1(s) + σ2(y(s), a(s))dW 2(s),

y(t) = η,

dy1(s) = h(y(s))ds+ dW 2(s),

y1(t) = 0,(2.129)


where W 1 and W 2 are two independent Brownian motions in Rd and Rm respec-tively on some stochastic basis

Ω,F , Fss∈[t,T ],P

which is a complete probabil-

ity space with the filtration satisfying the usual conditions. The initial conditionη is assumed to be Ft-measurable and square integrable. The control set Λ ⊂ Rn,and admissible controls are the processes a(·) : [t, T ]×Ω → Λ that are progressivelymeasurable with respect to the filtration F y1

s : s ∈ [t, T ], which is the augmentedfiltration of the filtration

F y1,0

s : s ∈ [t, T ]

generated by the observation processy1. We assume the following.

Hypothesis 2.46 The set Λ is a closed subset of Rn. The functions

b1 : Rd × Λ → Rd, h : Rd → Rm

are uniformly continuous and b1(·, a), h have their C2(Rd) norms bounded, uni-formly for a ∈ Λ. Moreover the functions

σ1 : Rd × Λ → LRd,Rd

, σ2 : Rd × Λ → L

Rm,Rd

are uniformly continuous and σ1(·, a),σ2(·, a) have their C3(Rd) norms bounded,uniformly for a ∈ Λ, and

σ1(ξ, a)σ1(ξ, a)

T ≥ λI

for some λ > 0 and all ξ ∈ Rd, a ∈ Λ.

This assumption in particular guarantees the existence of a unique strong solu-tion of the state equation (2.129), see e.g. Theorem 1.121. We denote its solutionat time s by (y(s; t, η, a(·)), y1(s; t, a(·))) or simply by (y(s), y1(s)).

We now consider the problem of minimizing the cost functional

I(t, η; a(·)) = E T

tl1(y(s), a(s))ds+ g1(y(T ))

(2.130)

over all admissible controls, where the cost functions

l1 : Rd × Λ → R; g1 : Rd → R

are suitable continuous functions, say with at most polynomial growth at infinityin the variable y, uniformly with respect to the variable a. A control strategy a∗(·)minimizing the cost I in (2.130) is called an optimal control in the strict sense.

2.6.6.2. The separated problem. The optimal control of partially observed dif-fusions is a very difficult problem with still many open questions (e.g the existenceof optimal controls in the strict sense, see e.g. [193], p. 261). One way of deal-ing with it is through the so called “separated” problem where one looks at theassociated problem of controlling the unnormalized conditional probability densityY (·) : [t, T ] → L1

Rd

of the state process y given the observation y1. This idea,

that arises from well known results in nonlinear filtering (see e.g. [143, 350, 456]),has been introduced first in [193] to prove existence of optimal controls in a suit-able weak sense. Here, following mainly [370], we briefly and informally explain theseparated problem and how it arises.

To introduce the new state Y and to compute the equation for it we considerfor each s ∈ [t, T ] the conditional law Πs of the random variable y(s) given the pathof y1 up to time s i.e., in our setting, given the σ-field F y1

s , and look at its densitywith respect to the Lebesgue measure in Rd. This density, up to a normalizingfactor, will be the new state Y (s) at time s. The conditional law Πs is a measurevalued process such that for every f ∈ Cb(Rd), the conditional expectation

E [f(y(s))|F y1s ] =

Rd

f(y)dΠs(y) P a.s..


Using the notation of [370], the above expression will be denoted by Πs(f). Theprocess Πs exists if there exists a regular conditional probability given F y1,0

s . Tocompute it, it is more convenient (as explained e.g. in [370] at the end of SectionI.5) to change the probability measure. We define the new probability measure Pby

dP = κ−1(T )dP,where

κ(s) = exp

s

th(y(r)), dy1(r)Rm − 1

2

s

t|h(y(r))|2Rmdr

.

Since κ(s) is a martingale we have

dP = κ−1(s)dP, on Fs. (2.131)It follows from the Girsanov Theorem (see e.g. [283], Section 3.5) that the processesW 1 and y1 become two independent Brownian motions respectively in Rd and Rm

in the new probability spaceΩ,F , (Fs)s∈[t,T ],P

. In this space the equation for

the process y(·) becomes

dy(s) =b1(y(s), a(s))− σ2(y(s), a(s))h(y(s))

ds

+σ1(y(s), a(s))dW 1(s) + σ2(y(s), a(s))dy1(s),

y(t) = η.

(2.132)

It follows from Bayes formula (see [406], page 225 or [364], Proposition 1.3, page18), see Lemma 3.1 in [370], that for s ∈ [t, T ] and f ∈ Cb(Rd),

Πs(f) = E [f(y(s))|F y1s ] =

E [f(y(s))κ(s)|F y1s ]

E [κ(s)|F y1s ]

.

So, if we are able to compute, for every s ∈ [t, T ] and for every f ∈ Cb(Rd), thequantity E [f(y(s))κ(s)|F y1

s ], then we can also find Πs(f) for every such f .By Itô’s formula we have, for f ∈ C2

b (Rd), s ∈ [t, T ],

f(y(s))κ(s) = f(η) +

s

tκ(r)La(r)f(y(r))dr

+

s

tκ(r)∇f(y(r)),σ1(y(r), a(r))dW 1(r)Rd

+

s

tκ(r)∇f(y(r)),σ2(y(r), a(r))dy1(r)Rd +

s

tκ(r)f(y(r))h(y(r)), dy1(r)Rm ,

where for every a ∈ Λ, La : C2b (Rd) → Cb(Rd) is given by

(Laf)(ξ) = b1(ξ, a),∇f(ξ)Rd

+ 12Tr

σ1(ξ, a)[σ1(ξ, a)]T + σ2(ξ, a)[σ2(ξ, a)]T

D2f(ξ)

.

Now, computing the conditional expectation (see [370], Section I.4) we have, fors ∈ [t, T ],

E [f(y(s))κ(s)|F y1s ] = E[f(η)] +

s

tEκ(r)La(r)f(y(r))|F y1

r

dr

+

s

tE

κ(r)[σ2(y(r), a(r))]T∇f(y(r)) + κ(r)f(y(r))h(y(r))|F y1

r

, dy1(r)Rm .

If Πs is a measure-valued process such that, for every f ∈ Cb(Rd), Πs(f) =E [κ(s)f(y(s))|F y1

s ], (which exists if Πs exists) then the equation above impliesthat Πs must satisfy the equation

Πs(f) = Πt(f) +

s

tΠr(La(r)f)dr +

s

tΠr(Ba(r)f), dy1(r)Rm , s ∈ [t, T ],


where for every a ∈ Λ, Ba : C1b (Rd) → Cb(Rd;Rm) is given by

(Baf)(ξ) = [σ2(ξ, a)]T∇f(ξ) + f(ξ)h(ξ).

(To justify that E [κ(r)F (y(r), a(r))|F y1r ] = Πt(F (·, a(r)), P a.s. for a bounded

and continuous function F (ξ, a), one can use approximation by step functions,Proposition 1.40-(vii) and the Lebesgue dominated convergence theorem, since theequality is true for functions F (ξ, a) = 1A(a)1B(ξ), where A,B are Borel subsetsof Λ,Rd respectively, as a(r) is F y1

r measurable.)If Πs has a density Y (s) ∈ L1(Rd) with respect to the Lebesgue measure, then

the process Y (·) should satisfy, at least in a weak sense, the so-called Duncan-Mortensen-Zakai (DMZ) equation (introduced in [143, 350, 456])

dY (s) = L∗a(s)Y (s)ds+ B∗

a(s)Y (s), dy1(s), Y (t) = x, (2.133)

where x is the density of the law of the initial datum η of equation (2.129). Theprocess Y (·) is called the unnormalized conditional density of the state with respectof the observation process. If one can prove that equation (2.133) has a solution,it is the density with respect to the Lebesgue measure of Πs, see [370], Section I.4for more on this.

Now it is possible to rewrite the functional I in (2.130) in terms of the newprobability space and infinite dimensional state Y (·). Indeed, assuming that theprocess Y (·) takes values in L2(Rd), using (2.131) we have

I(t, η; a(·)) = T

tE [κ(s)l1(y(s), a(s))] ds+ E [κ((T )g1(y(T ))]

= E T

tE [κ(s)l1(y(s), a(s))|F y1

s ] ds

+ E

E [κ(T )g1(y(T ))|F y1

s ]

= E T

tl1(·, a(s)), Y (s)L2 ds+ g1(·), Y (T )L2

=: J(t, x; a(·)).(2.134)

Computing the adjoint operators, L∗a, B

∗a, we can rewrite (2.133) in an explicit and

more familiar form

dY (s) = Aa(s)Y (s)ds+m

k=1

Ska(s)Y (s)dy1,k(s), Y (t) = x, (2.135)

where for every a ∈ Λ, Aa and Ska (k = 1, . . . ,m) are the following differential

operators

(Aax) (ξ) =d

i,j=1

∂i [ai,j(ξ, a)∂jx(ξ)] +d

i=1

∂i [bi(ξ, a)x(ξ)] , (2.136)

andSkax

(ξ) =

d

i=1

dik(ξ, a)∂ix(ξ) + ek(ξ, a)x(ξ); k = 1, ...,m, (2.137)

wherea(ξ, a) = σ1(ξ, a)

σ1(ξ, a)

T+ σ2(ξ, a)

σ2(ξ, a)

T,

bi(ξ, a) = −b1i (ξ, a) + ∂jai,j(ξ, a); i = 1, ...d,

d(ξ, a) = −σ2(ξ, a),

ek(ξ, a) = hk(ξ)− ∂iσ2ik(ξ, a); k = 1, ...m.

The separated problem is thus the problem of maximizing the functional J overall admissible controls a(·), with state equation (2.135). It is an infinite dimensionaloptimal control problem which can be studied within the framework of this book


in the state space H = L2(Rd) or other spaces. In Section 3.11 we will investigateit in suitable weighted spaces. It is worth noticing that even though the originalcontrol problem was nonlinear, the DMZ equation (2.135) is linear and the costfunctional J is also linear in the state variable x.

The mild solution approach cannot be applied to (2.135). Existence and unique-ness of solutions in a variational sense was proved in [297] (see also [369]). Wediscuss variational solutions in Section 3.11 where we show that under suitable as-sumptions equation (2.135) is well posed in L2(Rd) and in weighted versions of it.We also explain how to show that Hypotheses 2.11 and 2.12 hold.

The HJB equation for the infinite dimensional problem has the form

vt + infa∈Λ

12

mk=1

D2vSk

ax, Skax

+ Aax,Dv+ f(x, a)

= 0,

v(T, x) = g(x).(2.138)

This is a fully nonlinear equation with unbounded first and second order terms andup to know only a viscosity solution approach has given some results on existenceand uniqueness of solutions [15, 245, 259, 320]. In [320] the equation was studied ina standard L2 space when the operators Sk

a were bounded multiplication operators.In [259] it was shown that the value function is a viscosity solution in a very weaksense when the HJB equation was considered in the space of measures (see also[261], [190]). The results of [245] are presented in Section 3.11, where (2.138) willbe studied in a weighted L2 space. The optimal control problem for the DMZequation and the HJB equation (2.138) are also discussed in [364].

2.6.7. Super-hedging of forward rates. We now present a stochastic opti-mal control problem arising in finance, in pricing derivatives. When the market isincomplete there is not a unique way to price a derivative product. It is then useful,in some cases, to find the range of all possible prices, i.e. the maximum and the min-imum of possible prices, called the super-hedging and the sub-hedging price. Suchprices are defined as value functions of suitable optimal control problems and thefinite dimensional theory of this problem has been widely studied: see e.g. [14, 151]for the one dimensional case and [248, 249, 324, 403], [87, 415, 416, 417, 418] in themultidimensional case; see also [440] for a first idea of the method in the infinitedimensional case.

When the underlying asset is a forward rate the natural model for it is theso-called Musiela model introduced in [352] that describes the dynamics of for-ward rates in terms of evolution of an infinite dimensional diffusion process. Con-sequently, the super-hedging problem in such case is naturally formulated as astochastic optimal control problem in infinite dimensions. We present now suchproblem, taken from the paper [287].

The Musiela model of interest rates [352] is a reparametrization of the Heath-Jarrow-Morton (HJM) model. In this model the forward rate process r(t,σ)σ,t≥0

evolves according to a stochastic differential equation

dr(t,σ) =

∂

∂σr(t,σ) +

d

i=1

τi(t,σ)

σ

0τi(t, µ)dµ

dt+

d

i=1

τi(t,σ)dw(t)i,

where W = (w1, ..., wd) is a standard d-dimensional Brownian motion, and τi arecertain functions. Using the notation A = d

dσ , τ(t)(σ) = (τ1(t,σ), ..., τd(t,σ)),

b(τ(t))(σ) =d

i=1

τi(t,σ)

σ

0τi(t, µ)dµ,


the above equation can be written as an abstract infinite dimensional stochasticdifferential equation

dr(t) = (Ar(t) + b(τ(t)))dt+ τ(t) · dW (t), r(0) ∈ H, (2.139)

where H is some separable Hilbert space of functions on R+ (for instance H =L2(R+), H = H1(R+) or their weighted versions), and · is the inner product inRd (see [228, 352, 440, 441]). We call equation (2.139) the Heath-Jarrow-Morton-Musiela (HJMM) equation. Given the right choice of the space H and properassumptions on τ , the equation has a unique mild solution, see Section 3.10. Us-ing the process r(t,σ), the price at time t of a zero-coupon bond (see [353]) withmaturity T is

BT (t) = e−

T−t

0 r(t,σ)dσ.

This model can be used to price swaptions, caps, and other interest rates andcurrency derivatives.

Consider first a case of European options. Given a contingent claim with thepayoff function g : H → R and an initial curve at time t, r(t)(σ) = x(σ), therational price of the option maturing at time T is

V (t, x) = Ee−

T

tr(s,0)dsg(r(T )) : r(t)(σ) = x(σ)

. (2.140)

For instance for a European swaption on a swap with cash-flows Ci, i = 1, ..., n, attimes T < T1 < ... < Tn,

g(z) =

K −

n

i=1

Cie Ti−T

0 z(σ)dσ

+

(2.141)

for some K > 0. The function V given by the (Feynman-Kac) formula (2.140)should satisfy the partial differential equation

∂u

∂t+

1

2

d

i=1

D2u τi(t), τi(t)

+ b(τ(t)) +Ax,Du − x(0)u = 0

u(T, x) = g(x), (t, x) ∈ (0, T )×H,

(2.142)

where ·, · is the inner product in H. The above equation is called an infinitedimensional Black-Scholes equation. It was analyzed in [227] in the space H =L2((0,+∞)) where existence of smooth solutions was proved for smooth g and τindependent of t but possibly depending on the state variable x. The existenceof solutions was also shown for some non-smooth g when τ was a constant by anargument that allowed a parallel between (2.142) and a finite dimensional Black-Scholes equation (see also [221, 440]).

The problem of pricing of American options in the framework of the Musielamodel can be rephrased as an optimal stopping problem for the above infinitedimensional diffusion process and is connected to an obstacle problem

max

∂u

∂t+

1

2

d

i=1

D2u τi(t), τi(t)

+ b(τ(t)) +Ax,Du − x(0)u, u− ϕ

= 0

(2.143)for some function ϕ. This equation was studied in [454] from the point of view ofBellman’s inclusions and a similar obstacle problem for a related model was alsoinvestigated in [219].

One of the drawbacks of the Musiela model is that it does not guarantee thepositivity of rates and in some cases it is almost certain that they are not positive(see [440]). To avoid such possibilities the term x(0) was replaced by x+(0) in [454].


We do the same here and throughout the section we always take the positive partof the rates.

Let us explain now the super-hedging problem. Suppose that the dynamicsof the forward rates are given by equation (2.139), however we are not able todetermine precisely the process τ that describes the volatility of the market. Weonly know that it takes values in some set Λ ⊂ Hd. We consider an agent whowants to price and hedge a European contingent claim with payoff g(r(T )) thatdepends on the value of the forward rate curve at the maturity time T (note thatin cases of interest in finance the payoff function g is not even C1).

To find the super-hedging price, given an initial condition r(t) ∈ H, we try tomaximize the payoff

Ee−

T

tr+(s,0)dsg(r(T ))

(2.144)

with respect to all progressively measurable stochastic processes τ taking valuesin Λ. The processes τ become controls and the maximization of (2.144) gives thevalue function V which should provide the super-hedging price at time t as C(t) =V (t, r(t)). Following the standard finite dimensional theory for such problems (seee.g. [151, 324, 403]), a super-hedging strategy in such context (i.e. an investmentstrategy that replicates the super-hedging price) “should” then be given by theprocess π(t) := DV (t, r(t)), so in terms of the space-like derivative (i.e. withrespect to r) of the value function V . In the infinite dimensional case similarresults have not been proved yet as the method of proof requires strong regularityproperties of the value function V which are not known. However the problemprovides a strong motivation for studying the following optimal control problem:maximize (2.144) over all processes τ ∈ Ut, where the state equation is given by(2.139). For this optimal control problem Hypotheses 2.11 and 2.12 are satisfiedthanks to Proposition 2.16. Moreover, if g is locally uniformly continuous and hasat most polynomial growth, it can be proved that also Hypothesis 2.23 holds and sothe dynamic programming principle holds. This is explained in Section 3.10. Theassociated HJB equation is

∂u

∂t+ Ax,Du+ F (x, u,Du,D2u) = 0 in (0, T )×H

u(T, x) = g(x) in H,

(2.145)

where for x ∈ H, s ∈ R, p ∈ H and X ∈ S(H),

F (x, s, p,X) = supτ∈Λ

1

2

d

i=1

X τi, τi+ b(τ), p − x+(0)s

.

Equation (2.145) is called an infinite dimensional Black-Scholes-Barenblatt (BSB)equation associated to the contingent claim g. In cases of interest in finance thepayoff function g is not even C1 and a notion of a generalized solution is needed.It was studied in [287]) using viscosity solutions. In this context (2.145) has aunique viscosity solution that coincides with the value function provided by themaximization of (2.144). This is discussed in Section 3.10. The results are shownin the space H = H1(R+) which makes the term x+(0) continuous. One can alsoinvestigate the problem in weighted versions of H1(R+).

The problem of pricing derivatives in the HJMM model when the Gaussiannoise is replace by a Lévy noise, and the analysis of the associated non-local BSBequation is studied in [427].

2.6.8. Optimal control of stochastic delay equations. In this last ex-ample we consider finite dimensional stochastic controlled systems with delay in


the state and/or in the control variables. Such control systems arise in many ap-plications (for example in optimal advertising theory, see [238, 239, 330], optimalportfolio management of pension funds, see e.g. [172]) and can be rephrased, usinga well known procedure (see e.g. [34] for the deterministic case and [89, 238] forthe stochastic case), as infinite dimensional controlled systems without delay. Wepresent two cases: the first is a system with pointwise delay only in the state vari-able (taken from [223, 172], see also a special case in [238, 239]), while the secondone displays delays (pointwise or distributed) both in the state and in the controlvariable (taken from [238, 239]). We separate the two cases, since they give rise toa different settings with different mathematical difficulties.

2.6.8.1. Delay in the state variable only. Let us consider a simple controlledone dimensional linear stochastic differential equation with a delay r > 0 in thestate variable:

dy(s) = (β0y(s) + β1y(s− r) + α(s)) ds+ σdW0(s),

y(t) = x0,

y(t+ θ) = x1(θ), θ ∈ [−r, 0),

(2.146)

where σ > 0, β0,β1 ∈ R are given constants, W0 is a one-dimensional standardBrownian motion defined on a complete probability space (Ω,F ,P), and F t

s is theaugmented filtration generated by W0. The control α(·) is an F t

s -progressivelymeasurable process with values in an interval [0, R] for some R > 0. We assumethat x1(·) ∈ L2(−r, 0). This type of equations is used e.g. in optimal portfoliomanagement of pension funds (see e.g. [172], where the state variable is the wealthof the fund and the control variable is the investment strategy), and in optimaladvertising (see e.g. [330, 238, 239], where the state variable is the “goodwill” ofa given product and the control is the investment in advertising). Such equationsalso seem to be relevant for some models arising in studying economic growth in astochastic environment (see [18, 19] in the deterministic case).

Given three real functions ϕ0, h0, g0 : R → R, we consider the problem ofminimizing a functional

I(t, y0, y1;α(·)) = E T

t[ϕ0 (y (s)) + h0 (α (s))] ds+ g0(y(T ))

(2.147)

over all control strategies α(·) ∈ Ut. For example, in the optimal advertising prob-lem the function h0 represents the cost of advertising while −ϕ0, and −g0 representthe profit coming from the so-called “goodwill” associated to a given product. Insuch an applied problem it is reasonable to assume the positivity of the state vari-able (the “goodwill” y(·) ≥ 0) and of the control variable (the investment α(·) ≥ 0),and a constraint on the control space (for example sups∈[t,T ] α(s) ≤ R for someR > 0 as we have done for other examples).

Existence, uniqueness and properties of solutions of delay equations like (2.146)can be studied either directly (see e.g. [274] Section 5, [410], or the survey [275])or by introducing an equivalent infinite dimensional formulation. If we follow theformer direction, the dynamic programming approach can be used only for specialproblems where the HJB equation reduces to a finite dimensional differential equa-tion (see [307], one can find similar ideas in [158, 179]), while rephrasing the stateequation and therefore the whole optimization problem in infinite dimensions al-lows to study a larger class of problems. There are different possibilities to rewritestochastic delay differential equations in the form (2.146) as evolution equationsin Hilbert or Banach spaces. Here we present the approach of [89] which allowsto rewrite equation (2.146) in the Hilbert space R × L2 (−r, 0). Regarding otherchoices of state spaces we refer for example to [347, 348], where the state space is


C([−r, 0]), or to the recent paper [189] (see in particular Theorem 2.2), where moregeneral spaces are used.

The setting of [89] is the following. Denote by H the space R× L2 (−r, 0) andconsider the linear operator A1 on H defined by:

D (A1) =

x0

x1(·)

∈ R×W 1,2(−r, 0;R), x0 = x1(0)

A1

x0

x1(·)

=

β0x0 + β1x1(−r)

x1(·)

.

The operator A1 generates a strongly continuous semigroup S1 (t) on H and, forz = (z0, z1(·)) ∈ H, S1 (t) z can be written in terms of the solution of the lineardeterministic delay equation

y(t) = β0y(t) + β1y(t− r),

y(0) = z0, y(θ) = z1(θ), θ ∈ [−r, 0),(2.148)

as follows:

S1 (t) z =

y(t)

y(t+ ·)

∈ H, t ≥ 0

(see [89, 127] and also [32]). Set now Ξ = R,Λ = [0, R] for a suitable R > 0, anddefine Q : Ξ → Ξ and B : Λ → H, G : Ξ → H by

Qw0 = w0, B1w0 =

w0

0

, Gw0 =

σw0

0

.

Then, setting X(s) = (y(s), y(s+·)), a(s) = α(s), and WQ = W0, the controlled sto-chastic delay equation (2.146) can be rewritten (see again [89, 127]) as the followinglinear evolution equation in H:

dX(s) = [A1X(s) +B1a(s)] dt+GdWQ(s),

X(t) =

x0

x1

:=

y0

y1

∈ H.

(2.149)

Thanks to Theorem 1.121 the state equation (2.149) admits a unique mild solution(denoted by X(s; t, x, a(·)) or simply by X(s)), and thus, thanks to Proposition2.16, Hypotheses 2.11 and 2.12 are satisfied.

Moreover the functional (2.147) can be rewritten as follows. Set

ϕ (x0, x1) := ϕ0 (x0)

g (x0, x1) := g0 (x0)

so that, for a given initial datum x ∈ H, the functional I becomes

J(t, x; a) := E T

t[ϕ (X (s)) + h0 (a (s))] ds+ g(X(T ))

. (2.150)

If ϕ0, h0 and g0 satisfy proper continuity and growth conditions that ensure Hy-pothesis 2.23, then the dynamic programming principle holds (see Section 3.6 onthis).

The associated Hamilton-Jacobi-Bellman equation is

vt +

1

2Tr(GG∗D2v) + Dv,A1x+ F0(Dv) + ϕ(x) = 0,

v(T, x) = g(x),(2.151)


where

F0(p) := inf0≤a≤R

h0(a) + p,B1a = inf0≤a≤R

h0(a) + p0a . (2.152)

Since the second component of B1 is always zero, the Hamiltonian F0 only dependson the one dimensional component p0 (i.e. on the first derivative D0v of v withrespect to the “present” component). Similarly in the second order term, the factthat the second component of G is zero implies that this term only depends on thesecond derivative D2

00 of v with respect to the “present” component. Thus we canjust write 1

2Tr(GG∗D2v) = 12γ

2D200v.

This kind of equations was studied in [223] by the L2 approach, where exis-tence of weakly differentiable solutions in Sobolev spaces with respect to a suitableinvariant measure µ was proved (see Chapter 5 and, in particular, Section 5.6).Also the BSDE approach, which produces Gâteaux differentiable solutions can beapplied here, since the so-called structure condition ImB1 ⊂ ImG holds. It isdeveloped, also for ore general equations, first in [209, 337], then in [466, 467], andfinally in [464, 465] including also boundary control/noise. See on this Section 6.5.Some results for the viscosity solution approach have been obtained, also for moregeneral cases, in [172, 173, 462, 463], see also the bibliographical notes in Section3.14.

We also mention, that in the deterministic case, using the fact that the Hamil-tonian F0 only depends on the derivative with respect to the “present”, in [175] aregularity result was proved for the viscosity solution of a first order HJB equationof type (2.151) for a case with nonlinear state equation and with state constraints.

2.6.8.2. Delay in the state and control. We now consider a stochastic optimalcontrol problem whose state equation has delay in both the state and the control.Such equations are used, for example, to model the evolution of the goodwill stockin advertising models (see e.g. [238, 239]). Suppose we have a controlled stochasticdelay differential equation

dy(s) =

β0y(s) +

0

−rβ1(ξ)y(s+ ξ) dξ + γ0α(s) +

0

−rγ1(ξ)α(s+ ξ) dξ

ds

+σ dW0(s), t ≤ s ≤ T,

y(t) = y0,

y(t+ ξ) = y1(ξ), α(t+ ξ) = δ(ξ), ξ ∈ [−r, 0],(2.153)

where σ > 0, β0, γ0 ∈ R are given real numbers, β1(·), γ1(·) ∈ L2(−r, 0), andW0,α(·) are as in the previous subsection. Since β1(·), γ1(·) are functions, we ruleout the case of pointwise delay. In fact also the pointwise delay case can be studied,however it gives rise to an unbounded control operator B2 in the state equation(2.154) below. For the moment we do not consider this case: we will say moreabout it in the comments after the HJB equation.

The initial data (y0, y1, δ) are taken in R×L2(−r, 0)×L2(−r, 0). We again tryto minimize the functional I defined by (2.147), over all controls α(·) ∈ Ut.

The problem can be rewritten in an infinite dimensional setting using a tech-nique which is slightly different from the one of the previous subsection; the resultswe use are proved in [238] and they generalize these proved in the deterministicsetting in [442].


We take, as before H := R× L2(−r, 0), Ξ = R,WQ = W0 and Λ = [0, R] for asuitable R > 0. We define the operator A2 : D(A2) ⊂ H → H as follows:

D(A2) :=x ∈ H : x1 ∈ W 1,2(−r, 0), x1(−r) = 0

A2 : (x0, x1(·)) →β0x0 + x1(0),β1(·)x0 − dx1(·)

dξ

.

Moreover, we define the bounded linear control operator B2 by

B2 : R → H

B2 : a →γ0a, γ1(·)a

,

and the operator G : R → H by G : w0 → (σw0, 0), as in the case of delay onlyin the state. The control variable will remain the same in the new system, soa(s) := α(s). The state variable is called the structural state and is defined usingthe following proposition proved in [238].

Proposition 2.47 Let X(·) be the mild solution of the abstract evolution equa-tion

dX(s) = (AX(s) +B2a(s)) dt+GdWQ(s)

X(t) = x ∈ H,(2.154)

with arbitrary initial datum x ∈ H and control a(·) ∈ M2µ(t, T ;R). Then, for s ≥ t,

one has, P-a.s.,X(s) = M(X0(s), X0(s+ ·), a(s+ ·)),

where M : H × L2(−r, 0) → H

M : (x0, x1(·), v(·)) → (x0,m(·)),(X0(s) is the first component of X(s)) and

m(ξ) :=

ξ

−rβ1(ζ)x1(ζ − ξ) dζ +

ξ

−rγ1(ζ)v(ζ − ξ) dζ.

Moreover, let y(s), s ≥ t be a continuous solution of the stochastic delay dif-ferential equation (2.153), and X(·) be the mild solution of the abstract evolutionequation (2.154) with initial condition

x = M(y0, y1, δ(·)).Then, for s ≥ t, one has, P-a.s.,

X(s) = M(y(s), y(s+ ·), a(s+ ·)),hence y(s) = X0(s), P-a.s., for all s ≥ 0.

Using this equivalence result, we can now give a reformulation of our problemin the Hilbert space H. The state equation is (2.154) with initial condition x :=M(y0, y1, δ(·)) and we denote its mild solution (which exists and is unique thanksto Theorem 1.121) by X(s) := X(s; t, x, a(·)). The objective functional to minimizeis the same J given by (2.150), where g and ϕ have the same meaning. Therefore,in this setup, Hypotheses 2.11 and 2.12 are satisfied thanks to Proposition 2.16.Thus, again, if ϕ0, h0 and g0 satisfy proper continuity and growth conditions, wecan ensure that the dynamic programming principle holds.

The Hamilton-Jacobi-Bellman equation in the infinite-dimensional setting is

vt +12Tr(GG∗D2v) + Dv,A2x+ inf0≤a≤R h0(a) + Dv,B2a = 0,

v(T, x) = g(x).(2.155)


This kind of HJB equations is more difficult than (2.151) since the so-called struc-ture condition (ImB2 ⊂ ImG) is no longer true, and thus it is impossible to usethe BSDE approach of [209, 337] and the approach of strong solutions in Sobolevspaces is used in [223]. However in a special case with no delay in the state, a clevervariant of the perturbation approach to mild solutions can be applied, see [240].

Concerning a viscosity solution approach we are not aware of any results in thestochastic case. For the deterministic case, please see [178] where also regularityresults for viscosity solutions are proved (see also [175] for such results).

Finally, we remark that (as it can be seen e.g. in [238]) in the case of pointwisedelay (i.e. when β1 is the Dirac delta at −r), the operator B2 above is unbounded.This unboundedness is similar to the one arising in boundary control problems, andup to now HJB equations of this kind have been investigated only in a special casein [240].


The stochastic optimal control problem introduced in Section 2.1 is an abstractinfinite dimensional version of problems studied in the literature. We refer to [38,47, 149, 194, 195, 294, 317, 318, 349, 357, 364, 382, 449] for the finite dimensionaltheory.

For deterministic optimal control problems and their connection with HJBequations the reader may consult [30, 40, 41, 69, 95, 96, 316, 455] and the books[23, 312] for the infinite dimensional case. Some aspects of the theory of stochasticoptimal control in infinite dimensions and second-order HJB equations can be foundin the books [129, 364].

We present the optimal control problem in its weak (Section 2.1.2) and strong(Section 2.1.1) formulations. The two distinct forms appeared already in the sixties,in the early days of the studies of finite-dimensional stochastic optimal controlproblems (see e.g. [191] and [303]); we follow the terminology of [449]. We recallthat for us the “weak” in “the weak formulation” is referred only to the fact thatthe generalized reference probability spaces vary with the controls and not to theconcept of solution that in this context is always strong in the probabilistic sense(see Remark 2.4). The notion of weak formulation is also different from that usedin [197] where the word “weak” is meant in the sense of the convex duality.

In Section 2.2 and more precisely in Subsection 2.2.1 we introduce a third for-mulation that we use to prove the DPP. We can call this third setup weak DPPformulation. In this framework, as in the weak formulation, we allow the probabilityspaces and Q-Wiener processes WQ to vary but we only consider the (augmented)filtration generated by the Q-Wiener processes. Thus the difference is that we passfrom generalized reference probability spaces (Definition 1.95) to reference probabil-ity spaces (Definition 2.7). Other formulations of stochastic optimal control prob-lems have been proposed in the literature with various notions of control processes.Markov (feedback) controls, i.e. controls of the form a(t,X(t)), where X(t) is thestate of the system at time t, have been considered e.g. in [47, 194, 195, 294, 304].So called natural strategies i.e. controls that can be expressed at time t as func-tions of the state trajectory up to time t are considered in [294] where it is alsoshown that, under suitable hypotheses, the value function of the problem for nat-ural strategies equals the value function of the problem in the strong formulation(Theorem 7, page 132 of [294]). Relaxed controls have been considered (see e.g.[47, 153, 254, 301]) mostly to prove existence of optimal controls. Our formulationsof control problems follow most closely these of [449]. In Theorem 2.22 we showthat the weak DPP formulation and the strong formulation if a reference proba-bility space is used are equivalent in the sense that the problems in the two forms


have the same value function. For similar results in the finite-dimensional caseone can see [155, 195, 294]. Nisio in [364] proves that the weak formulations usingthe reference and the generalized reference probability spaces give the same valuefunctions assuming that the control set is a convex subset of Rq.

The DPP proved in Section 2.3 (Theorem 2.24) is very abstract and general.We follow to a large extent the strategy from [449]. The proof uses the continuity ofthe value function in the spatial variable, however this assumption can be relaxedwith very little change in the proof (see e.g. [217]). Theorem 3.70, Section 3.6 (nextchapter) contains a version of the DPP for mild solutions in the formulation withstopping times when the value function is continuous. The DPP is often considereda standard result, however we included complete proofs since even in finite dimen-sions it is very technical and many of the proofs available in the literature miss alot of details.

Several other approaches to the proof of the DPP are available in the literature.Krylov [294] uses approximation of controls by step controls. A PDE based proof isin Fleming and Soner [195] where the DPP is first proved for a uniformly paraboliccase where the HJB equation has a smooth solution, and then the value functionis approximated by smooth value functions solving uniformly parabolic HJB equa-tions. The proof in [47] uses Markov controls. Nisio [357] uses approximations withswitching controls at binary times and a reduction to the canonical reference prob-ability space, while the proof in [356] uses so called non-anticipative controls andapproximations by controls with continuous trajectories. A proof based on discretetime dynamic programming principle and approximation by switching controls andis presented in [364]. In [364] the DPP is proved for the weak formulation of con-trol problem from Section 2.1.2. The proof in [349] is based on a reduction to thecanonical reference probability space. In [382] a sketch of the proof is given fora measurable value function, which however omits delicate measurability issues.Soner and Touzi [416] use deep measurable selection theorems to show the DPP forstochastic target problems without continuity assumptions on the value function.In a recent paper [155] El Karoui and Tan prove the DPP under general assump-tions. Other proofs (see e.g. [153, 254, 301]) use relaxed controls and compactnessof the set of admissible controls. In [244] the authors adapt to the infinite dimen-sional case the arguments used in [196], where the DPP was shown for a two player,zero sum stochastic differential game in finite dimensions, to prove the DPP for acontrol problem for stochastic Navier-Stokes equations in the canonical referenceprobability space.

A different approach to the DPP has been introduced, for the finite dimensionalcase, in [51]. In that paper the authors introduce the notion of weak dynamicprogramming : roughly speaking, instead of proving a result similar to (2.25) theyprove the following, weaker, fact: for any couple of continuous test functions φ andψ such that φ ≤ V ≤ ψ,

infa(·)∈Uφ

t

E η

te−

s


η

tc(X(τ))dτφ (η, X (η))

≤ V (t, x)

≤ infa(·)∈Uψ

t

E η

te−

s


η

tc(X(τ))dτψ (η, X (η))

.

(2.156)

In this way the difficulties due to the possible lack of continuity of the value functionV are avoided because the condition deals with test functions that are continuous.This formulation is of course tailored to the study of viscosity solutions of HJB


equations which are defined in terms of regular test functions (see Chapter 3). Theweak dynamic programming approach introduced in [51] has been generalized tothe case of expectation constraints and state constraints in [50], where an abstractdynamic programming result was stated. The weak dynamic programming was alsoused for a class of finite-dimensional impulsive problems in [49].

The verification theorem and construction of optimal feedback controls for asmooth value function presented in Section 2.5 follow similar standard results forthe finite-dimensional case which can be found for instance in Chapter 4 of [194],Chapter 3 of [195] or in Chapter 5 of [449]. In infinite dimensional Hilbert spaces,for the case of quadratic Hamiltonian, the reader is referred to Chapter 13 of [129].When the value function does not satisfy the strong regularity conditions of Section2.5 (C1,2 regularity and the derivative in the domain of A∗), only a few specificresults are available:

• There are no results in infinite dimensions for viscosity solutions thatare only continuous. A finite dimensional verification theorem can befound in [246, 247, 449]. In the deterministic case, some verificationresults for a Hilbert space case can be found in [66, 166, 312].

• For mild solutions in spaces of continuous functions there are severalcontributions [79, 81, 112, 113, 115, 231, 232, 235, 238, 239, 241, 333,334], some of which will be discussed in Chapter 4.

• For mild solutions in the space of L2 functions we refer to [3, 4, 223, 226],see also Chapter 5.

• For optimal synthesis obtained via backward stochastic differentialequations the reader is referred to [205, 209, 210, 211, 212] and to Chap-ter 6.

A different approach to optimal control problems that is not developed in thebook is the use of maximum principle; it is closely related to the study of back-ward stochastic differential equations (BSDEs). A general result for the finite-dimensional case is given in [376], see also [449]. A generalization to problems withnoises with jumps is addressed in [433, 434], where the authors first characterizethe adjoint process of the second variation as the solution of a BSDE in the Hilbertspace of Hilbert Schmidt operators.

In infinite dimensions the problem was initially studied in [33] and [263] for thecase of diffusion independent of the controls, and in [469] for a problem with linearstate equation and cost functional. Recently, thanks to the developments in thestudy of backward stochastic differential equations in infinite dimensions, new re-sults on maximum principle for stochastic infinite dimensional problems appeared.In [141, 142, 207] the second variation is characterized as a certain stochastic bi-linear form defined on L4(Ω;H), while in [322] a general case when the coefficientsare Fréchet-differentiable (twice for non-convex control domain) is treated and thesecond variation is characterized as a solution of a BSDE “in the sense of transpo-sition”. The approach of [141, 142, 207] is used in [208] where regularity conditionson the coefficients are weakened to study a large class of optimal control problemsdriven by stochastic PDE of parabolic type on a bounded open set of Rn. Otherresults for specific classes of equations include [253], for a one-dimensional heatequation with noise and control on the boundary, and [367], for a class of problemswith delay state equation (both with distributed and discrete delay). The papers[142] and [367] also include, respectively, an unbounded diffusion term and Lévynoise. In general the maximum principle approach only gives necessary conditionsfor optimality, however under suitable convexity assumptions also sufficiency can beproved. Such results for finite dimensional systems can be found in [16, 449, 470].A sufficiency result for a class of infinite dimensional systems is proved in [343],


while a sufficient condition for certain delay systems with diffusion independent ofthe controls is characterized in [266].

CHAPTER 3

Viscosity solutions

This chapter is devoted to the theory of viscosity solutions of Hamilton-Jacobi-Bellman equations in Hilbert spaces. At its core is the notion of the so calledB-continuous viscosity solution which was introduced for first order equations byM. G. Crandall and P. L. Lions in [103, 104] and later extended to second orderequations in [422]. The theory applies to fully nonlinear equations with variousunbounded terms. This is its main advantage over the notions of mild and strongsolutions discussed in Chapter 4, mild solutions in L2 spaces discussed in Chapter5 and the BSDE techniques of Chapter 6. After the introduction of the core theorywe discuss several special cases which require various adjustments in the definitionof viscosity solution. The material of the chapter is arranged in the following way:

• In Section 3.1 we introduce the notion of B-continuity, the spaces H−α

defined by a strictly positive self-adjoint operator B, and we presentseveral estimates involving | · |−1 norms for solutions of deterministicand stochastic evolution equations. We also discuss a smooth perturbedoptimization principle in Hilbert spaces.

• In Section 3.2 we present a maximum principle for B-upper semi-continuous functions in Hilbert spaces. This is a key technical resultneeded in the proofs of uniqueness of viscosity solutions.

• In Section 3.3 we introduce the definition of viscosity solution and inSection 3.4 we discuss basic convergence properties of viscosity solu-tions.

• Section 3.5 is devoted to uniqueness of viscosity solutions. We proveseveral comparison theorems for degenerate parabolic and elliptic equa-tions.

• In Sections 3.6 and 3.7 we present results on existence of viscosity solu-tions. In Section 3.6 we study properties of value functions of stochasticoptimal control problems and prove that they are viscosity solutions ofthe associated HJB equations. In Section 3.7 we discuss how to obtainexistence of viscosity solutions for more general equations, for instanceof Isaacs type, by the method of finite dimensional approximations.

• In Section 3.9 another method to prove existence of viscosity solutions,Perron’s method, is presented. In this section we also explain how incertain cases the method of half-relaxed limits of Barles-Perthame canbe adapted to viscosity solutions in Hilbert spaces. A classical limitingproblem of singular perturbations is discussed in Section 3.8.

• In Section 3.10 we explain how the theory redof viscosity solutions isapplied to the infinite dimensional Black-Scholes-Barenblatt equationoriginating in the theory of bond markets.

• Sections 3.11, 3.12 and 3.13 discuss three special cases, the HJB equa-tion related to the optimal control of Duncan-Mortensen-Zakai equa-tion, the HJB equation for a boundary optimal control problem andthe HJB equation for optimal control of stochastic Navier-Stokes equa-tions. These cases require modifications of the definition of viscosity

143

144 3. VISCOSITY SOLUTIONS

solution. We explain how the basic theory of Sections 3.5 and 3.6 canbe extended and adapted to equations containing special unboundedterms.

Throughout this chapter H is a real, separable Hilbert space with inner product·, · and norm | · |. We remind that we identify H with its dual. We denote byS(H) the set of bounded, self-adjoint operators on H.

3.1. Preliminary results

3.1.1. B-continuity and weak and strong B-conditions.

Definition 3.1 Given a strictly positive B ∈ S(H) and α > 0, we define thespace H−α as the completion of H with respect to the norm

|x|2−α := Bαx, x .

The strict positivity of B ensures that the operator Bα/2 extends to an isometry ofH−α onto H that we denote again by Bα/2. H−α is a Hilbert space when endowedwith the inner product induced by Bα/2:

x, y−α :=Bα/2x,Bα/2y

.

Definition 3.2 If B,α are as in Definition 3.1, we denote by Hα the spaceHα := Bα/2(H) endowed with the Hilbert space structure characterized by the fol-lowing inner product:

x, yα :=B−α/2x,B−α/2y

.

Thanks to the strict positivity of B, B−α/2 : Hα → H is an isometry onto H; Hα

can be identified with the dual of H−α.Of course, even if not explicitly emphasized by the notation, the spaces Hα

depend on the choice of B.

We will often use a notion of continuity, called B-continuity, which is strongerthan the usual continuity and weaker than weak sequential continuity.

Definition 3.3 (B-upper/lower semicontinuity) Let B ∈ S(H) be a strictlypositive operator on H. Given I ⊆ R and U ⊆ H, we say that a function u : I×U →R ∪ ±∞ is B-upper semicontinuous (respectively, B-lower semicontinuous) if,for any tnn∈N ⊆ I and xnn∈N ⊆ U , such that tn → t ∈ I, xn x ∈ U andBxn → Bx as n → ∞, we have

lim supn→∞

u(tn, xn) ≤ u(t, x) ( respectively, lim infn→∞

u(tn, xn) ≥ u(t, x) ).

Definition 3.4 (B-continuity) Given B, I and U as in Definition 3.3, we saythat a function u : I ×U → R is B-continuous if it is both B-upper semicontinuousand B-lower semicontinuous.

Remark 3.5 It is easy to see that one gets the same definition of B-upper/lower-semicontinuity if the condition xn x ∈ U in Definition 3.3 is replaced by therequirement that xnn∈N is bounded and x ∈ U .

Lemma 3.6 Let B be as in Definition 3.3. Then:(i) If B is compact then u is B-upper semi continuous (respectively, B-

lower semi continuous, B-continuous) if and only if u is weakly se-quentially upper semi continuous (respectively, weakly sequentially lowersemi continuous, weakly sequentially continuous).

3.1. PRELIMINARY RESULTS 145

(ii) Let α > 0. Then u is B-upper semi continuous (respectively, B-lowersemi continuous, B-continuous) if and only if u is Bα-upper semi con-tinuous (respectively, Bα-lower semi continuous, Bα-continuous).

(iii) Let U be weakly sequentially closed, and α > 0. Then u is B-continuouson I×U if and only if u is continuous in the |·|×|·|−α norm on boundedsubsets of I×U . If B is compact and I = [a, b], then u is B-continuouson [a, b]×U if and only if u is uniformly continuous in the | · |× | · |−α

norm on [a, b] × (U ∩ BR) for every R > 0. Finally, if u is weaklysequentially continuous on [a, b]× U then u is uniformly continuous inthe | · |× | · |−α norm on [a, b]× (U ∩BR) for every R > 0.

(iv) Let B1, B2 ∈ S(H) be two strictly positive operators on H such thatB1(H) = B2(H) and xnn∈N ⊆ H. Then B1xn → B1x if and onlyif B2xn → B2x. In particular the notions of B1-continuity and B2-continuity are equivalent.

Proof. Part (i) is obvious.(ii) We show that, for any α ≥ β > 0, u is Bα-continuous (resp. Bα-lower semi

continuous, Bα-upper semi continuous) if and only if it is Bβ-continuous (resp. Bβ-lower semi continuous, Bβ-upper semi continuous). To show this fact it is enoughto prove that for a given weakly convergent sequence xn x ∈ H we have that|Bα(xn − x)| → 0 if and only if |Bβ(xn − x)| → 0. Since α ≥ β the “if” part isobvious. For the “only if” part, assume that |Bα(xn − x)| → 0 and observe that,since xn is weakly convergent and hence bounded,

|Bα/2(xn − x)|2 = xn, Bα(xn − x) − x,Bα(xn − x) → 0.

So, if α/2 ≤ β, this fact and the “if” part allow to conclude the proof, otherwiseone can conclude iterating the argument.

(iii) The first statement follows from (ii). The only nontrivial statement ofthe second claim is the “only if” part of it. So let B be compact. From (ii) wecan assume α = 2. Assume by contradiction that, for some ε > 0, there exist twosequences (tn, xn) and (sn, yn) in [a, b]× (U ∩BR) s.t.

|tn − sn|+ |xn − yn|−2 → 0 and |u(tn, xn)− u(sn, yn)| > . (3.1)Since [a, b]× (U ∩BR) is weakly sequentially compact we can assume that xn xand yn y for some x, y ∈ U ∩ BR and that tn, sn → s for some s ∈ [a, b]. Sowe have B(xn − yn) → 0 and (xn − yn) (x − y) and thus (since the graph ofa continuous operator is weakly closed), B(x − y) = 0 which implies that x = y.Since B is compact we also have Bxn → Bx and Byn → By = Bx. So, since uis B-continuous, u(tn, xn) → u(s, x) and u(sn, yn) → u(s, x) and this contradicts(3.1). If the third claim is not true then again there must exist sequences (tn, xn)and (sn, yn) s.t. (3.1) holds. But then again, up to a subsequence, tn, sn → s forsome s ∈ [a, b] and xn, yn x for some x ∈ U ∩ BR and this, together with theweak sequential continuity of u, contradicts (3.1).

(iv). Let B1xn → B1x as n → ∞. It follows easily from the closed graphtheorem theorem that B−1

1 B2 is bounded. Thus B2B−11 = (B−1

1 B2)∗ on B1(H)and (B−1

1 B2)∗ is a bounded operator. Therefore

B2xn = B2B−11 B1xn = (B−1

1 B2)∗B1xn → (B−1

1 B2)∗B1x = B2B

−11 B1x = B2x.

The other implication is proved similarly. Definition 3.7 (B-closed set) We will say that a set U ⊂ H is B-closed ifwhenever xn ∈ U, xn x,Bxn → Bx then x ∈ U .

Remark 3.8 Every weakly sequentially closed subset of H is B-closed, in par-ticular every convex closed subset of H is B-closed.


The following weak and strong B-conditions were introduced in [103, 104].

Definition 3.9 (Weak B-condition) Let A be a linear, densely defined, closedoperator in H. We say that an operator B ∈ L(H) satisfies the weak B-conditionfor A if B is strictly positive, self-adjoint, A∗B ∈ L(H), and

−A∗B + c0B ≥ 0 for some c0 ≥ 0. (3.2)

Definition 3.10 (Strong B-condition) Let A be a linear, densely defined,closed operator in H. We say that an operator B ∈ L(H) satisfies the strongB-condition for A if B is strictly positive, self-adjoint, A∗B ∈ L(H), and

−A∗B + c0B ≥ I for some c0 ≥ 0. (3.3)

It is well known that if A is a densely defined closed operator in H then theoperator B = (I + AA∗)−1/2 is bounded, strictly positive, self-adjoint and A∗B ∈L(H). The strong and weak B-conditions require a little more. We will apply themwhen the operator A is maximal dissipative. The following result has been shownin [396].

Theorem 3.11 If A is a linear, densely defined maximal dissipative operator inH then the weak B condition is satisfied with B = ((µI − A)(µI − A)∗)−1/2 andc0 = µ, where µ ≥ 0 is any constant such that µI −A∗ ≥ δI for some δ > 0.

Proof. Denote C = (µI − A)(µI − A)∗. By our assumptions, C−1 existsand C−1 ∈ L(H). It is also easy to see that C = C∗ > 0. We set B = C−1/2. ThenB = B∗ > 0, and we have, for x ∈ H,

|Bx|2 = (µI −A)−1

∗(µI −A)−1x, x =

(µI −A)−1x2 .

Therefore, by Proposition B.2-(i) (see also [130], Proposition B.1, p. 429 or[455], Theorem 2.2, p. 208), it follows that R(B) = R

(µI −A)−1

∗=

R(µI −A∗)−1

= D(A∗).

Denote S = (µI − A)∗B. Then S ∈ L(H) and it is unitary. In fact SB−1 isthe polar decomposition of µI −A∗. It remains to show that S ≥ 0.

To this end we complexify the space and the operators. Let Hc = x = x+ iy :x, y ∈ H with standard operations (x+ iy) + (z + iw) = (x + z) + i(y + w), (a+ib)(x + iy) = (ax − by) + i(bx + ay) and the inner product (x + iy), (z + iw)c =x, z + y, w + iy, z − ix,w. An operator T in H is complexified by settingTc(x+ iy) = Tx+ iTy and then (Tc)∗ = (T ∗)c. It is easy to see that we still haveµIc − A∗

c ≥ 0 in the sense that Re(µIc − A∗c)x, xc ≥ 0, Bc = B∗

c > 0 and Sc isunitary (and thus normal). It is enough to show that Sc ≥ 0.

Suppose that Sc is not nonnegative. Since Sc is normal it then follows fromthe spectral representation theorem that there is a nontrivial closed subspace K ofHc which is invariant for Sc and Sc is strongly dissipative on K, i.e. ReScx, xc ≤−ν|x|2c for some ν > 0. Let PK be the orthogonal projection onto K. Then theoperator PKBc : K → K is self adjoint and strictly positive. We choose λ > 0 inthe spectrum of PKBc. Then there exists y = 0 in K such that

|PKBcy − λy|c ≤λν

2Sc|y|c.

We set x = Bcy. Then

µIc −A∗c x, xc = Scy, xc = Scy, PK xc = Scy, PKBcyc

= λScy, yc + Scy, PKBcy − λyc


Taking the real part of the above relation and using that Re(µIc − A∗c)x, xc ≥ 0

we obtain

0 ≤ −λν|y|2c + Sc|y|cλν

2Sc|y|c = −λν

2|y|2c

which is a contradiction. Therefore Sc ≥ 0 and thus S ≥ 0. In fact one can show(see [396]) that S > 0.

The following are two concrete examples of operators satisfying the weak B-condition:

Example 3.12 If the operator A is maximal dissipative and skew-adjoint, i.e.A∗ = −A, the above implies that we can take B = (µI − A2)−1/2 for every µ > 0.However, in such case a compact B cannot satisfy the strong B-condition, sinceif it did, then for every eigenvalue λ of B with an eigenvector e we would have|e|2 ≤ (−A∗B + c0B)e, e = λ(A + c0I)e, e ≤ λc0|e|2 which is impossible sincethe eigenvalues accumulate at zero.

Example 3.13 (Operators coming from hyperbolic equations) Let A be amaximal dissipative, self-adjoint operator in a Hilbert space H with a boundedinverse. It is then well known (see e.g. A.5.4 in [130]) that the operator

D(A) =

D(A)×

D((−A)1/2)

, A =

0 IA 0

,

is maximal dissipative in the Hilbert space H =

D((−A)1/2)

×H

, equipped with

the following “energy” type inner product

uv

,

uv

H= (−A)1/2u, (−A)1/2uH + v, vH ,

uv

,

uv

∈ H.

Moreover, A∗ = −A.It is easy to check that the operator

B =

(−A)−1/2 0

0 (−A)−1/2

is bounded, positive, self-adjoint on H, and such that A∗B is bounded and the weakB-condition holds with constant c0 = 0. In fact

−A∗B

uv

,

uv

H= 0.

Moreover we have

uv

−1

=|(−A)1/4u|2 + |(−A)−1/4v|2

1/2.

Let us now examine the strong B-condition in some examples.

Example 3.14 If A is maximal dissipative and self-adjoint in H, it satisfies thestrong B-condition with B = (I −A)−1 and c0 = 1.

Example 3.15 Suppose now that A0 be a densely defined, closed operator in Hwhich satisfies the strong B-condition for some operator B0 and constant c0. Let


A1 be another densely defined, closed operator in H such that A∗1B0 is bounded

and−A∗

1B0 + c1B0 ≥ −νI (3.4)for some ν ∈ (0, 1) and some constant c1. It is then clear that A = A0 + A1

satisfies the strong B-condition with B = (1/(1 − ν))B0 and the new constantc := c0 + c1. Obviously (3.4) holds if A∗

1B0 < 1. Also rather standard argumentsshow that (3.4) is satisfied for every ν ∈ (0, 1) and some constant c1 if A∗

1B0 iscompact. To see this let e1, e2, ... be an orthonormal basis of H. For N ≥ 1we denote HN = spane1, ..., eN, PN be the orthogonal projection onto HN , andQN := I − PN . For x ∈ H we will write xN := PNx, x⊥

N := QNx. Since A∗1B0 is

compact, there is N1 ≥ 1 such that A∗1B0 − PN1A

∗1B0PN1 ≤ ν/2. Therefore it is

enough to prove that there is c1 such that −PN1A∗1B0PN1 + c1B ≥ −ν/2I which,

since PN1A∗1B0PN1x, x ≤ C|xN1 |2 will be true if

C|xN1 |2 ≤ c1B0x, x+ ν|x|2/2,i.e. if

C ≤ c1B0x, x+ ν|x|2/2 for any x such that |xN1 | = 1.

The above is certainly satisfied if |x⊥N1

| ≥ (2C/ν)1/2. Moreover, it is easy to seethat

infx:|xN1 |=1,|x⊥

N1|≤(2C/ν)1/2

B0x, x = δN1 > 0.

Thus it is enough to take c1 = C/δN1 .

Example 3.16 (Operators coming from elliptic equations) Let O be a bounded(regular enough) domain in Rn. Let

A :=

ni,j ∂i(aij∂j) +

ni bi∂i + c

D(A) := H10 (O) ∩H2(O),

where aij = aji, bi, c ∈ L∞(O) for i, j ∈ 1, .., n, and there exists θ > 0 such thatn

i,j

aijξiξj ≥ θ|ξ|2 ∀ξ ∈ Rn

a.e. in O. We observe that if A0 is the operator A with c = bi = 0, i = 1, ..., n,then A0 is maximal dissipative and self-adjoint in H = L2(O) and the strong B-condition holds for A0 with B0 = (I −A0)−1 and c0 = 1. Moreover B0 is compactas an operator from L2(O) to H1

0 (O) and thus, if A1 = A − A0, it follows thatA∗

1B0 is compact. Thus the strong B-condition is satisfied for A with B = λB0 forsome constant λ.

If in addition aij ∈ W 1,∞(O), bi = 0, i, j = 1, ..., n one can also take B0 =

λ(A)−1 above, where

Af := −∆fD(A) := H1

0 (O) ∩H2(O)

for λ big enough. This follows from an application of the Sobolevskii inequality(see for instance Theorem 1.1 in [315], see also [305, 414]).

Lemma 3.17 Let B ∈ S(H) be a strictly positive operator on H and A be alinear, densely defined, maximal dissipative operator. Then:

(i) If D(A∗) = D(B−1), then the operator S = −A∗B + c0B is invertiblefor any c0 > 0, and S−1 ∈ L(H).

(ii) If B satisfies the strong B-condition for A, then D(A∗) = D(B−1).


Proof. (i) The statement is obvious since B−1(−A∗ + c0I)−1 is boundedand it is the inverse of S.

(ii) Let S be defined as in part (i) but with c0 being the constant from thestrong B-condition for A. Since S is bounded and S ≥ I, S−1 exists and it isbounded. Moreover we have B = (−A∗ + c0I)−1S which, by the invertibility of Simplies that D(B−1) = R(B) = R(−A∗ + c0I)−1 = D(A∗).

We refer the reader to [396] for an abstract condition involving interpolationspaces which ensures that the strong B-condition is satisfied and to [103, 104] forother comments about B-continuity and strong and weak B-conditions.

3.1.2. Estimates for solutions of stochastic differential equations. LetT > 0. Let A be a linear, densely defined, maximal dissipative operator in H, andQ ∈ L+(Ξ). Let (Ω,F , Fss∈[0,T ],P,WQ) be a generalized reference probabilityspace. Let Λ be a Polish space. Let b : [0, T ]×H × Λ → H be B([0, T ])⊗ B(H)⊗B(Λ)/B(H) measurable, and σ : [0, T ]×H ×Λ → L2(Ξ0, H) be B([0, T ])⊗B(H)⊗B(Λ)/B(L2(Ξ0, H)) measurable. Let a(·) : [0, T ] × Ω → Λ be Fs-progressivelymeasurable. For x ∈ H we consider the following SDE

dX(s) = (AX(s) + b(s,X(s), a(s))) dt+ σ(s,X(s), a(s))dWQ(s)X(0) = x

(3.5)

and its approximation

dXn(s) = (AnXn(s) + b(s,Xn(s), a(s))) dt+ σ(s,Xn(s), a(s))dWQ(s)Xn(0) = x,

(3.6)

where An is the Yosida approximation of A defined in (B.8). The approximatingequation (3.6) was already introduced in Chapter 1. Here we discuss some morespecific results that will be needed in later chapters.

Let C ≥ 0 and γ ∈ [0, 1]. We will make use of the following assumptions.

|b(s, x, a)− b(s, y, a)| ≤ C|x− y| ∀x, y ∈ H, s ∈ [0, T ], a ∈ Λ, (3.7)σ(s, x, a)− σ(s, y, a)L2(U0;H) ≤ C|x− y| ∀x, y ∈ H, s ∈ [0, T ], a ∈ Λ, (3.8)|b(s, x, a)| ≤ C(1 + |x|) ∀x ∈ H, s ∈ [0, T ], a ∈ Λ, (3.9)σ(s, x, a)L2(U0;H) ≤ C(1 + |x|γ) ∀x ∈ H, s ∈ [0, T ], a ∈ Λ. (3.10)

Recall that, thanks to Theorem 1.121, assumptions (3.7), (3.8), (3.9) and (3.10),ensure the existence of unique mild solutions X(·) and Xn(·) of (3.5) and (3.6).

Proposition 3.18 Let T > 0 and γ ∈ [0, 1]. Assume that (3.7), (3.8), (3.9)and (3.10) hold. Let X(·) be the mild solution of (3.5). Then there exist constantsc1 > 0, c2 > 0 (depending only on T,C, γ) such that

E

sup0≤s≤T

ec1(1+|X(s)|2)(1−γ)

≤ c2e

(1+|x|2)(1−γ)

if γ ∈ [0, 1) (3.11)

and

E

sup0≤s≤T

ec1(log(2+|X(s)|2))2

≤ c2e(log(2+|x|2))2 if γ = 1. (3.12)

Proof. We first consider the case 0 ≤ γ < 1. Let Xn be the mild solutionof the approximating equation (3.6) and τk be the minimum of T and the first exittime of Xn from the set |z| ≤ k. Let β > 0 and α > 0 be numbers to be specified


later. Since An is bounded, Xn solves the integral equation

Xn(s) = x+

s

0AnX

n(r) + b(r,Xn(r), a(r))dr

+

s

0σ(r,Xn(r), a(r))dWQ(r).

(3.13)

Thus we can apply Ito’s formula (see Theorem 1.154) to the function

Φ : [0, T ]×H → RΦ(s, x) = eβe

−αs(1+|x|2)1−γ

and obtain,

eβe−α(s∧τ

k)(1+|Xn(s∧τk)|2)1−γ

= eβ(1+|x|2)1−γ − s∧τk

0αβe−αr(1 + |Xn(r)|2)1−γeβe

−αr(1+|Xn(r)|2)1−γ

dr

+

s∧τk

02(1− γ)βe−αreβe

−αr(1+|Xn(r)|2)1−γ

(1 + |Xn(r)|2)−γ

× AnXn(r) + b(r,Xn(r), a(r)), Xn(r)dr

+

s∧τk

02(1− γ)βe−αreβe

−αr(1+|Xn(r)|2)1−γ

(1 + |Xn(r)|2)−γ

× Xn(r),σ(r,Xn(r), a(r))dWQ(r)

+1

2

s∧τk

0eβe

−αr(1+|Xn(r)|2)1−γ

Tr

σ(r,Xn(r), a(r))Q

12

σ(r,Xn(r), a(r))Q

12

∗

× 22β2e−2αr(1 + |Xn(r)|2)−2γ(1− γ)2Xn(r)⊗Xn(r)

− 2βγe−αr(1 + |Xn(r)|2)−γ−1(1− γ)Xn(r)⊗Xn(r)

+ βe−αr(1 + |Xn(r)|2)−γ(1− γ)I

dr

≤ eβ(1+|x|2)1−γ

+

s∧τk

0(1 + |Xn(r)|2)1−γeβe

−αr(1+|Xn(r)|2)1−γ

× (−α+ C(β))βe−αrdr

+ 2

s

01[0,τk](1− γ)βe−αreβe

−αr(1+|Xn(r)|2)1−γ

(1 + |Xn(r)|2)−γ

× Xn(r),σ(r,Xn(r), a(r))dWQ(r)

(3.14)

for some absolute constant C(β), nondecreasing in β and also depending on C, γ,where we used Lemma 1.105 in the last line of (3.14) .

Therefore, choosing α = C(β) + 1 in (3.14) we obtain

eβe−α(s∧τ

k)(1+|Xn(s∧τk)|2)1−γ

+

s∧τk

0βe−αr(1 + |Xn(r)|2)1−γeβe

−αr(1+|Xn(r)|2)1−γ

dr

≤ eβ(1+|x|2)1−γ

+ 2

s

01[0,τk](1− γ)βe−αreβe

−αr(1+|Xn(r)|2)1−γ

(1 + |Xn(r)|2)−γ

× Xn(r),σ(r,Xn(r), a(r))dWQ(r).(3.15)


Therefore, taking expectation in (3.15), yields

Eeβe−α(s∧τ

k)(1+|Xn(s∧τk)|2)1−γ

+ E s∧τk

0βe−αr(1 + |Xn(r)|2)1−γeβe

−αr(1+|Xn(r)|2)1−γ

dr

≤ eβ(1+|x|2)1−γ

. (3.16)

Now we choose α = C(2) + 1 in (3.14) so that (3.15) and (3.16) are satisfiedfor β = 2. Moreover we can observe that, since C(β) is an increasing function of βand since the term βe−αreβe

−αr(1+|Xn(r)|2)1−γ

(1 + |Xn(r)|2)1−γ is always positive,(3.15) and (3.16) are satisfied also when we choose β = 1 and α = C(2) + 1. Using(3.15) with this last choice of α and β and observing that the integral in the lefthand side of (3.15) is positive, we get

sup0≤u≤s

ee−α(u∧τ

k)(1+|Xn(u∧τk)|2)1−γ ≤ e(1+|x|2)1−γ

+ sup0≤u≤s

u

02e−αree

−αr(1+|Xn(r)|2)1−γ

1[0,τk]

× (1− γ)(1 + |Xn(r)|2)−γXn(r),σ(r,Xn(r), a(r))dWQ(r),

and therefore, using Burkholder-Davis-Gundy inequality (see Theorem 1.106), wehave

E sup0≤u≤s

ee−α(u∧τ

k)(1+|Xn(u∧τk)|2)1−γ ≤ e(1+|x|2)1−γ

+

E sup

0≤u≤s

u

02e−αree

−αr(1+|Xn(r)|2)1−γ

1[0,τk]

× (1− γ)(1 + |Xn(r)|2)−γXn(r),σ(r,Xn(r), a(r))dWQ(r)

≤ e(1+|x|2)1−γ

+

E s

0C1e

−2αre2e−αr(1+|Xn(r)|2)1−γ

(1 + |Xn(r)|2)1−γ1[0,τk]dr

12

.

(3.17)

Using (3.16) with β = 2 we see that the last two lines of (3.17) are less than orequal to

C2e(1+|x|2)1−γ

.

Above, the constants Ci, i = 1, 2, only depend on C, γ, T . Thus we have obtained

E sup0≤s≤T

ee−αT (1+|Xn(s∧τk)|2)1−γ ≤ C2e

(1+|x|2)1−γ

. (3.18)

Since limk→+∞ τk = T a.s., letting k → +∞ in (3.18) and using Fatou’s lemma, weobtain

E sup0≤s≤T

ee−αT (1+|Xn(s)|2)1−γ ≤ C2e

(1+|x|2)1−γ

.

It now remains to use (see Theorem 1.125) thatlim

n→∞E

sup0≤s≤T

|Xn(s)−X(s)|2= 0.

It implies the existence of a subsequence Xnk satisfying limn→∞ sup0≤s≤T |Xnk(s)−X(s)|2 = 0 almost surely and then it ensures that, a.s.,

limnk→∞

sup0≤s≤T

ee−αT (1+|Xn

k (s)|2)1−γ

= sup0≤s≤T

ee−αT (1+|X(s)|2)1−γ

.


We can then apply Fatou’s lemma again to obtain the claim.For γ = 1 we can repeat the same arguments applied to the function

eβe−αs(log(2+|x|2))2 .

Lemma 3.19 Let A be a linear, densely defined maximal dissipative operator in Hand B an operator satisfying the weak B-condition for A for some constant c0 > 0.Then:

(i) For any R > 0 there exists a constant C(R) such that, for x ∈ H, |x| ≤R and t ≥ 0,

|etAx− x|−1 ≤ C(R)√t. (3.19)

(ii) If B satisfies the strong B-condition for A with constant c0 then, forx ∈ H and t ≥ 0,

|etAx|2−1 + 2t|etAx|2 ≤ e2c0t|x|2−1. (3.20)

Proof. (i) Denote Z(t) = etAx. If x ∈ D(A), using that A is maximaldissipative, Theorem B.45, and (3.2) we have

|Z(t)− x|2−1 =

t

02B(Z(s)− x), AZ(s)ds

≤ 2

t

0A∗BZ(s), Z(s)ds+ 2A∗B|x|2t

≤ 2c0

t

0|Z(s)|2−1ds+ 2A∗B|x|2t ≤ (2c0B|x|2 + 2A∗B|x|2)t.

The estimate now follows by density of D(A).(ii) Again it is enough to show the estimate for x ∈ D(A). We then have by

(3.3)d

ds|Z(s)|2−1 = 2A∗BZ(s), Z(s) ≤ 2c0|Z(s)|2−1 − 2|Z(s)|2,

and thusd

ds

e−2c0s|Z(s)|2−1

= −2c0e

−2c0s|Z(s)|2−1

+ e−2c0s d

ds|Z(s)|2−1 ≤ −2e−2c0s|Z(s)|2. (3.21)

Integrating we obtain

e−2c0t|Z(t)|2−1 + 2

t

0e−2c0s|Z(s)|2ds ≤ |x|2−1.

The inequality now follows upon noticing that e−2c0t|Z(t)|2 ≤ e−2c0s|Z(s)|2 for0 ≤ s ≤ t since esA is a semigroup of contractions. Lemma 3.20 Let A be a linear, densely defined maximal dissipative operator inH and B an operator satisfying the weak B-condition for A for some c0 ≥ 0. Let(3.7), (3.9) and (3.10) with γ = 1 hold, and let

b(s, x, a)− b(s, y, a), B(x− y) ≤ C|x− y|2−1 (3.22)σ(s, x, a)− σ(s, y, a)L2(U0,H) ≤ C|x− y|−1, (3.23)

for all x, y ∈ H, s ∈ [0, T ] and a ∈ Λ. If X(·) and Y (·) are the mild solutionsof (3.5) with initial conditions X(0) = x and Y (0) = y respectively, driven by theprocess a : [0, T ]× Ω → Λ, then

sups∈[0,T ]

E|X(s)− Y (s)|2−1

≤ C(T )|x− y|2−1 (3.24)


where C(T ) is a constant depending only on T,C, c0, B.Proof. We define the function

F : H → RF (z) = |z|2−1 = Bz, z .

We notice that DF (z) = 2Bz and D2F (z) = 2B. We will apply Ito’s for-mula to F along the trajectories of the process Z(s) := X(s) − Y (s), whichis a mild solution of dZ(s) = (AZ(s) + f(s)) dt + Φ(s)dWQ(s), where f(s) :=b(s,X(s), a(s)) − b(s, Y (s), a(s)) and Φ(s) := σ(s,X(s), a(s)) − σ(s, Y (s), a(s)).Thanks to (3.7), (3.9), (3.10) and (3.23), the assumptions of Theorem 1.124 aresatisfied and thus we have (1.38) and the hypotheses of Proposition 1.155 are sat-isfied. Therefore we have

E|Z(s)|2−1

= |x− y|2−1 +

s

0E [2A∗BZ(r), Z(r)+ 2BZ(r), f(r)] ds

+

s

0ETr

Φ(r)Q

12

Φ(r)Q

12

∗B

dr, (3.25)

and using (3.2), (3.22) and (3.23), we find

E|Z(s)|2−1

≤ |x− y|2−1 +

s

0E2c0|Z(r)|2−1 + 2C|Z(r)|2−1

dr

+

s

0EBC2|Z(r)|2−1

dr. (3.26)

Applying Gronwall’s lemma we obtain (3.24).

Remark 3.21 Condition (3.22) is obviously satisfied if

|b(s, x, a)− b(s, y, a)|−1 ≤ C|x− y|−1

for all x, y ∈ H, s ∈ [0, T ] and a ∈ Λ.

Lemma 3.22 Let A be a linear, densely defined maximal dissipative operator inH. Let B be a bounded, strictly positive, self-adjoint operator on H such thatA∗B is bounded. Let (3.7), (3.8), (3.9) and (3.10) with γ = 1 hold. If X(·) isthe mild solution of (3.5) with initial condition X(0) = x driven by the processa : [0, T ]× Ω → Λ, then

E|X(s)− x|2−1

≤ C(|x|, T )s (3.27)

where C(|x|, T ) is a constant depending only on T, |x|, C, B, A∗B.Proof. We define the function

F : H → RF (z) = |z − x|−1 = B(z − x), z − x .

We have DF (z) = 2B(z − x) and D2F (z) = 2B, and applying Proposition 1.156yields

E|X(s)− x|2−1

= 2

s

0E [X(r), A∗B(X(r)− x)+ b(r,X(r), a(r)), B(X(r)− x)] dr

+

s

0ETr

σ(r,X(r), a(r))Q

12

σ(r,X(r), a(r))Q

12

∗B

dr. (3.28)

Using (3.9), (3.10), the boundedness of A∗B and (1.38), we easily deduce usingCauchy-Schwarz inequality, that the absolute values of the integrands in the right


hand side of (3.28) remain bounded by some constant C(T, |x|) depending only onT, |x|, C, B, A∗B. This concludes the proof of (3.27).

Lemma 3.23 Let A be a linear, densely defined maximal dissipative operator inH and B an operator satisfying the strong B-condition for A for some c0 ≥ 0. Let(3.7), (3.9), (3.10) and (3.23) hold. If X(·) and Y (·) are the mild solutions of (3.5)with initial conditions X(0) = x and Y (0) = y respectively, driven by the processa : [0, T ]× Ω → Λ, then

sups∈[0,T ]

E|X(s)− Y (s)|2−1

+ E

s

0|X(r)− Y (r)|2dr

≤ C(T )|x− y|2−1 (3.29)

and, for any s ∈ (0, T ),

E|X(s)− Y (s)|2

≤ C(T )

s|x− y|2−1, (3.30)

where C(T ) is a constant depending only on T,C, c0, B.Proof. Following the proof of Lemma 3.20 if Z(s) = X(s) − Y (s), f(s) =

b(s,X(s), a(s)) − b(s, Y (s), a(s)) and Φ(s) = σ(s,X(s), a(s)) − σ(s, Y (s), a(s)) wehave (as in (3.25)):

E|Z(s)|2−1

= |x− y|2−1 +

s

0E [2A∗BZ(r), Z(r)+ 2BZ(r), f(r)] ds

+

s

0ETr

Φ(r)Q

12

Φ(r)Q

12

∗B

dr. (3.31)

We observe that (3.3) implies

2A∗BZ(r), Z(r)+ 2Z(r), Z(r) ≤ 2c0BZ(r), Z(r) ,which, together with (3.7), gives

2A∗BZ(r), Z(r)+ 2BZ(r), f(r)≤ 2c0|Z(r)|2−1 − 2|Z(r)|2 + 2CB1/2|Z(r)|−1|Z(r)| ≤ c1|Z(r)|2−1 − |Z(r)|2

(3.32)

where c1 depends on c0, B and C. Using (3.23) we have

Tr

Φ(r)Q

12

Φ(r)Q

12

∗B

≤ BC2|Z(r)|2−1 = c2|Z(r)|2−1. (3.33)

It thus follows from (3.31), (3.32), (3.33) that

E|Z(s)|2−1

+

s

0E|Z(r)|2

dr ≤ |x− y|2−1 + (c1 + c2)

s

0E|Z(r)|2−1

dr.

Such an inequality holds, of course, also dropping the positive term s0 E

|Z(r)|2

dr

and then (3.29) follows easily from Gronwall’s lemma. Regarding (3.30), using thedefinition of mild solution, (3.20), (3.29), and elementary computations, we have

E|X(s)− Y (s)|2

≤ C1(|esA(x− y)|2 + |x− y|2−1) ≤

C2

s|x− y|2−1,

where C1 and C2 only depend on T,C, c0, B.

Proposition 3.24 Let m > 0. Let (3.7) and (3.8), (3.9) and (3.10) with γ = 1hold. Let X(·) be the mild solution of (3.5), driven by the process a(·) : [0, T ]×Ω →Λ and let C be the constant appearing in (3.9) and (3.10). Denote

λ = Cm+1

2C2m(m− 1) if m ≥ 2,


andλ = Cm+

1

2C2m if 0 < m < 2.

Then for every λ > λ there exists a constant Cλ such that

ECλ + |X(s)|2

m

2

≤ (Cλ + |x|2)m

2 eλs for all s ≥ 0. (3.34)

Proof. Let λ > λ and let = (λ) > 0 be such that λ(1 + ) = λ. We setCλ > 0 to be a number such that 2r ≤ Cλ − 1 + r2 for all r ≥ 0. It is then easyto see that

Cmr(1 + r) +1

2C2m(m− 1)(1 + r)2 ≤ λ(K + r2) for all r ≥ 0.

Define F (z) = (Cλ + |z|2)m

2 . Then DF (z) = m(Cλ + |z|2)m−22 z and D2F (z) =

m(m− 2)(Cλ + |z|2)m−42 z ⊗ z +m(1 + |z|2)m−2

2 I.Assume first that m ≥ 2. Using Proposition 1.157 and (3.9), (3.10) we then

have

ECλ + |X(s)|2

m

2

≤ (Cλ + |x|2)m

2

+

s

0Em(Cλ + |X(r)|2)m−2

2 X(r), b(r,X(r), a(r))

+1

2Tr

σ(r,X(r), a(r))Q

12

σ(r,X(r), a(r))Q

12

∗

×m(m− 2)(Cλ + |X(r)|2)m−4

2 X(r)⊗X(r) +m(Cλ + |X(r)|2)m−22 I

dr

≤ (Cλ + |x|2)m

2 + λ

s

0E1 + |X(r)|2

m

2

dr (3.35)

and we conclude applying Gronwall’s lemma.For 0 < m < 2 the first term in the fourth line of (3.35) can be dropped and

we argue as before since

Cmr(1 + r) +1

2C2m(1 + r)2 ≤ λ(K + r2) for all r ≥ 0.

3.1.3. Perturbed optimization. The following is a classical result of Eke-land and Lebourg [147], see also [420] and [312], Lemma 4.2 page 245, for a moregeneral formulation.

Theorem 3.25 Let D be a bounded closed subset of a real Hilbert space K andf : D → R ∪ −∞ be upper semicontinuous and such that dom (f) := x ∈ D :f(x) ∈ R = ∅. Suppose that f is bounded from above. Then, for any δ > 0, thereexist y ∈ K, x ∈ D such that |y|K < δ and the function

x → f(x) + y, xKhas a strict maximum over D at x.

Corollary 3.26 Let H be a real, separable Hilbert space with inner product·, ·, , let B be a strictly positive operator in S(H) and D ⊆ H be a bounded, B-closed subset of H. Let f : D → R ∪ −∞ be a B-upper semicontinuous function,bounded from above. Then, for any δ > 0, there exist p ∈ H, x ∈ D such that|p| < δ, and the function

x → f(x) + Bp, xattains a maximum over D at x, which is strict in the topology of H−2.


Proof. We want to apply Theorem 3.25 to D endowed with the topologyinduced by H−2.

D is obviously bounded in H−2 and it is easy to see that D is closed in H−2. Toshow it, let (xn)n∈N be a sequence in D such that xn

n→∞−−−−→H−2

x ∈ H−2, i.e. Bxn → z

for some z ∈ H. Since D is bounded in H, there is a subsequence, still denotedby xn, such that xn

H x ∈ H for some x ∈ D. But the graph of B is weakly

sequentially closed, so we obtain Bx = z, which implies that |xn − x|−2 → 0, andthus x = x. Since D is B-closed, we thus have x ∈ D.

In particular we showed that if (xn) is a sequence in D such that xnn→∞−−−−→H−2

x,

then x ∈ D and xn x. Since f is B-upper semicontinuous, this shows that f isupper semicontinuous on D considered as a subset H−2.

We can now apply Theorem 3.25 to obtain that for all δ > 0 there existsy ∈ H−2 with |y|H−2 < δ such that x → f(x) + y, xH−2

attains a strict maximum(in the topology of H−2) on D at some point x. Denote p := By ∈ H. SinceB : H−2 → H is an isometry, we have that |p| = |y|−2 < δ. Therefore for x ∈ H,y, xH−2

=B2y, x

H

= Bp, xH , which completes the proof.

3.2. Maximum principle

From now on, throughout the rest of Chapter 3, unless stated otherwise, A isa linear, densely defined, maximal dissipative operator in H.

In this section B is any strictly positive operator in S(H). Let e1, e2, ...be an orthonormal basis in H−1 (see Definition 3.1) made of elements of H. ForN > 2 we denote HN = spane1, ..., eN. Let PN : H−1 → H−1 be the orthogonalprojection onto HN . It is clear that PN is also a bounded operator on H andtherefore so is QN := I − PN , i.e. PN , QN ∈ L(H). It is also easy to see thatBPN = P ∗

NBPN = P ∗NB,BQN = Q∗

NBQN = Q∗NB, where P ∗

N , Q∗N are adjoints of

PN , QN as operators in L(H). For x ∈ H we will write xN := PNx, x⊥N := QNx.

We remark that if B is compact then BγQN → 0 as N → +∞ for every γ > 0.Also in this case a natural choice for the basis e1, e2, ... is to take ei = B− 1

2 fi,where f1, f2, ... is an orthonormal basis of H composed of eigenvectors of B. Thischoice of basis has a property that it is orthogonal in H−1 and H.

For a function w ∈ C2(H−1) we will write DH−1w, D2H−1

w to denote theFréchet derivatives of w as a function in C2(H−1) whereas Dw,D2w mean theFréchet derivatives of w as a function in C2(H). We remark that the spaces H1, H2

in Theorem 3.27 are the spaces introduced in Section 3.1.1, not one of the spacesHN defined above. This is why we put the restriction N > 2.

Theorem 3.27 (Maximum Principle) Let B ∈ S(H) be strictly positive andlet N > 2, κ > 0. Let u, v : H → R ∪ −∞ be B-upper semi-continuous, boundedfrom above functions and such that

lim sup|x|→+∞

u(x)

|x| < 0 and lim sup|x|→+∞

v(x)

|x| < 0. (3.36)

Let Φ ∈ C2(HN ×HN ) be such that

u(xN + x⊥N ) + v(yN + y⊥N )− Φ(xN , yN )

has a strict global maximum over H×H at a point (x, y). Then there exist functionsϕk,ψk ∈ C2(H) for k = 1, 2, ... such that ϕk, B−1Dϕk, D2ϕk,ψk, B−1Dψk, D2ψk

are bounded and uniformly continuous, and such that

u(x)− ϕk(x)

3.2. MAXIMUM PRINCIPLE 157

has a global maximum at some point xk,

v(y)− ψk(y)

has a global maximum at some point yk, andxk, u(xk), Dϕk(xk), D

2ϕk(xk) k→+∞−−−−−→

x, u(x), DxΦ(xN , yN ), XN

in H × R×H2 × L(H−1;H1), (3.37)

yk, v(yk), Dψk(yk), D

2ψk(yk) k→+∞−−−−−→

y, v(y), DyΦ(xN , yN ), YN

in H × R×H2 × L(H−1;H1), (3.38)

where XN , YN ∈ S(H), XN = P ∗NXNPN , YN = P ∗

NYNPN ,

−1

κ+ CL(H−1×H−1)

BPN 00 BPN

≤

XN 00 YN

≤

B 00 B

C + κC2

(3.39)

and C = D2H−1×H−1

Φ(xN , yN ).

We remark that in fact ϕk,ψk ∈ C2(H−1).

Proof. Define

u(xN ) := supx⊥

N∈QNH

u(xN + x⊥N ),

v(yN ) := supy⊥

N∈QNH

v(yN + y⊥N ),

the partial sup-convolutions of u and v respectively, and let u∗ and v∗ be theirupper semi-continuous envelopes (see Definition D.9). We remark that u, v do notneed to be upper semi-continuous (see [102]). Since u + v − Φ had a strict globalmaximum at (x, y) it easily follows that

u∗(xN ) + v∗(yN )− Φ(xN , yN ) (3.40)

has a strict global maximum over HN × HN at (xN , yN ). Moreover we haveu∗(xN ) = u(x), v∗(yN ) = v(y).

We can now apply the finite dimensional maximum principle (see TheoremE.10, that is a particular case of Theorem 3.2 in [101]) when we consider HN asa subspace of H−1. (We remind that in HN the topology of H−1 is equivalent tothe topology of H). Denote HN with this topology by HN . We also note thatΦ ∈ C2(HN × HN ) and thus we can consider it as a function in C2(H−1×H−1) bysetting Φ(x, y) := Φ(PNx, PNy).

Therefore there exist bounded functions ϕk,ψk ∈ C2(HN ) with bounded anduniformly continuous derivatives (which we can consider as functions in C2(H−1)by setting ϕk(x) := ϕk(PNx) and ψk(y) := ψk(PNy)) such that u∗(xN ) − ϕk(xN )has a strict global maximum at some point xk

N , v∗(yN )−ψk(yN ) has a strict globalminimum at some point ykN , and such thatxkN , u∗(xk

N ), DH−1ϕk(xkN ), D2

H−1ϕk(x

kN )

k→∞−−−−→xN , u(x), DH−1,xΦ(xN , yN ), XN

, (3.41)

ykN , v∗(ykN ), DH−1ψk(y

kN ), D2

H−1ψk(y

kN )

k→∞−−−−→yN , v(y), DH−1,yΦ(xN , yN ), YN

, (3.42)


and

−1

κ+ CL(H−1×H−1)

PN 00 PN

≤

XN 00 YN

≤ C + κC2 in H−1 ×H−1 (3.43)

for some XN , YN ∈ S(H−1) that satisfy XN = PN XNPN , YN = PN YNPN asoperators in L(H−1) and, since

DH−1ϕk(xkN ) = PNDH−1ϕk(x

kN ), D2

H−1ϕk(x

kN ) = PND2

H−1ϕk(x

kN )PN ,

DH−1ψk(xkN ) = PNDH−1ψk(y

kN ), D2

H−1ψk(y

kN ) = PND2

H−1ψk(y

kN )PN ,

and in HN the topology of H−1 is equivalent to the topology of H, the convergences(3.41), (3.42) hold in H × R×H × L(H−1).

It is easy to see that

Dϕk(x) = BDH−1ϕk(x), D2ϕk(x) = BD2H−1

ϕk(x),Dψk(y) = BDH−1ψk(y), D2ψk(y) = BD2

H−1ψk(y).

(Note that if X ∈ S(H−1) then BX ∈ S(H) since for x, y ∈ H we have BXx, y =Xx, y−1 = x,Xy−1 = x,BXy.) Therefore, setting XN = BXN , YN = BYN ,we obtain from (3.41) and (3.42) that

xkN , u∗(xk

N ), Dϕk(xkN ), D2ϕk(x

kN )

k→∞−−−−→xN , u(x), DxΦ(xN , yN ), XN

in H × R×H2 × L(H−1;H1), (3.44)

ykN , v∗(ykN ), Dψk(y

kN ), D2ψk(y

kN )

k→∞−−−−→yN , v(y), DyΦ(xN , yN ), YN

in H × R×H2 × L(H−1;H1), (3.45)

XN = P ∗NXNPN , YN = P ∗

NYNPN , and (3.39) is satisfied.Now, using Corollary 3.26 and (3.36), for every k and j big enough we can find

pkj , qkj ∈ H, such that |pkj |+ |qkj | ≤ 1/j, and

u(x)− ϕk(x)− Bpkj , x has a global maximum at some point xkj , (3.46)

and

v(y)− ψk(y)− Bqkj , y has a global maximum at some point ykj , (3.47)

where, because of (3.36), |xkj |+ |ykj | ≤ Rk for some Rk > 0. Then if |x|, |y| ≤ R for

R ≥ Rk

u∗((xkj )N ) + v∗((ykj )N )− ϕk(x

kj )− ψk(y

kj )

≥ u(xkj ) + v(ykj )− ϕk(x

kj )− ψk(y

kj )

≥ u(x) + v(y)− ϕk(x)− ψk(y)− Bpkj , x− xkj − Bqkj , y − ykj

≥ u(x) + v(y)− ϕk(x)− ψk(y)−4RB

j.

(3.48)

Since by (3.36) if j is big enough, u(x)−ϕk(x)−Bpkj , x−xkj → −∞ as |x| → +∞,

and v(y)− ψk(y)− Bqkj , y − ykj → −∞ as |y| → +∞, choosing R big enough and

3.2. MAXIMUM PRINCIPLE 159

taking suprema over x⊥N , y⊥N in (3.48) and then envelopes at xk

N , ykN we obtain forsufficiently big j that

u∗((xkj )N ) + v∗((ykj )N )− ϕk(x

kj )− ψk(y

kj )

≥ u(xkj ) + v(ykj )− ϕk(x

kj )− ψk(y

kj )

≥ u∗(xkN ) + v∗(ykN )− ϕk(x

k)− ψk(yk)− 4RB

j.

(3.49)

Since u∗(xN ) + v∗(yN ) − ϕk(xN ) − ψk(yN ) has a strict global maximum at(xk

N , ykN ), we deduce from (3.49) that

(xkj )N → xk

N , (ykj )N → ykN , u∗((xkj )N ) → u∗(xk

N ), v∗((ykj )N ) → v∗(ykN ) (3.50)

as j → +∞ and then also

u(xkj ) → u∗(xk

N ), v(ykj ) → v∗(ykN ) as j → +∞. (3.51)

Using these and (3.44)-(3.45) we can therefore select a subsequence jk suchthat(xk

jk)N , u(xkjk), Dϕk(x

kjk), D

2ϕk(xkjk)

k→∞−−−−→xN , u(x), DxΦ(xN , yN ), XN

in H × R×H2 × L(H−1;H1),

(ykjk)N , v(ykjk), Dψk(y

kjk), D

2ψk(ykjk)

k→∞−−−−→yN , v(y), DyΦ(xN , yN ), YN

in H × R×H2 × L(H−1;H1).

It remains to show that xkjk → x and ykjk → y. This is however now obvious since

u(xkjk) + v(xk

jk)− Φ((xkjk)N , (ykjk)N ) → u(x) + v(y)− Φ(xN , yN )

and by assumption this function has a strict global maximum at (x, y). Thereforethe lemma holds with ϕk(x) := ϕk(x) + Bpkjk , x, ψk(y) := ψk(y) + Bqkjk , y andxk := xk

jk , yk := ykjk .

Theorem 3.27 applied to Φ(xN , yN ) = 12 |xN − yN |2−1 for > 0 yields the

following result which will be used in the proofs of comparison theorems.

Corollary 3.28 Let B ∈ S(H) be strictly positive and let N ≥ 1, > 0. Letu,−v : H → R∪ −∞ be B-upper semi-continuous, bounded from above functionssatisfying (3.36). Let

u(xN + x⊥N )− v(yN + y⊥N )− |xN − yN |2−1

2

have a strict global maximum over H × H at a point (x, y). Thenthere exist functions ϕk,ψk ∈ C2(H) for k = 1, 2, ... such thatϕk, B−1Dϕk, D2ϕk,ψk, B−1Dψk, D2ψk are bounded and uniformly continuous, andsuch that

u(x)− ϕk(x)

has a global maximum at some point xk,

v(y)− ψk(y)

has a global minimum at some point yk, and

xk, u(xk), Dϕk(xk), D

2ϕk(xk) k→∞−−−−→

x, u(x),

B(xN − yN )

, XN

in H × R×H2 × L(H−1;H1), (3.52)


yk, v(yk), Dψk(yk), D

2ψk(yk) k→∞−−−−→

y, v(y),

B(xN − yN )

, YN

in H × R×H2 × L(H−1;H1), (3.53)

where XN = P ∗NXNPN , YN = P ∗

NYNPN ,

−3

BPN 00 BPN

≤

XN 00 −YN

≤ 3

BPN −BPN

−BPN BPN

. (3.54)

Proof. Observe that if Φ(xN , yN ) = 12 |xN − yN |2−1 then

C = D2H−1×H−1

Φ(xN , yN ) =1

PN −PN

−PN PN

and thus κC2 = 2κ C and CL(H−1×H−1) = 2

. Then (3.54) follows from (3.39)choosing κ = .

We remark that the convergence in L(H−1;H1) in particular implies conver-gence in L(H).

The time dependent analogue of Corollary 3.28 is the following.

Corollary 3.29 Let B ∈ S(H) be strictly positive and let N ≥ 1, ,β > 0. Letu,−v : (0, T ) ×H → R ∪ −∞ be B-upper semi-continuous, bounded from abovefunctions satisfying

lim sup|x|→+∞

supt∈(0,T )

u(t, x)

|x| < 0 and lim sup|x|→+∞

supt∈(0,T )

−v(t, x)

|x| < 0. (3.55)

Let

u(t, xN + x⊥N )− v(s, yN + y⊥N )− |xN − yN |2−1

2− (t− s)2

2β

have a strict global maximum over (0, T ) × H × (0, T ) × H at a point (t, x, s, y).Then there exist functions ϕk,ψk ∈ C2((0, T ) × H) for k = 1, 2, ... such thatϕk, (ϕk)t, B−1Dϕk, D2ϕk,ψk, (ψk)t, B−1Dψk, D2ψk are bounded and uniformlycontinuous, and such that

u(t, x)− ϕk(t, x)

has a global maximum at some point (tk, xk),

v(s, y)− ψk(s, y)

has a global minimum at some point (sk, yk), andtk, xk, u(tk, xk), (ϕk)t(tk, xk), Dϕk(tk, xk), D

2ϕk(tk, xk)

k→∞−−−−−−−−−−−−−−−−−→R×H×R×H2×L(H−1;H1)

t, x, u(t, x),

t− s

β,B(xN − yN )

, XN

(3.56)

sk, yk, v(sk, yk), (ψk)t(sk, yk), Dψk(sk, yk), D

2ψk(sk, yk)

k→∞−−−−−−−−−−−−−−−−−→R×H×R×H2×L(H−1;H1)

s, y, v(s, y),

t− s

β,B(xN − yN )

, YN

(3.57)

where XN = P ∗NXNPN , YN = P ∗

NYNPN and they satisfy (3.54).

Proof. We can obviously extend u, v to R×H preserving all the propertiesof the functions and the strict global maximum at (t, x, s, y). We consider thespace H = R×H and the operator B := IR ×B. Writing (t, x) for elements of this

3.3. VISCOSITY SOLUTIONS 161

extended space we now consider the function Φ(t, xN , s, yN ) = 12 |xN − yN |2−1 +

12β (t− s)2. We now rescale time by setting

u(t, x) = u

β

12

t, x

, v(s, y) = u

β

12

s, y

,

Φ(t, xN , s, yN ) = Φ

β

12

t, xN ,

β

12

s, yN

=

|xN − yN |2−1

2+

(t− s)2

2.

Thenu(t, x)− v(s, y)− Φ(t, xN , s, yN )

has a strict global maximum over H × H at the point (( β )

12 t, x, (

β )12 s, y). We can

now apply Corollary 3.28 to produce the required functions ϕk,ψk. We now havehave operators XN , YN satisfying a version of (3.54) on H × H. However it is easyto see that its restriction to 0 × H × 0 × H gives (3.54). The claim followsafter rescaling time back to the original variables which will only change the timederivatives of ϕk,ψk.

Remark 3.30 A different type of time dependent maximum principle can beobtained which relies on the finite dimensional parabolic maximum principle pre-sented in Theorem E.11. Such a result can be found in [102], Theorem 3.2, howeverit is stated there in a version which uses second order parabolic jets and is applica-ble to equations with bounded terms (see Section 3.3.1). Theorem 3.2 in [102] alsoimposes an additional condition on the functions, which is satisfied when they areviscosity sub- and supersolutions of bounded time depended second order equationsin a Hilbert space. The maximum principle stated in Corollary 3.29 does not im-pose extra conditions and thus it is more universal and can be applied more easilywhich is why we prefer it here. However the other maximum principle has certainadvantages. For instance one can use it to prove comparison principles for boundedtime dependent second order equations in a Hilbert space without the assumptionthat the viscosity subsolutions and supersolutions attain the initial/terminal val-ues locally uniformly. Such results can be found in [290]. This type of maximumprinciple was also implicitly used in [421, 422].

3.3. Viscosity solutions

Throughout this section U is an open subset of H and the operator B satisfiesthe following assumption (see Section 3.1.1).

Hypothesis 3.31 B ∈ S(H) is a strictly positive operator such that A∗B isbounded.

Contrary to the finite dimensional case, in infinite dimensions there is no oneuniversal definition of viscosity solution. The basic idea of using pointwise max-imum principle and replacing nonexistent derivatives of a solution by derivativesof test functions is still the same. However, because of the presence of unboundedterms and operators, the choice of test functions and the interpretation of un-bounded terms must be often adjusted for different types of equations. In thissection we present a generic definition of viscosity solution for a general class ofstationary equations and time dependent Cauchy problems. The solutions definedbelow are called B-continuous viscosity solutions.

We consider the following boundary and terminal boundary value problems:

−Ax,Du+ F (x, u,Du,D2u) = 0 in Uu(x) = f(x) on ∂U

(3.58)


and

ut − Ax,Du+ F (t, x, u,Du,D2u) = 0 in (0, T )× Uu(0, x) = g(x) for x ∈ U,u(t, x) = f1(t, x) for (t, x) ∈ (0, T )× ∂U,

(3.59)

where F : (0, T ) × U × R × H × S(H) → R, g : U → R, f : ∂U → R, andf1 : (0, T )× ∂U → R are continuous.

Definition 3.32 A function ψ is a test function if ψ = ϕ+ h(t, |x|), where:(i) ϕ ∈ C1,2 ((0, T )× U) is locally bounded, and is such that ϕ is B-lower

semicontinuous, and ϕt, A∗Dϕ, Dϕ, D2ϕ are uniformly continuous on(0, T )× U .

(ii) h ∈ C1,2((0, T )×R) and is such that for every t ∈ (0, T ), h(t, ·) is evenand h(t, ·) is non-decreasing on [0,+∞).

For stationary equations ϕ and h are independent of t.

We remark that even though |x| is not differentiable at 0, the functionh(t, |x|) ∈ C1,2((0, T ) × H). The requirement that ϕt, A∗Dϕ, Dϕ, D2ϕ are uni-formly continuous (and hence grow at most linearly at infinity) is a little arbitrary.It can be replaced by a requirement that they are locally uniformly continuousand have some prescribed growth at infinity, for instance at most polynomial. Thegrowth restriction can also be removed, however it is useful in applications to sto-chastic optimal control since it is not clear if one can modify a test function ϕoutside a fixed set so that the modification has a required growth at infinity, whilepreserving the property that A∗Dϕ is continuous. Thus the radial part h of testfunctions, which can be modified at will, plays the role of a cut-off function whichtakes care of the growth at infinity. We can thus require ϕ to be as nice as wewant as long as our choice gives us enough test functions which are needed to builda good theory. The requirement that A∗Dϕ is uniformly continuous can also bereplaced by a requirement that B−1Dϕ is continuous. This however would makethe definition more dependent on the choice of B. The reader can experiment withvarious modifications of the above definition and we will later see how the choiceof test functions must be adjusted to particular cases.

Definition 3.33 A locally bounded B-upper semi-continuous (see Definition3.3) function u on U is a viscosity subsolution of (3.58) if u ≤ f on ∂U andwhenever u−ψ has a local maximum at a point x for a test function ψ = ϕ+h(|x|)then

−x,A∗Dϕ(x)+ F (x, u(x), Dψ(x), D2ψ(x)) ≤ 0. (3.60)A locally bounded B-lower semi-continuous function u on U is a viscosity superso-lution of (3.58) if u ≥ f on ∂U and whenever u+ψ has a local minimum at a pointx for a test function ψ = ϕ+ h(|x|) then

x,A∗Dϕ(x)+ F (x, u(x),−Dψ(x),−D2ψ(x)) ≥ 0. (3.61)

A viscosity solution of (3.58) is a function which is both a viscosity subsolution anda viscosity supersolution of (3.58).

Definition 3.34 A locally bounded B-upper semi-continuous function u on[0, T )×U is a viscosity subsolution of (3.59) if u(0, y) ≤ g(y) for y ∈ U , u ≤ f1 on(0, T )× ∂U and whenever u−ψ has a local maximum at a point (t, x) ∈ (0, T )×Ufor a test function ψ(s, y) = ϕ(s, y) + h(s, |y|) then

ψt(t, x)− x,A∗Dϕ(t, x)+ F (t, x, u(t, x), Dψ(t, x), D2ψ(t, x)) ≤ 0. (3.62)

A locally bounded B-lower semi-continuous function u on [0, T ) × U is a viscositysupersolution of (3.59) if u(0, y) ≥ g(y) for y ∈ U , u ≥ f1 on (0, T ) × ∂U and


whenever u + ψ has a local minimum at a point (t, x) ∈ (0, T ) × U for a testfunction ψ(s, y) = ϕ(s, y) + h(s, |y|) then

−ψt(t, x) + x,A∗Dϕ(t, x)+ F (t, x, u(t, x),−Dψ(t, x),−D2ψ(t, x)) ≥ 0. (3.63)


The main idea behind this definition of solution is the following. Test functionsare split into two categories. Good test functions ϕ provide enough functions toapply the doubling argument in the proof of comparison and produce maxima andminima using perturbed optimization by functions in this class. Radial functions hare needed as cut-off functions to be able to produce local/global maxima and min-ima and to confine the region of their possible locations. As always in the theory ofviscosity solutions, non-existing derivatives of u are replaced by existing derivativesof test functions. The term Ax,Dϕ(t, x) is interpreted as x,A∗Dϕ(t, x). Wecannot do the same with the term Ax,Dh(t, |x|) since Dh(t, |x|) = hr(t, |x|) x

|x|(where hr is the partial derivative of h with respect to the second variable) andwe cannot hope in general that x ∈ D(A∗) (nor that x ∈ D(A)). Therefore thisterm is dropped. This can be done effectively since the term hr(t,|x|)

|x| Ax, x (orhr(t,|x|)

|x| A∗x, x) would be non-positive if it were well defined. Thus the definitionis consistent with what the definition of viscosity solution should be under idealconditions.

In applications to control problems it is more natural to work with terminalvalue problems instead of the initial value problems. A terminal value problem canbe converted into an initial value problem by a change of variable t := T − t. Thusa terminal boundary value problem corresponding to (3.59) is

ut + Ax,Du − F (t, x, u,Du,D2u) = 0 in (0, T )× Uu(T, x) = g(x) for x ∈ U,u(t, x) = f(t, x) for (t, x) ∈ (0, T )× ∂U,

(3.64)

where f(t, x) = f1(T − t, x). Since we will be working with terminal value problemswe state below the definition of viscosity solution adapted to this case, which is aconsequence of Definition 3.34. (We keep the minus sign in front of the HamiltonianF since we will formulate the conditions for F that will apply to both stationaryand time dependent terminal value problems.)

Definition 3.35 A locally bounded B-upper semi-continuous function u on(0, T ]×U is a viscosity subsolution of (3.64) if u(T, y) ≤ g(y) for y ∈ U , u ≤ f on(0, T )× ∂U and whenever u−ψ has a local maximum at a point (t, x) ∈ (0, T )×Ufor a test function ψ(s, y) = ϕ(s, y) + h(s, |y|) then

ψt(t, x) + x,A∗Dϕ(t, x) − F (t, x, u(t, x), Dψ(t, x), D2ψ(t, x)) ≥ 0. (3.65)

A locally bounded B-lower semi-continuous function u on (0, T ] × U is a viscositysupersolution of (3.64) if u(T, y) ≥ g(y) for y ∈ U , u ≥ f on (0, T ) × ∂U andwhenever u + ψ has a local minimum at a point (t, x) ∈ (0, T ) × U for a testfunction ψ(s, y) = ϕ(s, y) + h(s, |y|) then

−ψt(t, x)− x,A∗Dϕ(t, x) − F (t, x, u(t, x),−Dψ(t, x),−D2ψ(t, x)) ≤ 0. (3.66)



Remark 3.36 It is easy to see that if u is a viscosity subsolution (respectively,supersolution) of (3.64) on (0, T ]×U then it is a viscosity subsolution (respectively,supersolution) of (3.64) on (T1, T ]× U for every 0 < T1 < T .

Lemma 3.37 Without loss of generality the maxima and minima in Definitions3.33, 3.34, and 3.35 can be assumed to be global and strict.

Proof. We will only show it for the case of subsolution in Definition 3.33as the other cases are similar. Let

u(x)− ϕ(x)− h(|x|) ≥ u(y)− ϕ(y)− h(|y|) for y ∈ BR(x) ⊂ U

for some R > 0.We will show that there exist test functions ϕ and h(| · |) such that Dϕ(x) =

Dϕ(x), D2ϕ(x) = D2ϕ(x), Dh(|x|) = Dh(|x|), D2h(|x|) = D2h(|x|), and u − ϕ −h(| · |) has a strict global maximum at x. Let η ∈ C2([0,∞)) be an increasingfunction such that

r + sup|y|≤r, y∈U

|u(y)|+ |ϕ(y)| ≤ η(r).

Let g1 ∈ C2((0,∞)) be a function such that

g1(r) =

0 if r ≤ |x|(r − |x|)4 if |x| < r < |x|+ 1increasing if |x|+ 1 ≤ r ≤ |x|+ 2η(r) if r > |x|+ 2.

Let ϕ1 ∈ C2([0,∞)) be defined by

ϕ1(r) =

r4 r ≤ 1,increasing 1 < r < 2,2 r ≥ 2.

Now for n ≥ 1 consider the function

Φn(y) = u(y)− ϕ(y)− nϕ1(|x− y|−1)− h(|y|)− g1(|y|).Obviously we have

Φn(x) = u(x)− ϕ(x)− h(|x|).Suppose there is a subsequence nk → +∞ and ynk

∈ U , ynk= x such that

Φnk(ynk

) ≥ Φnk(x). Then we must have |x− ynk

|−1 → 0 as k → ∞ and |ynk| ≤ C1

for some C1 > 0, i.e. ynk x and Bynk

→ Bx. Since u,−ϕ are B-upper semi-continuous, and h+ g1 is increasing on [|x|,+∞), this implies that |yn| → |x|, andtherefore ynk

→ x in H. But then ynk∈ BR(x) for big k and so we get

Φnk(ynk

) < u(ynk)− ϕ(ynk

)− h(|ynk|) ≤ u(x)− ϕ(x)− h(|x|)

which is a contradiction. Therefore there must exist n such that Φn(y) < Φn(x)for all y ∈ U, y = x. It then easily follows that Φn+1 has a strict global maximumat x. Therefore the conclusion follows with ϕ(y) = ϕ(y)+ (n+1)ϕ1(|x− y|−1) andh(|y|) = h(|y|) + g1(|y|).

It follows from the proof of Lemma 3.37 that if we know a priori that u hascertain growth at ∞ (at least quadratic) and U = H, we can then obtain the samegrowth for h. (We notice that if U = H then ϕ has at most quadratic growth atinfinity.) For instance if u has a polynomial growth at ∞ we can have h which is apolynomial of some special form for big |x|. This can be important in applicationsto stochastic optimal control where we may want to impose additional conditions ontest functions to be able to apply stochastic calculus. In these applications it mayalso be useful to assume that h(r)/r is globally bounded away from 0 for the radial


test functions h. To avoid technical difficulties it may then be more convenientto choose h belonging to one particular class of functions, say certain polynomialswith growth depending on the growth of sub- and super-solutions we are dealingwith. However when using such narrow classes of radial test functions one may beforced to require that the maxima and minima in the definition of viscosity solutionbe global as the definitions using global and local maxima and minima may nolonger be equivalent.

Remark 3.38 We assumed in Definitions 3.33, 3.34, and 3.35 that A was alinear, densely defined, maximal dissipative operator in H, i.e. that it generateda C0-semigroup of contractions etA. The definitions can be used to cover the casewhen A− ωI is maximal dissipative for some ω > 0, i.e. if

etA ≤ eωt for all t ≥ 0. (3.67)

One way to do it is to replace A by A = A − ωI and F by F (t, x, r, p,X) =F (t, x, r, p,X)− ωx, p. Another way is by making a change of variables u(t, x) =u(t, eωtx) in the equation which reduces equation (3.64) to an equation with Areplaced by A− ωI and F replaced by F (t, x, r, p,X) = F (t, eωtx, e−ωtp, e−2ωtX).

Lemma 3.39 Let F : [0, T )× U ×R×H × S(H) → R be continuous. Definition3.35 is equivalent to the definition in which we only require that the maxima andminima be one-sided (i.e. that u∓ψ has a local maximum/minimum at a point (t, x)restricted to [t, T )× U), if we also require that the subsolutions and supersolutionsare continuous. In particular the equation is also satisfied at t = 0 if we in additionrequire in the case when a one-sided maximum/minimum is attained at (0, x) thatthe test functions ϕ ∈ C1,2([0, T )× U) and h ∈ C1,2([0, T )× R).

Proof. Suppose that u is a viscosity subsolution of (3.64) in the sense ofDefinition 3.35 and let u(s, y)−ϕ(s, y)−h(s, |y|) = u(s, y)−ψ(s, y) has a one sidedlocal maximum at (t, x) over [t, t+ ) × B(x). Arguing as in the proof of Lemma3.37, we can assume that the one-sided local maximum is strict. Then for big nthere exist by Corollary 3.26 (which we can apply because any closed convex subsetof H is B-closed, see Remark 3.8) an ∈ R, pn ∈ H, |an|+ |pn| ≤ 1/n such that thefunction

u(s, y)− ψ(s, y)− 1

n(s− t)− ans− Bpn, y

has a local (two-sided) maximum at (sn, yn) ∈ (t, t + ) × B(x). Since the initiallocal maximum was strict we have that (sn, yn) → (t, x). Without loss of generalitywe can assume that is 1

n(s−t) is a test function by modifying it around s = t andthen extending it to (0, T ). Therefore we obtain using Definition 3.35 that

ψt(sn, yn) + an − 1

n(sn − t)2+ yn, A∗(Dϕ(sn, yn) +Bpn)

− F (sn, yn, u(sn, yn), Dψ(sn, yn) +Bpn, D2ψ(sn, yn)) ≥ 0

which gives us

ψt(t, x) + x,A∗Dϕ(t, x) − F (t, x, u(t, x), Dψ(t, x), D2ψ(t, x)) ≥ 0.

after letting n → +∞.

3.3.1. Bounded equations. If A = 0 there is no need to use the notion ofB-continuity. Viscosity solutions can then be defined in the same way as for finite


dimensional problems. We present the definition for the time independent problem

F (x, u,Du,D2u) = 0 in Uu(x) = f(x) on ∂U.

(3.68)

The definition for time dependent problems is similar. We call such equations“bounded” since they do not contain any unbounded terms.

Definition 3.40 A locally bounded upper semi-continuous function u on U isa viscosity subsolution of (3.68) if u ≤ f on ∂U and whenever u − ϕ has a localmaximum at a point x for a test function ϕ ∈ C2(U) then

F (x, u(x), Dϕ(x), D2ϕ(x)) ≤ 0.

A locally bounded lower semi-continuous function u on U is a viscosity supersolutionof (3.68) if u ≥ f on ∂U and whenever u−ϕ has a local minimum at a point x fora test function ϕ ∈ C2(U) then

F (x, u(x), Dϕ(x), D2ϕ(x)) ≥ 0.


Equation (3.68) and its parabolic version were studied, together with the as-sociated control problems, in [319, 321]. In [319] a stronger definition of viscositysolution was introduced, allowing for more general test functions which are notnecessarily twice Fréchet differentiable. One can also replace Definition 3.40 witha definition using second-order jets (see [101, 321]). Regularity results for boundedequations and their obstacle problems have been obtained in [319, 426]. Existenceand uniqueness results for such equations can also be found in [290].

3.4. Consistency of viscosity solutions

Consistency property of viscosity solutions, i.e. the ability to pass to limits inthe equations under minimal assumptions on the solutions, is one of the greateststrengths of the notion of viscosity solution.

Let B be an operator satisfying Hypothesis 3.31. Let An, n = 1, 2, ..., be linear,densely defined, maximal dissipative operators in H such that D(A∗) ⊂ D(A∗

n).We consider equations

ut + Anx,Du − Fn(t, x, u,Du,D2u) = 0 in (0, T )× U, (3.69)

where U is an open subset of H. We assume that viscosity sub- and super-solutionsof (3.69) are B-upper (respectively, lower) semicontinuous with the same fixed B.We note that if ϕ is a test function in Definition 3.32-(i), then

A∗nDϕ = A∗

n(I −A∗)−1(I −A∗)Dϕ,

and thus, since A∗n(I −A∗)−1 ∈ L(H), ϕ is a test function of type (i) for equation

(3.69).We have the following result.

Theorem 3.41 Let the above assumptions about An, n = 1, 2, ... be satisfied. Letun, n = 1, 2, ... be a viscosity subsolutions (respectively, supersolutions) of (3.69)(with some terminal and boundary conditions which are not essential here). Supposethat Fn : (0, T )× U × R×H × S(H) → R, n = 1, 2, ... are continuous, and

whenever x, xn ∈ D(A∗), xn → x, and A∗xn → A∗x, then A∗nxn → A∗x. (3.70)

Let un converge locally uniformly to a function u on (0, T )×U . Then u is a viscositysubsolution of

ut + Ax,Du − F−(t, x, u,Du,D2u) = 0 in (0, T )× U

3.4. CONSISTENCY OF VISCOSITY SOLUTIONS 167

(respectively, supersolution of

ut + Ax,Du − F+(t, x, u,Du,D2u) = 0 in (0, T )× U, )

where

F−(t, x, r, p,X) = limi→+∞

inf

Fn(τ, y, s, q, Y ) : n ≥ i,

|t− τ |+ |x− y|+ |r − s|+ |p− q|+ X − Y ≤ 1

i

, (3.71)

F+(t, x, r, p,X) = limi→+∞

sup

Fn(τ, y, s, q, Y ) : n ≥ i,

|t− τ |+ |x− y|+ |r − s|+ |p− q|+ X − Y ≤ 1

i

. (3.72)

Notation 3.42 We denote the right-hand side of (3.71) and (3.72) respectivelyby

lim infn→+∞∗Fn(t, x, r, p,X),

andlim supn→+∞

∗Fn(t, x, r, p,X).

Obviously

lim supn→+∞

∗(−Fn(t, x, r, p,X)) = −lim infn→+∞∗Fn(t, x, r, p,X).

Proof. We will only do the proof for the subsolution case. The function uis obviously locally bounded and B-upper semi continuous. Suppose that u(s, y)−ψ(s, y) = u(s, y) − ϕ(s, y) − h(s, |y|) has a local maximum at a point (t, x). ByLemma 3.37 the maximum can be assumed to be strict. Let D = (s, y) : |t− s| ≤δ, |x − y| ≤ δ for some δ > 0. Applying Corollary 3.26 on D we obtain, for everyn, an ∈ R, pn ∈ H such that |an|+ |pn| ≤ 1

n and such that

un(s, y)− (ψ(s, y) + ans+ Bpn, y)

has maximum over D at some point (tn, xn). Since the original maximum at (t, x)was strict and the un converge uniformly on D to u, it is easy to see that we musthave (tn, xn) → (t, x) as n → +∞. Since we have

ψt(tn, xn) + an + xn, A∗n(Dϕ(tn, xn) +Bpn)

−Fn(tn, xn, un(tn, xn), Dψ(tn, xn) +Bpn, D2ψ(tn, xn)) ≥ 0,

the claim follows passing to the lim supn→+∞

∗ in the above inequality.

The result for time independent equations is similar. In finite dimensionalspaces one can pass to weaker limits with viscosity solutions. In particular, themethod of half-relaxed limits of Barles-Perthame (see [30], [101] and [195]) allowsto conclude that for a family of subsolutions (respectively, supersolutions) un, thefunction u+ = lim sup

n→+∞∗un is a subsolution and u− = lim inf

n→+∞ ∗un is a supersolution.

Unfortunately this is no longer true in infinite dimensions due to lack of localcompactness. The following simple example from [424] illustrates this phenomenon.Half-relaxed limits in a special case are discussed in Section 3.9.


Example 3.43 Let H be the real l2 space. Let F (p) = 1− |p|, and un(x) = xn,where x = (x1, ..., xn, ...). Then the functions un are classical (and thus viscosity)solutions of

F (Dun) = 0.

However u+ ≡ 0 and therefore F (Du+) = 1, i.e. u+ is not a subsolution ofF (Du+) = 0. To see that u+ ≡ 0 we notice that if n ≥ i and |x − y| ≤ 1/ithen

|un(y)| = |yn| ≤ |xn|+1

i→ 0 as n, i → ∞.

3.5. Comparison theorems

In this section we present comparison results for viscosity solutions. They areproved under either the weak or the strong B-condition for A (see Definitions 3.9and 3.10).

We use the notation from Section 3.2. In particular, we recall that e1, e2, ...is an orthonormal basis in H−1 made of elements of H, and for N > 2, HN =spane1, ..., eN, PN is the orthogonal projection in H−1 onto HN , and QN :=I − PN .

We will make the following assumptions about the function F : (0, T ) × U ×R×H × S(H) → R.

Hypothesis 3.44 F is uniformly continuous on bounded subsets of (0, T )×U ×R×H × S(H).

Hypothesis 3.45 There exists ν ≥ 0 such that for every (t, x, p,X) ∈ (0, T ) ×U ×H × S(H)

F (t, x, r, p,X)− F (t, x, s, p,X) ≥ ν(r − s) when r ≥ s.

Hypothesis 3.46 For every (t, x, r, p) ∈ (0, T )× U × R×H

F (t, x, r, p,X) ≥ F (t, x, r, p, Y ) when X ≤ Y.

Hypothesis 3.47 For all t ∈ (0, T ), r ∈ R, x ∈ U , p ∈ H, R > 0,

sup

|F (t, x, p,X + λBQN )− F (t, x, p,X)| :

X, |λ| ≤ R,X = P ∗NXPN

N→+∞−−−−−→ 0. (3.73)

Hypothesis 3.48 For every R > 0 there exists a modulus ωR such that, for all(t, x, y, r) ∈ (0, T ) × U × U × R such that |r|, |x|, |y| ≤ R, for any > 0, for allX,Y ∈ S(H) such that X = P ∗

NXPN , Y = P ∗NY PN for some N and satisfying

(3.54), we have

F

t, x, r,

B(x− y)

, X

− F

t, y, r,

B(x− y)

, Y

≥ −ωR

|x− y|−1

1 +

|x− y|−1

.

Hypothesis 3.49 There exist γ ∈ [0, 1] and a constant MF ≥ 0 such that

|F (t, x, r, p+ q,X + Y )− F (t, x, r, p,X) |≤ MF

(1 + |x|)|q|+ (1 + |x|γ)2Y

for all (t, x, r) ∈ (0, T )× U × R, p, q ∈ H, X,Y ∈ S(H).

3.5. COMPARISON THEOREMS 169

Hypothesis 3.45 guarantees that F is nondecreasing in the zeroth order variable.If ν > 0 we say that F is proper. Hypothesis 3.46 ensures that F is monotone inthe second order variable. When it is satisfied we say that F (and therefore theequation) is degenerate elliptic/parabolic.

3.5.1. Degenerate parabolic equations. In Theorem 3.50, the boundaryand terminal value functions f and g are not explicitly mentioned since they arenot relevant. However the subsolution function u and the supersolution functionv are defined on (0, T ] × U and conditions (3.74) and (3.75) describe their jointbehavior along the boundary ∂U and the terminal value T .

Theorem 3.50 (Comparison under weak B-condition) Let U ⊂ H be openand U be B-closed (see Definition 3.7). Let (3.2) hold and let F satisfy Hypotheses3.44, 3.46, 3.47, 3.48 and 3.49 and 3.45 with ν = 0. Let u be a viscosity subsolutionof (3.64), and v be a viscosity supersolution of (3.64). Suppose that for every R > 0there exists a modulus ωR such that

(u(t, x)− v(s, y))+ + (u(t, y)− v(s, x))+ ≤ ωR(|t− s|+ |x− y|−1) (3.74)

for t, s ∈ (0, T ), x ∈ ∂U, y ∈ U, |x|, |y| ≤ R, and that

limR→+∞

limr→0

limη→0

supu(t, x)− v(s, y) : |x− y|−1 < r

x, y ∈ U ∩BR, T − η ≤ t, s ≤ T≤ 0. (3.75)

Moreover suppose that there exist constants C, a > 0 such that

u,−v ≤ Cea|x|2−2γ

(t, x) ∈ (0, T )×H, if γ ∈ [0, 1), (3.76)

andu,−v ≤ Cea(log(1+|x|))2 (t, x) ∈ (0, T )×H, if γ = 1. (3.77)

Then for every κ > 0

limR→+∞

limr→0

limη→0

supu(t, x)− v(s, y) : |x− y|−1 < r, |t− s| < η

x, y ∈ U ∩BR,κ < t, s ≤ T≤ 0. (3.78)

In particular u ≤ v.

Remark 3.51 It is easy to see that (3.78) implies that for every κ > 0 and R > 0there exists a modulus ωR such that

u(t, x)− v(s, y) ≤ ωκ,R(|x− y|−1+ |t− s|) for x, y ∈ U ∩BR,κ < t, s ≤ T. (3.79)

Proof of Theorem 3.50. Let 0 < τ < 1 be such that a < 1/√τ . Addi-

tional condition on τ will be given later. Set T1 = T − τ . The proof will be donein several steps. We will first show (3.78) for T1 ≤ s, t ≤ T and then reapply theprocedure to intervals [T − 3τ/2, T − τ/2], [T − 4τ/2, T − 2τ/2],.... We will first dothe proof for the case γ = 1.

We argue by contradiction and assume that (3.78) is not true. Then there isκ > 0 such that

m = limR→+∞

limr→0

limη→0


x, y ∈ U ∩BR, T1 + κ ≤ t, s ≤ T> 0.


We note that m can be +∞. Denote

mδ := limr→0

limη→0

sup

u(t, x)− v(s, y)− δe

(log(2+|x|2))2√

t−T1 − δe(log(2+|y|

2))2√s−T1 :

|x− y|−1 < r, |t− s| < η, x, y ∈ U, T1 ≤ t, s ≤ T

,

mδ, := limη→0

sup


(log(2+|x|2))2√


2))2√s−T1

− |x− y|2−1

2: |t− s| < η, x, y ∈ U, T1 ≤ t, s ≤ T

,

mδ,,β := sup


(log(2+|x|2))2√


2))2√s−T1

− |x− y|2−1

2− (t− s)2

2β: x, y ∈ U, T1 ≤ t, s ≤ T

.

It is very easy to see thatm ≤ lim

δ→0mδ, (3.80)

mδ = lim→0

mδ,, (3.81)

mδ, = limβ→0

mδ,,β . (3.82)

Setting u(t, x) = −∞ if x ∈ U and v(t, x) = +∞ if x ∈ U we can consider u andv to be defined on (T1, T ] × H. Since U is B-closed such extended u is B-uppersemi-continuous on (T1, T ]×H and v is B-lower semi-continuous on (T1, T ]×H.

Define

Ψ(t, s, x, y) = u(t, x)−v(s, y)−δe(log(2+|x|

2))2√t−T1 −δe

(log(2+|y|2))2√

s−T1 − |x− y|2−1

2− (t− s)2

2β.

We notice that by (3.77) we obtain for instanceΨ(t, s, x, y) ≤ −(|x|2 + |y|2) if |x|+ |y| ≥ Kδ (3.83)

for some Kδ > 0. Therefore, using Corollary 3.26, for every n ≥ 1 we can findan, bn ∈ R, and pn, qn ∈ H such that |an|+ |bn|+ |pn|+ |qn| ≤ 1

n such thatΨ(t, s, x, y) + ant+ bns+ Bpn, x+ Bqn, y

achieves a strict global maximum at some point (t, s, x, y) ∈ [T1, T ]×[T1, T ]×H×H.(The maximum is initially strict in the | · |−2 norm but since the radial functionsare strictly increasing the maximum is in fact strict in the | · | norm.) Moreover fora fixed δ

|x|, |y|, |u(t, x)|, |v(s, y)| ≤ Rδ (3.84)for some Rδ independently of ,β, n. Obviously (t, s, x, y) ∈ (T1, T ]×(T1, T ]×U×U .It follows from (3.83) that

mδ,,β ≤ Ψ(t, s, x, y) +Cδ

n(3.85)

for some constant Cδ > 0. Therefore it follows that

mδ,,β +|t− s|24β

≤ Ψ(t, s, x, y) +|t− s|24β

+Cδ

n≤ mδ,,2β +

Cδ

n(3.86)

andmδ,,β +

|x− y|2−1

4+

|t− s|24β

≤ mδ,2,2β +Cδ

n. (3.87)


Inequalities (3.86) and (3.82) imply

limβ→0

lim supn→+∞

|t− s|2β

= 0 for every δ, > 0, (3.88)

and then (3.88), (3.87) and (3.81) imply

lim→0

lim supβ→0

lim supn→∞

|x− y|2−1

= 0 for every δ > 0. (3.89)

In particular it follows from (3.80), (3.81), (3.82), (3.85), (3.88) and (3.89) thatthere exists δ0 > 0 such that for all δ < δ0

lim inf→0

lim infβ→0

lim infn→∞

(u(t, x)− v(s, y)) ≥ m = minm2, 1. (3.90)

Conditions (3.74), (3.75), together with (3.88) and (3.89), imply that if δ, ,βare small enough and n is sufficiently big we must have (t, s, x, y) ∈ (T1, T ) ×(T1, T )× U × U .

We now have for N > 2

|x− y|2−1 = |PN (x− y)|2−1 + |QN (x− y)|2−1

and

|QN (x− y)|2−1 ≤ 2BQN (x− y), x− y+ 2|QN (x− x)|2−1

+ 2|QN (y − y)|2−1 − |QN (x− y)|2−1

with equality at x, y. Therefore, defining

u1(t, x) = u(t, x)− δe(log(2+|x|

2))2√t−T1 − BQN (x− y), x

− |QN (x− x)|2−1

+|QN (x− y)|2−1

2+ ant+ Bpn, x

and

v1(s, y) = v(s, y) + δe(log(2+|y|

2))2√s−T1 − BQN (x− y), y

+

|QN (y − y)|2−1

− bns− Bqn, y.

we see that

u1(t, x)− v1(s, y)−1

2|PN (x− y)|2−1 −

1

2β|t− s|2

has a strict global maximum at (t, s, x, y) over [T1, T ]×[T1, T ]×H×H. We can there-fore apply Corollary 3.29 to obtain test functions ϕk,ψk and points (tk, xk), (sk, yk)such that u1(t, x) − ϕk(t, x) has a maximum at (tk, xk), v1(s, y) − ψk(s, y) has aminimum at (sk, yk), and such that (3.56), (3.57) are satisfied for u1, v1 respectively.In particular (tk, xk), (sk, yk) ∈ (T1, T )× U for big k.

Define

ϕ(t, x) = ϕk(t, x) +BQN (x− y), x

+

|QN (x− x)|2−1

− |QN (x− y)|2−1

2− ant− Bpn, x, (3.91)

and

h(t, |x|) = δe(log(2+|x|

2))2√t−T1 .


Since u is a viscosity subsolution of (3.64) on (T1, T ] × U , using the definition ofviscosity subsolution we have

ϕt(tk, xk) + ht(tk, |xk|) + xk, A∗Dϕ(tk, xk)

− F (tk, xk, u(tk, xk), Dϕ(tk, xk) +Dh(tk, |xk|), D2ϕ(tk, xk) +D2h(tk, |xk|)) ≥ 0.(3.92)

Letting k → +∞ in (3.92) and using (3.56) yields

t− s

β− an + ht(t, |x|) +

x, A∗

B(x− y)

−Bpn

−F

t, x, u(t, x),

B(x− y)

−Bpn +Dh(t, |x|), XN +

2BQN

+D2h(t, |x|)

≥ 0.

(3.93)

We now compute

ht(t, |x|) = −δ(log(2 + |x|2))2

2(t− T1)32

e(log(2+|x|

2))2√t−T1 ,

Dh(t, |x|) = e(log(2+|x|

2))2√t−T1

4δ log(2 + |x|2)√t− T1

x

2 + |x|2 ,

and

D2h(t, |x|) = 4δ√t− T1

e(log(2+|x|

2))2√t−T1

4(log(2 + |x|2))2√t− T1(2 + |x|2)2

+2

(2 + |x|2)2 − 2 log(2 + |x|2)(2 + |x|2)2

x⊗ x+

log(2 + |x|2)2 + |x|2 I

.

We have

|Dh(t, |x|)|1 + |x| + D2h(t, |x|) ≤ C1δ(log(2 + |x|2))2

(t− T1)(1 + |x|)2 e(log(2+|x|

2))2√t−T1

for some absolute constant C1. Therefore we obtain from Hypothesis 3.49F

t, x, u(t, x),

B(x− y)

−Bpn +Dh(t, |x|), XN +

2BQN

+D2h(t, |x|)

− F

t, x, u(t, x),

B(x− y)

−Bpn, XN +

2BQN

≤ MF ((1 + |x|)|Dh(t, |x|)|+ (1 + |x|)2D2h(t, |x|))

≤ MFC1δ(log(2 + |x|2))2t− T1

e(log(2+|x|

2))2√t−T1 ≤ −1

2ht(t, |x|)

if

τ ≤ 1

(4MFC1)2. (3.94)

Hence if (3.94) is satisfied, using that 12ht(t, |x|) ≤ −Cτδ for some Cτ > 0, it follows

from (3.93) that

− Cτδ +t− s

β− an +

x, A∗

B(x− y)

−Bpn

− F

t, x, u(t, x),

B(x− y)

−Bpn, XN +

2BQN

≥ 0. (3.95)


Arguing similarly we obtain from the fact that v1(s, y) − ψk(s, y) has a minimumat (sk, yk) that

t− s

β+ bn +

y, A∗

B(x− y)

+Bqn

− F

s, y, v(s, y),

B(x− y)

+Bqn, YN − 2BQN

≤ 0. (3.96)

Therefore subtracting (3.95) from (3.96) and using (3.84), Hypothesis 3.47 yield

Cτδ −x− y,

A∗B(x− y)

+ F

t, x, u(t, x),

B(x− y)

, XN

− F

s, y, v(s, y),

B(x− y)

, YN

≤ ω1(δ, ,β;n,N), (3.97)

where limn→+∞ limN→+∞ ω1(δ, ,β;n,N) = 0 for fixed δ, ,β. Now Hypothesis3.45, (3.88), (3.90) and (3.97) imply

Cτδ −x− y,

A∗B(x− y)

+ F

t, x, u(t, x),

B(x− y)

, XN

− F

t, y, u(t, x),

B(x− y)

, YN

≤ ω2(δ; ,β, n,N), (3.98)

where lim sup→0 lim supβ→0 lim supn→+∞ lim supN→+∞ ω2(δ; ,β, n,N) = 0 forsufficiently small δ. We recall that XN , YN satisfy (3.54). We can now use (3.2),Hypothesis 3.48, (3.84) and then invoke (3.89) to get

Cτδ ≤ c0|x− y|2−1

+ ωRδ

|x− y|−1

1 +

|x− y|−1

+ ω2(δ; ,β, n,N) ≤ ω3(δ; ,β, n,N),

where lim sup→0 lim supβ→0 lim supn→+∞ lim supN→+∞ ω3(δ; ,β, n,N) = 0 forsufficiently small δ. This yields a contradiction for small δ.

Thus we obtain that m ≤ 0 and this allows us to reapply the procedure tointervals [T − 3τ/2, T − τ/2], [T − 4τ/2, T − 2τ/2], ..., [0, T − kτ/2], where k is suchthat T − kτ/2 > 0 ≥ T − (k + 2)τ/2.

For γ ∈ [0, 1) the proof is the same but we have to replace the functions

δe(log(2+|x|

2))2√t−T1 and δe

(log(2+|y|2))2√

s−T1

by

δe(1+|x|

2)1−γ√t−T1 and δe

(1+|y|2)1−γ√

s−T1 ,

respectively.

The assumptions of the comparison theorems can be weakened if we replacethe weak B condition by the strong B-condition i.e. if we replace (3.2) by (3.3). Inthis case we will use the following assumption instead of Hypothesis 3.48.

Hypothesis 3.52 For every R > 0 there exists a modulus ωR such that, for all(t, x, y, r) ∈ (0, T ) × U × U × R such that |r|, |x|, |y| ≤ R, for any > 0, for all


X,Y ∈ S(H) such that X = P ∗NXPN , Y = P ∗

NY PN for some N and satisfying(3.54), we have

F

t, x, r,

B(x− y)

, X

− F

t, y, r,

B(x− y)

, Y

≥ −ωR

|x− y|

1 +

|x− y|−1

.

Hypothesis 3.53 For every R > 0 there exists a modulus ωR such that|g(x)− g(y)| ≤ ωR(|x− y|) if x, y ∈ H, |x|, |y| ≤ R.

Theorem 3.54 (Comparison under B-strong condition) Let U = H. Let(3.3) hold and let F satisfy Hypotheses 3.44, 3.46, 3.47, 3.52, 3.49 and 3.45 withν = 0. Let g satisfy Hypothesis 3.53. Let u be a viscosity subsolution of (3.64) in(0, T ]×H, and v be a viscosity supersolution of (3.64) in (0, T ]×H. Suppose that

limt→T

(u(t, x)− g(e(T−t)Ax))+ + (v(t, x)− g(e(T−t)Ax))−

= 0 (3.99)

uniformly on bounded subsets of H and that either of (3.76) or (3.77) is satisfied.Then for every 0 < µ < T

mµ = limR→+∞

limr→0

limη→0


x, y ∈ BR, µ < t, s ≤ T − µ≤ 0. (3.100)


Proof. Let us again assume that γ = 1 and that (3.76) is satisfied. As inthe proof of Theorem 3.50 we take 0 < τ ≤ min(1, 1/

√a, 1/(4MFC1)2, where C1 is

the constant appearing in that proof, and we set T1 = T − τ . Define for 0 < µ < τ

mµ = limR→+∞

limr→0

limη→0


x, y ∈ BR, T1 + µ ≤ t, s ≤ T − µ.

If there is µ0 such that mµ0 > m > 0 then mµ > m > 0 for all 0 < µ < µ0. Defining

Ψ(t, s, x, y) = u(t, x)−v(s, y)− δe(log(2+|x|

2))2√t−T1 − δe

(log(2+|y|2))2√

s−T1 − |x− y|2−1

2− (t− s)2

2β

we again have that for every n ≥ 1 there exist an, bn ∈ R, pn, qn ∈ H such that|an|+ |bn|+ |pn|+ |qn| ≤ 1

n and thatΨ(t, s, x, y) + ant+ bns+ Bpn, x+ Bqn, y

achieves a strict global maximum over [T1, T−µ]×[T1, T−µ]×H×H at some point(tµ, sµ, xµ, yµ) ∈ (T1, T − µ]× (T1, T − µ]×H ×H, and that (3.84), (3.88), (3.89)hold (note that the constant Rδ in (3.84) is independent of µ). Moreover, since forµ < µ0, Ψ(tµ, sµ, xµ, yµ) ≥ Ψ(tµ0 , sµ0 , xµ0 , yµ0)− Cδ

n for some Cδ > 0 independentof µ it is easy to see that for every 0 < µ < µ0

lim inf→0

lim infβ→0

lim infn→+∞

(u(tµ, xµ)− v(sµ, yµ))

= lim inf→0

lim infβ→0

lim infn→+∞

Ψ(tµ, sµ, xµ, yµ)

≥ lim inf→0

lim infβ→0

lim infn→+∞

Ψ(tµ0 , sµ0 , xµ0 , yµ0)

= lim inf→0

lim infβ→0

lim infn→+∞

(u(tµ0 , xµ0)− v(sµ0 , yµ0)) ≥ m (3.101)

if δ < δ0 for some δ0 > 0 (depending only on µ0).


Thus we obtain that mµ ≤ 0 for all µ < τ and this allows us to reapply theprocedure to intervals [T − 3τ/2, T − τ/2], [T − 4τ/2, T − 2τ/2],... directly as in theproof of Theorem 3.50 since we now have

limR→+∞

limr→0

limη→0

supu(t, x)− v(s, y) : |x− y|−1 < r

x, y ∈ BR, T − τ/2− η ≤ t, s ≤ T − τ/2≤ 0. (3.102)

For γ ∈ [0, 1) the proof again uses the same modifications as indicated in theproof of Theorem 3.50.

3.5.2. Degenerate elliptic equations. In this subsection we consider thedegenerate elliptic case. We first introduce a slightly different version of Hypothesis3.49.

Hypothesis 3.55 For γ ∈ [0, 1] there exist MF , NF ≥ 0 such that|F (x, r, p+ q,X + Y )− F (x, r, p,X) |

≤ MF (1 + |x|γ)|q|+NF (1 + |x|γ)2Y for all (x, r) ∈ U × R, p, q ∈ H, X,Y ∈ S(H).

Theorem 3.56 (Comparison under B-weak condition) Let U ⊂ H be openand U be B-closed. Let (3.2) hold and let F satisfy Hypotheses 3.44, 3.46, 3.47,3.48, 3.55 and 3.45 with ν > 0. Let u and v be, respectively, a viscosity subsolutionand a viscosity supersolution of (3.58). Suppose that for every R > 0 there exists amodulus ωR such that

(u(x)− v(y))+ + (u(y)− v(x))+ ≤ ωR(|x− y|−1) (3.103)

for x ∈ ∂U , y ∈ U , |x|, |y| ≤ R. Moreover suppose that there exist constantsC, a > 0 such that one of the following conditions is satisfied

1. γ ∈ (0, 1), ∃ k ≥ 0 s.t. u,−v ≤ C(1 + |x|k) ∀x ∈ H, (3.104)

2. γ = 0, 2MFa+ 4NF (a+ a2) < ν, and u,−v ≤ Cea|x| ∀x ∈ H, (3.105)3. γ = 1, ∃k ≥ 0 s.t. MF k +NF k(k − 1) < ν if k ≥ 2,

k(MF +NF ) < ν if k < 2, and u,−v ≤ C(1 + |x|k) ∀x ∈ H. (3.106)Then

m = limR→+∞

limr→0

supu(x)− v(y) : |x− y|−1 < r, x, y ∈ U ∩BR

≤ 0. (3.107)


Proof. We will first do the proof in the case γ ∈ (0, 1). We argue bycontradiction and assume that m > 0. Let k > k, k ≥ 2. Denote

mδ := limr→0

supu(x)− v(y)− δ|x|k − δ|y|k : |x− y|−1 < r, x, y ∈ U

,

mδ, := sup

u(x)− v(y)− δ|x|k − δ|y|k − |x− y|2−1

2: x, y ∈ U

.

As in the proof of Theorem 3.50 it is easy to see thatm = lim

δ→0mδ, (3.108)

mδ = lim→0

mδ,. (3.109)

Again, setting u(x) = −∞ if x ∈ U and v(x) = +∞ if x ∈ U we can consideru and v to be defined on H. Since U is B-closed such extended u is B-uppersemi-continuous on (0, T ]×H and v is B-lower semi-continuous on (0, T ]×H.


Define

Ψ(x, y) = u(x)− v(y)− δ|x|k − δ|y|k − |x− y|2−1

2.

By (3.104) we can apply Corollary 3.26 to produce for every n ≥ 1 elements pn, qn ∈H such that |pn|+ |qn| ≤ 1

n and such that

Ψ(x, y) + Bpn, x+ Bqn, yachieves a strict global maximum over H×H at some point (x, y) ∈ U×U . Moreoverwe have

mδ, ≤ Ψ(x, y) +Cδ

nfor some constant Cδ > 0. Therefore it follows that

mδ, +|x− y|2

4≤ mδ,2 +

Cδ

n. (3.110)

Inequalities (3.110) and (3.109) imply

lim→0

lim supn→∞

|x− y|2−1

= 0 for every δ > 0. (3.111)

Condition (3.103), together with (3.111), now imply that if δ, are small enoughand n is sufficiently big we must have (x, y) ∈ U × U .

Similarly to the proof of Theorem 3.56 we now have for N > 2 that if we define

u1(x) = u(x)− δ|x|k − BQN (x− y), x

− |QN (x− x)|2−1

+|QN (x− y)|2−1

2+ Bpn, x

and

v1(y) = v(y) + δ|y|k − BQN (x− y), y

+|QN (y − y)|2−1

− Bqn, y.

thenu1(x)− v1(y)−

1

2|PN (x− y)|2−1

has a strict global maximum at (x, y) over H×H. We can therefore apply Corollary3.28 to obtain test functions ϕk,ψk and points xk, yk such that u1(x)− ϕk(x) hasa maximum at xk, v1(y)−ψk(y) has a minimum at yk, and such that (3.52), (3.53)are satisfied for u1, v1 respectively. In particular xk, yk ∈ U for big k.

Therefore, since u is a viscosity subsolution of (3.58) in U , using the definitionof viscosity subsolution, letting k → +∞ and using (3.52) we obtain

−x, A∗

B(x− y)

−Bpn

+ F (x, u(x), pn,δ,, Xn,δ,) ≤ 0 (3.112)

where we denoted

pn,δ, =B(x− y)

−Bpn + δk|x|k−2x,

andXn,δ, = XN +

2BQN

+ δk|x|k−2((k − 2)

x⊗ x

|x|2 + I)).

Hence we obtain from Hypothesis 3.55 that

−x, A∗

B(x− y)

−Bpn

+ F

x, u(x),

B(x− y)

−Bpn, XN +

2BQN

− δMF

(1 + |x|γ)k|x|k−1 + (1 + |x|γ)2k(k − 1)|x|k−2

≤ 0. (3.113)


Arguing similarly we obtain that

−y, A∗

B(x− y)

+Bqn

+ F

y, v(y),

B(x− y)

+Bqn, XN − 2BQN

+ δMF ((1 + |y|γ)k|y|k−1 + (1 + |y|γ)2k(k − 1)|y|k−2) ≥ 0. (3.114)

Therefore subtracting (3.114) from (3.113) and using Hypothesis 3.47 yield

−x− y,

A∗B(x− y)

+ F

x, u(x),

B(x− y)

, XN

− F

y, v(y),

B(x− y)

, YN

− Cδ1 + |x|k−1+γ + |y|k−1+γ

≤ ω1(δ, ;n,N), (3.115)

for some C = C(MF , k, γ), where limn→+∞ limN→+∞ ω1(δ, ;n,N) = 0 for fixedδ, .

It now follows from (3.108), (3.109), (3.110) that

lim infδ→0

lim inf→0

lim infn→+∞

(u(x)− v(y)− δ|x|k − δ|y|k) > m = minm2, 1> 0. (3.116)

This, Hypothesis 3.45 and (3.115) imply

−x− y,

A∗B(x− y)

+ ν(u(x)− v(y))

+ F

x, v(y),

B(x− y)

, XN

− F

y, v(y),

B(x− y)

, YN

− Cδ1 + |x|k−1+γ + |y|k−1+γ

≤ ω1(δ, ;n,N), (3.117)

We recall that XN , YN satisfy (3.54). We can now use (3.2), Hypothesis 3.48,and the fact that |x|, |y|, |u(x)|, |v(y)| ≤ Rδ for some Rδ independent of , n to get

ν(u(x)− v(y))− Cδ(1 + |x|k−1+γ + |y|k−1+γ)

≤ c0|x− y|2−1

+ ωRδ

|x− y|−1

1 +

|x− y|−1

+ ω1(δ, ;n,N)

≤ ω2(δ; , n,N),

(3.118)

where lim sup→0 lim supn→+∞ lim supN→+∞ ω2(δ; , n,N) = 0 for sufficiently smallδ. Therefore we have from (3.116) and (3.118) that

m ≤ −νδ(|x|k + |y|k) + Cδ(1 + |x|k−1+γ + |y|k−1+γ) + ω3(δ, , n,N), (3.119)

where lim supδ→0 lim sup→0 lim supn→+∞ lim supN→+∞ ω3(δ, , n,N) = 0. Since

maxr≥0

−νδrk + Cδrk−1+γ

≤ C1δ,

taking lim supδ→0 lim sup→0 lim supn→+∞ lim supN→+∞ in (3.119), we conclude

νm ≤ 0,

which is a contradiction unless m ≤ 0.For γ = 0 the proof is almost the same. We replace the functions

δ|x|k and δ|x|k

by

δeb√

1+|x|2 and δeb√

1+|y|2 ,


respectively, where b > a is such that ν > 2MF b + 4NF (b + b2). We then obtain,in place of (3.118),

ν(u(x)− v(y))− δ(2MF b+ 4NF (b+ b2))eb√

1+|x|2 + eb√

1+|y|2≤ ω2(δ; , n,N)

which, using ν > 2MF b+ 4NF (b+ b2) and the fact that now

lim infδ→0

lim inf→0

lim infn→+∞

u(x)− v(y)− δeb

√1+|x|2 − δeb

√1+|y|2

> m > 0,

produces againνm ≤ ω4(δ, , n,N), (3.120)

where lim supδ→0 lim sup→0 lim supn→+∞ lim supN→+∞ ω4(δ, , n,N) = 0.For γ = 1 the proof is also very similar. Let k ≥ 2. We take k1 > k > k such

that ν > MF k1 + NF k1(k1 − 1) and replace the functions δ|x|k and δ|x|k in thedefinition of mδ respectively by h(x) = δ(1 + |x|2) k

2 and h(y) = δ(1 + |y|2) k

2 . It iseasy to check that

|Dh(x)| ≤ δk(1 + |x|2) k

2−1|x|, |D2h(x)| ≤ δk(k − 1)(1 + |x|2) k

2−1

and so there exists r > 0 such that

(1 + |x|)|Dh(x)| ≤ δk1(1 + |x|2) k

2 if |x| ≥ r,

(1 + |x|)2D2h(x) ≤ δk1(k1 − 1)(1 + |x|2) k

2 if |x| ≥ r,

If we now repeat the arguments of the proof and use the above estimates we obtain,in place of (3.118),

ν(u(x)− v(y))− δ(MF k1 +NF k1(k1 − 1))(1 + |x|2) k

2 + (1 + |y|2) k

2

≤ ω2(δ; , n,N) + ω3(δ) (3.121)

for some modulus ω3 which depends on r. The result now follows upon noticingthat

lim infδ→0

lim inf→0

lim infn→+∞

(u(x)− v(y)− δ(1 + |x|2) k

2 + δ(1 + |y|2) k

2 ) > m > 0

and using ν > MF k1 +NF k1(k1 − 1).If k < 2 we proceed the same as for k ≥ 2. We take k1 > k > k such that

ν > MF k1 +NF k1 and as before take h(x) = δ(1 + |x|2) k

2 and h(y) = δ(1 + |y|2) k

2 .However now

D2h(x) = δk(k − 2)(1 + |x|2) k

2−2x⊗ x+ δk(1 + |x|2) k

2−1I ≤ δk(1 + |x|2) k

2−1I

Thus when we plug the derivatives of h into the equation in the proof of comparison,using Hypothesis 3.46 we can replace D2h(x) by δk(1 + |x|2) k

2−1I and also dosimilarly for D2h(y). The rest of the arguments are the same. Remark 3.57 The conditions in (3.104)-(3.106) may not be optimal for someequations due to a rather general assumption Hypothesis 3.55 and the way it iswritten. However they are optimal in some cases. Consider a simple first orderequation u − xu = 0 in R which has two obvious classical solutions u1 ≡ 0 andu2(x) = x, and the second order equation 2u−x2u = 0 which has solutions u1 ≡ 0and u2(x) = x2. For u − xu = 0, (3.106) produces k < 1, and for 2u − x2u = 0we obtain k < 2. Equation u − µu = 0, µ > 0, has two classical solutions u1 ≡ 0and u2(x) = ex/µ. Notice that here MF = µ/2 and (3.105) gives a < 1/µ.

Theorem 3.58 (Comparison under B-strong condition) The conclusions ofTheorem 3.56 hold if (3.2) is replaced by (3.3) and Hypothesis 3.48 is replaced byHypothesis 3.52.


Proof. The proof is exactly the same as the proof of Theorem 3.56 with onemodification. Using the notation of this proof, instead of (3.118) (for γ ∈ (0, 1)),by (3.3) and Hypothesis 3.52 we now have

ν(u(x)− v(y))− Cδ(1 + |x|k−1+γ + |y|k−1+γ) ≤ c0|x− y|2−1

− |x− y|2

+ ωRδ

|x− y|

1 +

|x− y|−1

+ ω1(δ, ;n,N). (3.122)

If ωRδ(r) ≤ νm/4 +Kδs for some Kδ > 0 we obtain

ωRδ

|x− y|

1 +

|x− y|−1

≤ νm

4+Kδ|x− y|

1 +

|x− y|−1

≤ νm

2+

|x− y|2

+ Kδ|x− y|2−1

(3.123)

for some Kδ > 0 and small enough . Therefore putting this in (3.122) and applying(3.111) yields

ν(u(x)− v(y))− Cδ(1 + |x|k−1+γ + |y|k−1+γ) ≤ νm

2+ ω2(δ; , n,N), (3.124)

where lim sup→0 lim supn→+∞ lim supN→+∞ ω2(δ; , n,N) = 0 for sufficiently smallδ. This allows us to continue and conclude the proof using the same arguments asthose used in the proof of Theorem 3.56. Other cases are done similarly.

Remark 3.59 We remark that all results of this section extend to equations ofthe form

ut + infα∈A

supβ∈B

Aα,βx,Du − Fα,β(t, x, u,Du,D2u)

= 0

andinfα∈A

supβ∈B

−Aα,βx,Du+ Fα,β(t, x, u,Du,D2u)

= 0,

where A,B are arbitrary sets, provided that all assumptions are satisfied byAα,β , Fα,β , uniformly in α and β. Since the definition of viscosity solution de-pended on the operators A and B, we have to assume that there exist a linear,densely, defined maximal dissipative operator A in H such that D(A∗) ⊂ D(A∗

α,β)for all α,β, and a bounded, strictly positive, self-adjoint operator B on H such thatA∗B is bounded. To ensure the uniformity of test functions, they are now definedby Definition 3.32 for A, and the notion of B-continuity is defined using our fixedB which works for all Aα,β . All operators Aα,β must then satisfy either the weakor strong B-condition with this B and a constant c0 independent of α and β.

3.6. Existence of solutions: Value function

In this section we investigate existence of viscosity solutions for Hamilton-Jacobi-Bellman equations associated with stochastic optimal control problems. Insuch cases the Hamiltonians F in equations (3.58) and (3.64) are convex/concave inu,Du,D2u. We show that, under suitable hypotheses, the unique viscosity solutionof (3.64) (respectively, (3.58)) is the value function of the associated finite horizon(respectively, infinite horizon) optimal control problem. A key ingredient in theproof will be the use of the Dynamic Programming Principle (Theorem 2.24). Werecall briefly the weak formulation of a stochastic optimal control problem that hasbeen introduced in Chapter 2.

We fix a final time 0 < T ≤ +∞, a Polish space Λ (the control space), a real,separable Hilbert space Ξ (the space of the noise) and Q ∈ L+

1 (Ξ).

3.6. EXISTENCE OF SOLUTIONS: VALUE FUNCTION 181

Following Definition 2.7, for t ∈ [0, T ), we say that the 5-tuple ν :=Ων ,F ν ,F ν,t

s ,Pν ,W νQ

is a reference probability space if:

(i) (Ων ,F ν ,Pν) is a complete probability space.(ii) W ν

Q = W νQ(s) : t ≤ s ≤ T is a Ξ-valued Q-Wiener process on

(Ων ,F ν ,Pν) (with W νQ(t) = 0, Pν a.s.).

(iii) The filtration F ν,ts = σF ν,t,0

s ,N, where F ν,t,0s = σW ν

Q(τ) : t ≤ τ ≤s and N are the Pν-null sets in F ν .

We say that a process a(·) is an admissible control on [t, T ] (respectivelyon [t,+∞) if T = +∞) if there exists a reference probability space ν =Ων ,F ν ,F ν,t

s ,Pν ,W νQ

such that a(·) : [t, T ]×Ων → Λ (respectively a(·) : [t,+∞)×

Ων → Λ) is F ν,ts -progressively measurable. To indicate the dependence of

a(·) on the reference probability space we will write aν(·) and, with a slightabuse the notation, we will often write aν(·) to denote the whole 6-tupleΩν ,F ν ,F ν,t

s ,Pν ,W νQ, a

ν(·). We denote the set of all admissible controls aν(·)

by Ut.

The finite horizon problem: Let T < +∞. For any aν(·) ∈ Ut we consider thesystem evolving according to the following state equation

dX(s) = (AX(s) + b(s,X(s), aν(s))) ds+ σ(s,X(s), aν(s))dW ν

Q(s)X(t) = x,

(3.125)

where A is a linear, densely defined, maximal dissipative operator in H generatinga C0-semigroup of contractions etA. The functions b and σ satisfy conditions thatwill be specified below. Our hypotheses will guarantee that (3.125) admits, for anyaν(·) ∈ Ut, a unique mild solution (see Definition 1.113) denoted by X(s; t, x; aν(·)).We consider the problem of minimizing a cost functional

J(t, x; aν(·)) = Eν

T

te−

s

tc(X(τ ;t,x,aν(·)))dτ l(s,X(s; t, x; aν(·)), aν(s))ds

+ e−

T

tc(X(τ ;t,x,aν(·)))dτg(X(T ; t, x, aν))

(3.126)

over all aν(·) ∈ Ut. The value function of this minimization problem is defined asfollows:

V (t, x) := infaν(·)∈Ut

J(t, x; a(·)), (3.127)

while the associated Hamilton-Jacobi-Bellman equation is given by

vt + Ax,Dv − F (t, x, v,Dv,D2v) = 0v(T, x) = g(x),

(3.128)

where

F (t, x, r, p,X) := supa∈Λ

− 1

2Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗X

− p, b(t, x, a)+ c(x)r − l(t, x, a)

. (3.129)

Remark 3.60 We point out that if σ(t, x, a) ∈ L(Ξ, H), then the termσ(t, x, a)Q

12

σ(t, x, a)Q

12

∗

can be written in a more common and convenient form

σ(t, x, a)Qσ(t, x, a)∗.


The infinite horizon problem: We only study the case of constant discounting,i.e. we assume that c = λ > 0. However, under suitable assumptions, the results weprove could be adapted to a more general case of non-constant c. For any aν(·) ∈ U0

we consider a system described by a stochastic differential equation

dX(s) = (AX(s) + b(X(s), aν(s))) ds+ σ(X(s), aν(s))dW νQ(s)

X(0) = x.(3.130)

The mild solution of (3.130) will be denoted by X(s; 0, x; aν(·)). The infinite horizonproblem consists in minimizing a cost functional

J(x; aν(·)) = Eν

+∞

0e−λtl(X(s; 0, x; aν), aν(s))ds

. (3.131)

over all controls aν(·) ∈ U0. The value function is given by

V (x) := infaν(·)∈U0

J(x; a(·)), (3.132)

and the corresponding Hamilton-Jacobi-Bellman equation is

λv(x)− Ax,Dv+ F (x, v,Dv,D2v) = 0, (3.133)

where

F (x, r, p,X) := supa∈Λ

−1

2Tr

σ(x, a)Q

12

σ(x, a)Q

12

∗X− p, b(x, a) − l(x, a)

.

(3.134)

3.6.1. Finite horizon problem. In this subsection we prove that, undersuitable hypotheses, the value function (3.127) of the finite horizon problem isthe unique viscosity solution of (3.128). We obtain the results under two sets ofhypotheses: in the first the generator A satisfies the weak B-condition for someoperator B (see Definition 3.9), in the second it satisfies the strong B-condition(Definition 3.10) which allows to put milder assumptions on the coefficients of thestate equation and of the cost functional. Note that the words strong and weakused to describe the B-conditions have nothing to do with the strong and weakformulation of the optimal control problem (see Sections 2.1.2 and 2.1.2).

To avoid cumbersome notation we will drop the index “ν” whenever it does notcause and confusion.

Proposition 3.61 (Regularity of V under weak B-condition) Let B satisfythe weak B-condition for A (Definition 3.9) and b : [0, T ]×H ×Λ → H, σ : [0, T ]×H × Λ → L2(Ξ0, H), l : [0, T ] × H × Λ → R be continuous. Assume that b and σsatisfy (3.7), (3.9), (3.10) with γ = 1, (3.22) and (3.23), and let c be bounded frombelow. Suppose that there exist local moduli ωl(·, ·) and ω(·, ·) such that

|l(t, x, a)− l(s, y, a)| ≤ ωl(|x− y|−1 + |s− t|, R)

for all x, y ∈ B(0, R), a ∈ Λ, s, t ∈ [0, T ] (3.135)

and

|g(x)− g(y)|, |c(x)− c(y)| ≤ ω(|x− y|−1, R) for all x, y ∈ B(0, R). (3.136)

Moreover assume that there exist two nonnegative constants C,m such that

|c(x)|, |g(x)|, |l(t, x, a)| ≤ C(1 + |x|m) (3.137)

for all x ∈ H, a ∈ Λ and t ∈ [0, T ]. Then:(i) There exists a local modulus σ1(·, ·) such that

|J(t, x; a(·))− J(t, y; a(·))| ≤ σ1(|x− y|−1, R) (3.138)

for all x, y ∈ B(0, R), t ∈ [0, T ] and a(·) ∈ Ut.


(ii) There exists a nonnegative constant C, and a local modulus σ2(·, ·) suchthat

|J(t, x; a(·))|, |V (t, x)| ≤ C(1 + |x|m), (3.139)for all (t, x) ∈ [0, T ]×H and a(·) ∈ Ut. and

|V (t, x)− V (s, y)| ≤ σ2(|t− s|+ |x− y|−1, R) (3.140)for all x, y ∈ B(0, R), t, s ∈ [0, T ].

Proof. Part (i): Let L be a constant such that c(x) ≥ L for all x ∈ H. Wewill assume that L < 0. Choose x, y ∈ B(0, R), a(·) ∈ Ut and denote X(·; t, x; a(·))and X(·; t, y; a(·)) respectively by X(·) and Y (·). We have

|J(t, y; a(·))− J(t, x; a(·))| ≤ I1 + I2

:=

T

tEe

−

r

tc(X(τ))dτ l(r,X(r), a(r))− e−

r

tc(Y (τ))dτ l(r, Y (r), a(r))

dr

+E|e−

T

tc(X(τ))dτg(X(T ))− e−

T

tc(Y (τ))dτg(Y (T ))|

. (3.141)

We first consider I1.

I1 ≤ I11 + I12

:=

T

tEe−

r

tc(X(τ))dτ |l(r,X(r), a(r))− l(r, Y (r), a(r))|

dr

+

T

tE|l(r, Y (r), a(r))|

e−

r

tc(X(τ))dτ − e−

r

tc(Y (τ))dτ

dr. (3.142)

In the following we will denote M any absolute constant independent of R andof the control. Given > 0 we can find, thanks to (D.1), a positive constant K

(non-increasing in ) such that, for any s > 0, ωl(s,1 ) ≤ + Ks. Using (3.135)

and (3.137), we obtain

I11 ≤ e−TL

T

tE|l(r,X(r), a(r))− l(r, Y (r), a(r))|dr

≤ e−TL

T

t

|X(r)|< 1

and |Y (r)|< 1ωl

|X(r)− Y (r)|−1,

1

dP

+ e−TL

T

t

|X(r)|≥ 1

or |Y (r)|≥ 1M(2 + |X(r)|m + |Y (r)|m)dPdr.

≤ M

T

t(+KE|X(r)− Y (r)|−1)dr

+M

T

t

E(1 + |X(r)|2m + |Y (r)|2m)

12

P|X(r)| ≥ 1

+ P

|Y (r)| ≥ 1

12

dr.

It follows from (1.38) that we haveE( sup

t≤r≤T|X(r)|2m + sup

t≤r≤T|Y (r)|2m) ≤ CR, (3.143)

where CR is a constant independent of the control but depending on R. In partic-ular, this implies by Chebychev’s inequality, that

P

supt≤r≤T

|X(r)| ≥ 1

+ P

sup

t≤r≤T|Y (r)| ≥ 1

12

≤ γ(, R), (3.144)

for some local modulus γ. Thus, using (3.143), (3.144) and (3.24), we obtainI11 ≤ M(+K|x− y|−1) + γ1(, R) (3.145)


for some local modulus γ1. Taking the infimum of the right hand side of (3.145)over > 0 produces a local modulus (·, R) such that

I11 ≤ (|x− y|−1, R). (3.146)

To estimate I12 observe that T

tE|l(r, Y (r), a(r))|

e−

s


s

tc(Y (τ))dτ

dr

≤ T

tE|l(r, Y (r), a(r))|2

dr

12

× T

tEe−

s


s

tc(Y (τ))dτ

2dr

12

. (3.147)

We observe that for a, b ∈ R, a, b ≥ L one hase−a − e−b

≤ e−L|a− b|. Therefore,using (3.143), it follows similarly as before that, for some CR > 0 depending on Rbut independent of the choice of the control,

I12 ≤ CR

T

tE T

tω

|X(r)− Y (r)|−1,

1

dr

2

+ P

supt≤r≤T

|X(r)| ≥ 1

+ P

sup

t≤r≤T|Y (r)| ≥ 1

dr

12

.

We now use again (3.144), (3.24), and argue as for I11 to find that there exists alocal modulus (·, R) such that

I12 ≤ (|x− y|−1, R). (3.148)

The term I2 in (3.141) can be estimated similarly. Thus we obtain claim (i).

Part (ii): Estimate (3.139) follows directly from (3.137) and (1.38). Moreover,by (3.138), we have

|V (t, x)− V (t, y)| ≤ σ1(|x− y|−1, R) ∀x, y ∈ B(0, R), t ∈ [0, T ]. (3.149)

It remains to prove that

|V (t, x)− V (s, x)| ≤ σ(|t− s|, R) ∀x ∈ B(0, R), t, s ∈ [0, T ] (3.150)

for some local modulus σ.We notice that it follows from our assumptions, (3.138), (3.139), and Proposi-

tion 2.16, that the assumptions of Theorem 2.24 are satisfied and thus the dynamicprogramming principle (2.25) holds. Let now (3.150), let 0 ≤ t < s ≤ T andx ∈ B(0, R). Let X(·) be the solution of (3.125).

Using (2.25) we have, for some constant CR depending on R,

|V (s, x)− V (t, x)| ≤ supa(·)∈Ut

e−LTE s

t|l(r,X(r), a(r))|dr

+ supa(·)∈Ut

EV (s, x)− V (s,X(s))e−

s

tc(X(r))dr

≤ CR|t− s|+ e−LT supa(·)∈Ut

E |V (s, x)− V (s,X(s))|

+ supa(·)∈Ut

EV (s, x)

e−

s

tc(X(r))dr − 1

=: CR|t− s|+D1 +D2, (3.151)

where we have used (1.38), (3.137).


Since V satisfies (3.139) and (3.149), arguing as in the estimates for I11 andusing (3.27) and (3.144), we obtain for every > 0, a(·) ∈ Ut

E |V (s, x)− V (s,X(s))| ≤ Eσ1(|x−X(s)|−1,1

) + CR

P( sup

t≤r≤T|X(r)| ≥ 1

)

12

≤ + CR,|t− s| 12 + γ(, R), (3.152)

thus the same estimate holds for D1.To estimate D2, let C ≥ 0 be a constant such that c(y) ≤ C when |y| ≤ 1

.Then, for every > 0, a(·) ∈ Ut,

E|V (s, x)|

e−

s

tc(X(r))dr − 1

≤ CR max

e−|t−s|L − 1,

1− e−C|t−s|

+ P( sup

t≤r≤T|X(r)| ≥ 1

)

≤ CR maxe−|t−s|L − 1,

1− e−C|t−s|

+ γ(, R)

, (3.153)

and D2 satisfies the same estimate. Plugging (3.152) and (3.153) into (3.151) andtaking infimum over > 0 provides (3.150)

Proposition 3.62 (Regularity of V under strong B-condition) Let B satisfythe strong B-condition for A (Definition 3.10). Let b : [0, T ]×H×Λ → H, σ : [0, T ]×H ×Λ → L2(Ξ0, H), l : [0, T ]×H ×Λ → R be continuous, let b and σ satisfy (3.7),(3.9), (3.10) with γ = 1 and (3.23), and let c be bounded from below. Suppose thatthere exist local moduli ωl(·, ·) and ω(·, ·) such that

|l(t, x, a)− l(s, y, a)| ≤ ωl(|x− y|+ |s− t|, R), (3.154)

for all x, y ∈ B(0, R), a ∈ Λ, s, t ∈ [0, T ] and

|g(x)− g(y)|, |c(x)− c(y)| ≤ ω(|x− y|, R), (3.155)

for all x, y ∈ B(0, R), and that (3.137) holds.Then:

(i) The functions J and V satisfy (3.139) and there exists a local modulusσ(·, ·) such that

|J(t, x; a(·))− J(t, y; a(·))| ≤ σ(|x− y|, R) (3.156)

for all x, y ∈ B(0, R), t ∈ [0, T ], a(·) ∈ Ut.(ii) For any τ ∈ (0, T ), there exists a local modulus στ (·, ·) such that

|V (t, x)− V (s, y)| ≤ στ (|x− y|−1, R) (3.157)

for all x, y ∈ B(0, R), t ∈ [0, τ ].(iii) There exists a local modulus ρ(·, ·) such that

|V (t, x)− V (s, e(s−t)Ax)| ≤ ρ(s− t, R) (3.158)

for all x ∈ B(0, R), 0 ≤ t ≤ s ≤ T .

Proof. Obviously J and V satisfy (3.139) as in Proposition 3.61. Also(3.156) is proved exactly as (3.138) in Proposition 3.61. The only difference is that,since l, g, c are now continuous in the usual norm of H instead of the | · |−1 norm,we have to replace (3.24) by (1.40).

To show (3.157) we begin as in (3.141). The term I1 is estimated in exactlythe same way as in the proof of Proposition 3.61 using (3.29) instead of (3.24). For


the term I2 we have

I2 ≤ I21 + I22 := Ee−

T

tc(X(r))dr |g(X(T ))− g(Y (T ))|

+ E|g(Y (T ))|

e−

T

tc(X(r))dr − e−

T

tc(Y (r))dr

.

The term I22 is again standard if we use (3.29). If g satisfied (3.136) we could alsoproceed as before with the term I21 to obtain (3.160) (see Remark 3.63). Since gonly satisfies (3.155) we have to proceed slightly differently. We have, by (3.137),(3.155), (3.143), (3.144), (3.30)

I21 ≤ e−TLE |g(X(T ))− g(Y (T ))| ≤ Eω

|X(T )− Y (T )|, 1

+ e−TL

|X(T )|≥ 1

or |Y (T )|≥ 1C(2 + |X(T )|m + |Y (T )|m)dP.

≤ e−TL(+KE|X(T )− Y (T )|) + γ(, R)

≤ e−TL+ e−TL

C(T )

τ

12

|x− y|−1 + γ(, R),

where C(T ) is the constant from (3.30) and γ is a local modulus. It remains totake the infimum over all > 0

The proof of (3.158) is also very similar to the proof of (3.150). We can nowclaim that the dynamic programming principle is satisfied and thus, as in (3.159),if x ∈ B(0, R) and 0 ≤ t ≤ s ≤ T , we have

|V (t, x)− V (s, e(s−t)Ax)| ≤ supa(·)∈Ut

e−LTE s

t|l(r,X(r), a(r))|dr

+ e−LT supa(·)∈Ut

EV (s, e(s−t)Ax)− V (s,X(s))

+ supa(·)∈Ut

EV (s, e(s−t)Ax)

e−

s

tc(X(r))dr − 1

, (3.159)

where X(r) is the solution of (3.125). The first and the third term above areestimated as in (3.151) and (3.153). For the middle term we first notice that thereexists some constant CR depending on R but independent of the control such that

E|X(s)− e(s−t)Ax|2 ≤ CR(s− t).

Therefore, using (3.156),

EV (s, e(s−t)Ax)− V (s,X(s))

≤ Eσ

|X(s)− e(s−t)Ax|, 1

+ CR

P( sup

t≤r≤T|X(r)| ≥ 1

)

12

≤ + CR,|t− s| 12 + γ(, R),

which implies the claim as all the constants and the local modulus γ are independentof t and the controls.

Remark 3.63 It follows easily from the above proof that if g satisfies (3.136)instead of (3.155), then

|V (t, x)−V (s, y)| ≤ ω(|t−s|+|x−y|−1, R) ∀x, y ∈ B(0, R), t, s ∈ [0, T ] (3.160)

for some local modulus ω(·, ·).


Remark 3.64 It is clear from the proof that (3.156), and the same estimatefor V , still holds if (3.23) is replaced by (3.8) and the strong B-condition for A isreplaced by a standard requirement that A generates a C0-semigroup.

In the next lemma we provide Itô like formulae for test functions ψ = ϕ+h(t, |x|)introduced in Definition 3.32. As we remarked after this definition, even though|x| is not differentiable at 0, the function h0(t, x) := h(t, |x|) ∈ C1,2((0, T ) × H),so with a slight abuse of notation, in the following we will write h(t, x) instead ofh(t, |x|), Dh(t, x) instead of Dh0(t, x) =

x|x|

ddrh(t, r)|r=|x| (which is 0 when x = 0),

and D2h(t, x) instead of D2h0(t, x).

Lemma 3.65 Let b and σ be continuous, satisfy (3.7), (3.8), (3.9), (3.10), andlet c : H → R be continuous, bounded from below and satisfy (3.137). Consider atest function (in the sense of Definition 3.32) ψ = ϕ + h. Suppose that h satisfies(1.105) and consider the solution X(·) of (3.125) for a given control a(·) ∈ Ut.Then, for any s ∈ [t, T ],

Ee−

s

tc(X(τ))dτϕ(s,X(s))

= ϕ(t, x)

E s

te−

r

tc(X(τ))dτ

ϕt(r,X(r)) + X(r), A∗Dϕ(r,X(r))

+ b(r,X(r), a(r)), Dϕ(r,X(r)) − c(X(r))ϕ(r,X(r))

+1

2Tr

σ(r,X(r), a(r))Q

12

σ(r,X(r), a(r))Q

12

∗D2ϕ(r,X(r))

dr

(3.161)

and

Ee−

s

tc(X(τ))dτh(s,X(s))

≤ h(t, x) + E

s

te−

r

tc(X(τ))dτ

ht(r,X(r))

+ b(r,X(r), a(r)), Dh(r,X(r)) − c(X(r))h(r,X(r))

+1

2Tr

σ(r,X(r), a(r))Q

12

σ(r,X(r), a(r))Q

12

∗D2h(r,X(r))

dr

. (3.162)

Proof. We define cn(y) := min(c(y), n) and observe that, η(r) :=e−

r

tcn(X(τ))dτ is the unique solution of dηn(r) = bn(r)dr (and ηn(t) = 1), where

bn(r) = −ηn(r)cn(X(τ)) is bounded. We can thus use Propositions 1.156 and 1.157and send n → +∞ to obtain the claim. Theorem 3.66 (Existence under weak B-condition) Let the assumptions ofProposition 3.61 be satisfied, and let in addition b(·, x, a) and σ(·, x, a) be uniformlycontinuous on [0, T ], uniformly in (x, a) ∈ B(0, R) × Λ for every R > 0. Supposealso that, for every (t, x),

limN→+∞

supa∈Λ

Tr

σ(t, x, a)Q12

σ(t, x, a)Q

12

∗BQN

= 0. (3.163)

Then the value function V (t, x), defined in (3.127), is the unique viscosity solutionof (3.128) among functions in the set

S :=u : [0, T ]×H → R : |u(t, x)| ≤ C(1 + |x|k) for some k ≥ 0,

limt→T

|u(t, x)− g(x)| = 0 uniformly on bounded subsets of H.

Proof. Without loss of generality we can assume that c is positive since ifc ≥ L for L < 0 then V is a viscosity solution of (3.128) if and only if V = eL(T−t)Vis a viscosity solution of (3.128) with c replaced by c = c − L and l replaced byeL(T−t)l.


Existence: Proposition 3.61 ensures that V is B-continuous and that it belongsto S. We first prove that V is a viscosity supersolution of (3.128). Let V + ψhave a local minimum at (t, x) ∈ (0, T ) × H for a test function ψ = ϕ + h (insense of Definition 3.32). We can assume that h and its derivatives Dh, D2h, ht

have polynomial growth (see on this the discussion following Lemma 3.37), thatthe minimum is global (see Lemma 3.37), and that V (t, x) + ψ(t, x) = 0, so for all(s, y) we have V (s, y) ≥ −ψ(s, y).

By Proposition 2.16, Theorem 2.24, and Proposition 3.61, the dynamic pro-gramming principle (2.25) is satisfied. Thus for > 0 there exists a controlaν(·) ∈ Ut such that, denoting X(·) := X(·; t, x; aν(·)),

V (t, x) + 2 ≥ Eν

t+

te−

r

tc(X(τ))dτ l(r,X(r), aν(r))dr

+ e−

t+

tc(X(τ))dτV (t+ , X(t+ ))

.

This implies that

2 − ϕ(t, x)− h(t, x) ≥ Eν

t+

te−

r

tc(X(τ))dτ l(r,X(r), aν(r))dr

− e−

t+

tc(X(τ))dτϕ(t+ , X(t+ ))− e−

t+

tc(X(τ))dτh(t+ , X(t+ ))

. (3.164)

Using (3.161), (3.162) and (3.164) we find

0 ≤ +1

Eν

t+

te−

r

tc(X(τ))dτ

− l(r,X(r), aν(r))

+ ψt(r,X(r)) + b(r,X(r), aν(r)), Dψ(r,X(r))

+1

2Tr

σ(r,X(r), aν(r))Q

12

σ(r,X(r), aν(r))Q

12

∗D2ψ(r,X(r))

− c(X(r))ψ(r,X(r)) + X(r), A∗Dϕ(r,X(r))dr

. (3.165)

Now we observe that, thanks to (1.39), we can find a constant r > 0, dependingon > 0 but independent of the control aν(·), such that r

→0+−−−−→ 0, and the set

Ω1 =

ω ∈ Ων : sup

r∈[t,t+]|X(r)− x| ≤ r

,

satisfies

Pν(Ω1) → 1 as → 0. (3.166)

We set Ω2 = Ων\Ω

1. If we denote by Ψ(r) the integrand in (3.165), the assumptionsand properties of test functions imply,

|Ψ(r)| ≤ C(1 + |X(r)|N ) (3.167)


for some N ≥ 0. Thus, by (1.38), (3.166), (3.167), and the continuity of thefunctions in the integrand , we obtain

0 ≤ +1

Eν

t+

t

− l(t, x, aν(r)) + ψt(t, x) + b(t, x, aν(r)), Dψ(r,X(r))

+1

2Tr

σ(t, x, aν(r))Q

12

σ(t, x, aν(r))Q

12

∗D2ψ(t, x)

− c(x)ψ(t, x) + x,A∗Dϕ(t, x)1Ω

1dr

+ C1

t+

t(P(Ω

2)12 (E[1 + |X(r)|N ]2)

12 + γ1()

≤ 1

Eν

t+

t

− l(t, x, aν(r)) + ψt(t, x) + b(t, x, aν(r)), Dψ(r,X(r))

+1

2Tr

σ(t, x, aν(r))Q

12

σ(t, x, aν(r))Q

12

∗D2ψ(t, x)

+ c(x)V (t, x) + x,A∗Dϕ(t, x)dr

+ γ2()

≤ 1

t+

tEν

ψt(t, x) + x,A∗Dϕ(t, x)

+ F (t, x, V (t, x),−Dψ(t, x),−D2ψ(t, x))dr + γ2()

= ψt(t, x) + x,A∗Dϕ(t, x)+ F (t, x, V (t, x),−Dψ(t, x),−D2ψ(t, x)) + γ2(),

where γ1, γ2 above are such that lim→0 γi() = 0, i = 1, 2, and are independent ofthe control aν(r) and of the reference probability space ν. The claim follows afterwe let → 0.

To show the subsolution property, let V − ψ has a global maximum at(t, x), where h and its derivatives Dh, D2h, ht have polynomial growth, andV (t, x) = ψ(t, x). We choose a ∈ Λ and take a constant control a(s) ≡ a de-fined on some reference probability space, and we denote again X(·) := X(·; t, x; a).Using dynamic programming principle (2.25) we have

ψ(t, x) = V (t, x)

≤ E t+

te−

r

tc(X(τ))dτ l(r,X(r), a)dr + e−

t+

tc(X(τ))dτV (t+ , X(t+ ))

≤ E t+

te−

r

tc(X(τ))dτ l(r,X(r), a)dr + e−

t+

tc(X(τ))dτψ(t+ , X(t+ ))

and then as before we get

1

E t+

te−

r

tc(X(τ))dτ

l(r,X(r), a) + ψt(r,X(r)) + X(r), A∗Dϕ(r,X(r))

− c(X(r))ψ(r,X(r)) + b(r,X(r), a), Dϕ(r,X(r))

+1

2Tr

σ(r,X(r), a)Q

12

σ(r,X(r), a)Q

12

∗D2ψ(r,X(r))

dr

≥ 0.

The same argument as in the proof of the supersolution part now yields

l(t, x, a) + ψt(t, x) + x,A∗Dϕ(t, x) − c(x)V (t, x)

+ b(t, x, a), Dϕ(t, x)+ 1

2Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗D2ψ(t, x)

≥ 0

(3.168)


and the claim follows after we take the infa∈Λ in (3.168).

Uniqueness: To prove the uniqueness of the solution we need to show that thehypotheses of Theorem 3.50 are satisfied with the set U = H.

Hypothesis 3.44 follows from the local uniform continuity ofb(·, ·, a),σ(·, ·, a), l(·, ·, a), c(·), uniform in a ∈ Λ, (3.9), (3.10), and (3.137).Hypothesis 3.45 follows from the positivity of c. For Hypothesis 3.46 we can argueas follows: since

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗is a positive, self-adjoint, trace

class operator, it is obvious that, for X,Y ∈ S(H) with X ≤ Y ,

−Tr

σ(t, x, a)Q12

σ(t, x, a)Q

12

∗X≥ −Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗Y,

and then taking the supremum over a ∈ Λ we see that Hypothesis 3.46 is satisfied.Hypothesis 3.47 follows from (3.163) (see further comments about it after the endof the proof).

To show that Hypothesis 3.48 holds observe that, using (3.22), (3.136) and(3.135), we have, for |r|, |x|, |y| ≤ R,

F

t, x, r,

B(x− y)

, X

− F

t, y, r,

B(x− y)

, Y

≥ − supa∈Λ

l(t, x, a)− l(t, y, a) +

B(x− y)

, b(t, x, a)

−

B(x− y)

, b(t, y, a)

− r(c(x)− c(y)) +1

2Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗X

− 1

2Tr

σ(t, y, a)Q

12

σ(t, y, a)Q

12

∗Y

≥ − supa∈Λ

|l(t, x, a)− l(t, y, a)|−R supa∈Λ

|c(x)− c(y)|

− supa∈Λ

B(x− y)

, b(t, x, a)− b(t, y, a)

− supa∈Λ

1

2Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗X

− 1

2Tr

σ(t, y, a)Q

12

σ(t, y, a)Q

12

∗Y

≥ −ωl(|x− y|−1, R)− C|x− y|2−1

−Rωc(|x− y|−1, R)

− supa∈Λ

1

2Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗X

−1

2Tr

σ(t, y, a)Q

12

σ(t, y, a)Q

12

∗Y

.

To estimate the last term we use that X and Y satisfy (3.54). In particular we have

X 00 −Y

≤ 3

B −B−B B

.

Multiplying both sides of this inequality by the operator

Z =

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗ σ(t, x, a)Q

12

σ(t, y, a)Q

12

∗

σ(t, y, a)Q

12

σ(t, x, a)Q

12

∗ σ(t, y, a)Q

12

σ(t, y, a)Q

12

∗


and taking the trace preserves the inequality. This can be seen by evaluating thetrace on the basis of eigenvectors of Z as it is a compact, self-adjoint, and positiveoperator. Therefore, thanks to (3.23),

Tr

σ(t, x, a)Q12

σ(t, x, a)Q

12

∗X− Tr

σ(t, y, a)Q

12

σ(t, y, a)Q

12

∗Y

≤ 3

Tr

(σ(t, x, a)− σ(t, y, a))Q

12

(σ(t, x, a)− σ(t, y, a))Q

12

∗B

≤ C|x− y|2−1

, (3.169)

for all a ∈ Λ for some C. We thus conclude that

F

t, x, r,

B(x− y)

, X

− F

t, y, r,

B(x− y)

, Y

≥ −ωl(|x− y|−1, R)−Rωc(|x− y|−1, R)− C|x− y|2−1

(3.170)

for some constant C, and so Hypothesis 3.48 is satisfied. Hypothesis 3.49 withγ = 2 follows from (3.9) and (3.10). This concludes the proof of the uniqueness.

Let us analyze condition (3.163). Let u1, u2, ... be any orthonormal basis ofΞ. Let e1, e2, ... be an orthonormal basis in H−1 made of elements of H as inSection 3.2. Then f1, f2, ..., where fi = B

12 ei is an orthonormal basis of H.

We have

Tr

σ(t, x, a)Q12

σ(t, x, a)Q

12

∗BQN

= Tr(σ(t, x, a)Q

12 )∗BQN (σ(t, x, a)Q

12 )

= Tr(σ(t, x, a)Q

12 )∗Q∗

NBQN (σ(t, x, a)Q12 )

=∞

i=1

BQNσ(t, x, a)Q

12ui, QNσ(t, x, a)Q

12ui

=

∞

i=1

|QNσ(t, x, a)Q12ui|2−1

=∞

i=1

|B 12QNσ(t, x, a)Q

12ui|2 =

∞

i=1

|(σ(t, x, a)Q 12 )∗Q∗

NB12 fi|2Ξ

=∞

i=1

|(σ(t, x, a)Q 12 )∗Q∗

NBei|2Ξ =∞

i=N+1

|(σ(t, x, a)Q 12 )∗Bei|2Ξ

=∞

i=N+1

|(σ(t, x, a)Q 12 )∗B

12 fi|2Ξ =

∞

i=N+1

|(B 12σ(t, x, a)Q

12 )∗fi|2Ξ

(see also [286], page 33). Therefore, we have

fN (a) := Tr

σ(t, x, a)Q12

σ(t, x, a)Q

12

∗BQN

=

∞i=1 |QNσ(t, x, a)Q

12ui|2−1

=∞

i=N+1 |(B12σ(t, x, a)Q

12 )∗fi|2Ξ. (3.171)

The functions fN : Λ → R, N ≥ 1, are continuous, nonnegative, and since by(3.10),

∞

i=1

|(B 12σ(t, x, a)Q

12 )∗fi|2Ξ = Tr

σ(t, x, a)Q

12

σ(t, x, a)Q

12

∗B≤ C1,


it follows from (3.171) that for every a ∈ Λ, fN (a) ↓ 0 as N → +∞. Thus, if Λ iscompact, we must have fN (a) → 0 uniformly on Λ as N → +∞, which means that(3.163) is satisfied.

Another case when (3.163) is satisfied is when B is compact. This is an obviousconsequence of the fact that in this case BQN → 0 as N → +∞.

One can use (3.171) to obtain other criteria for (3.163) to hold. For instance itwill be satisfied if

∞

i=1

ai < +∞,

whereai := sup

a∈Λ|σ(t, x, a)Q 1

2ui|2−1,

and if for every i

limN→+∞

supa∈Λ

|QNσ(t, x, a)Q12ui|−1 = 0.

Theorem 3.67 (Existence under strong B-condition) Let the assumptions ofProposition 3.62 and (3.163) be satisfied, and let in addition b(·, x, a), σ(·, x, a) beuniformly continuous on [0, T ], uniformly for (x, a) ∈ B(0, R)×Λ for every R > 0.Then the value function V (t, x), defined in (3.127) is the unique viscosity solutionof (3.128) among functions in the set

S :=u : [0, T ]×H → R : |u(t, x)| ≤ C(1 + |x|k) for some k ≥ 0,

limt→T

|u(t, x)− g(e(T−t)Ax)| = 0 uniformly on bounded subsets of H.

Proof. The proof follows the lines of the proof for the weak case. To proveuniqueness we now use Theorem 3.54 instead of Theorem 3.50 so we need to verifyHypothesis 3.52 instead of Hypothesis 3.48. This can be done arguing as beforeusing (3.7) instead of (3.22).

Remark 3.68 B-continuity is built into the definition of viscosity solution, how-ever it is clear from the proof of existence that B-continuity of the value function isnot needed to show that it satisfies the sub- and supersolution conditions requiredby the definition of viscosity solution. Thus, if we disregard the requirement of B-continuity, we can still prove that the value function is a “viscosity solution” undermuch weaker sets of assumptions than these of Theorems 3.66 and 3.67.

Example 3.69 (Controlled stochastic wave equation) Consider a control problemfor the stochastic wave equation

∂2y∂s2 (s, ξ) = ∆y(s, ξ) + f(ξ, y(s, ξ), a(s))

+h(ξ, y(s, ξ), a(s)) ∂∂sWQ(s, ξ), s > t, ξ ∈ O,

y(s, ξ) = 0, s > t, ξ ∈ ∂O,y(t, ξ) = y0(ξ), ξ ∈ O,∂y∂t (t, ξ) = z0(ξ), ξ ∈ O,

(3.172)

where O is a bounded regular domain in Rd, y0 ∈ H10 (O), z0 ∈ L2(O), Q is an

operator in L+1 (L

2(O)) and WQ is a Q-Wiener process, Λ is a Polish space anda(·) ∈ Ut. In addition f, h : O×R×Λ → R. Suppose we want to minimize the costfunctional

I(t, y0, z0; a(·)) = E T

t

Oβ(s, y(s, ξ), a(s))dξds+

Oγ(y(T, ξ))dξ

over all a(·) ∈ Ut, where β : [0, T ]× R× Λ → R, γ : R → R.


Let ∆ξ be the Laplace operator with the domain D(∆ξ) = H2(O) ∩ H10 (O).

Then (see Section C.1 and in particular (C.11)) D((−∆ξ)12 ) = H1

0 (O). We set1

H =

H1

0 (O)×

L2(O)

equipped with the inner product

yz

,

yz

H

= (−∆ξ)1/2y, (−∆ξ)

1/2yL2(O) + z, zL2(O),

yz

,

yz

∈ H.

The operator

D(A) =

H2(O) ∩H1

0 (O)×

H10 (O)

, A =

0 I∆ 0

,

is maximal dissipative in H and A∗ = −A. Equation (3.172) can then be rewrittenas the following evolution equation

dX(s) = (AX(s) + b(X(s), a(s))) dt+ σ(X(s), a(s))dWQ(s), X(t) = x :=

y0z0

,

(3.173)in H, where

b

yz

, a

=

0

f(·, y(·), a)

, σ

yz

, a

yz

=

0

h(·, y(·), a)z

, (3.174)

WQ =

0

WQ

, Q

yz

=

0Qz

.

We consider the operator

B =

(−∆ξ)−1/2 0

0 (−∆ξ)−1/2

.

It is bounded, positive, self-adjoint on H, A∗B is bounded andA∗B

yz

,

yz

H

= 0.

Moreover we have

yz

−1

=|(−∆ξ)

1/4y|2 + |(−∆ξ)−1/4z|2

1/2.

Assume that f, h are continuous in all variables, f(ξ, ·, a), h(ξ, ·, a) are Lipschitzcontinuous with Lipschitz constant L independent of ξ, a, and f(·, 0, ·), h(·, 0, ·) arebounded. Thenb

yz

, a

− b

yz

, a

H

= |f(·, y(·), a)− f(·, y(·), a)|L2(O)

≤ L

O|y(ξ)− y(ξ)|2dξ

12

= L|y − y|L2(O)

≤ C|(−∆ξ)1/4(y − y)|L2(O) ≤ C

yz

−yz

−1

1We remark that if Tr(Q) = +∞ then the right choice of the state space for the stochasticwave equation is L

2(O) × D((−∆ξ)− 1

2 ), at least for additive noise, see [130] Example 5.8 page127.


for some constant C which follows from embeddings D((−∆ξ)1/4) → H1/2(O) →L2(O) (see Section C.1). Thus the function b satisfies (3.9) and (3.22). Unfortu-nately we need more in order for σ to satisfy (3.23) (or even (3.8)). We present onesufficient condition. Obviously other conditions are possible. Denote Ξ0 = Q1/2H.Suppose that d > 1 and (−∆ξ)(d−1)/4Q1/2 ∈ L2(L2(O)). Then we have by Propo-sition B.26

σ

yz

, a

− σ

yz

, a

L2(Ξ0,H)

=(h(·, y(·), a)− h(·, y(·), a))Q1/2

L2(L2(O))

≤(h(·, y(·), a)− h(·, y(·), a))(−∆ξ)

(1−d)/4L(L2(O))

(−∆ξ)(d−1)/4Q1/2

L2(L2(O))

≤ L sup|z|

L2(O)≤1

O|y(ξ)− y(ξ)|2|(−∆ξ)

(1−d)/4z(ξ)|2dξ1

2 (−∆ξ)(d−1)/4Q1/2

L2(L2(O))

≤ L|y− y|L2d/(d−1)(O) sup|z|

L2(O)≤1|(−∆ξ)

(1−d)/4z|L2d(O)

(−∆ξ)(d−1)/4Q1/2

L2(L2(O))

≤ C|(−∆ξ)1/4(y − y)|L2(O)

(−∆ξ)(d−1)/4Q1/2

L2(L2(O))

≤ C

yz

−yz

−1

(−∆ξ)(d−1)/4Q1/2

L2(L2(O))

, (3.175)

where we have used Sobolev embeddings H1/2(O) → L2d/(d−1)(O) and|(−∆ξ)(1−d)/4z|L2d(O) ≤ C1|z|L2(O) if d > 1 (see Section C.1). Thus σ sat-isfies (3.23). The same calculation shows that for d = 1 it is enough that(−∆ξ)αQ1/2 ∈ L2(L2(O)) for some α > 0. It is now easy to check that (3.10)is true with γ = 1.

We now define

l

s,

yz

, a

=

0

O β(s, y(ξ), a)dξ

, g

yz

=

0

O γ(y(ξ))dξ

,

and rewrite the cost functional I as

J(t, x; a(·)) = E T


.

It is easy to see by calculations similar to these for b that if β is continuous andβ(·, ·, a), γ are uniformly continuous with a modulus of continuity independent of athen

|l(s, x, a)−l(t, y, a)|+|g(x)−g(y)| ≤ ω(|x−y|−1+|s−t|), s, t ∈ [0, T ], x, y ∈ H, a ∈ Λ

for some modulus ω. If β(0, 0, ·) is bounded then l satisfies (3.137) with m = 1.

3.6.2. Improved version of dynamic programming principle. Once weknow that the value function is continuous in both variables, a stronger versionof the dynamic programming principle involving stopping times can be proved.We will only do it for the finite horizon problem discussed in the previous section,however the same result would be true for other optimal control problems, includingthese on infinite horizon.

We define the set Vt in the following way. For every a(·) ∈ Uνt for some reference

probability space ν = (Ω,F ,F ts ,P,WQ), we choose an F t

s -stopping time t ≤ τa(·) ≤T . The set Vt is the set of all such pairs

a(·), τa(·)

. We also define Vt to be the set of

these pairs in Vt for which the underlying reference probability space is standard,


i.e. Vt =a(·), τa(·)

, where a(·) ∈ Uν

t . To simplify notation we will just write(a(·), τ) instead of

a(·), τa(·)

.

Theorem 3.70 Let b : [0, T ] × H × Λ → H, σ : [0, T ] × H × Λ → L2(Ξ0, H),l : [0, T ] × H × Λ → R be continuous, let b and σ satisfy (3.7), (3.8), (3.9), and(3.10) with γ = 1. Let c be bounded from below, and let l, g, c satisfy (3.137),(3.154), (3.155). Then

V (t, x) = inf(a(·),τ)∈Vt

E τ

te−

s

tc(X(r))drl (s,X (s) , a (s)) ds+ e−

τ

tc(X(r))drV (τ, X (τ))

.

(3.176)

Proof. Without loss of generality we always assume that the Q-Wienerprocesses in the reference probability spaces have everywhere continuous paths.We recall that, by Proposition 3.62 and Remark 3.64, J satisfies (3.156) and Vsatisfies

|V (t, x)− V (s, y)| ≤ ω(|t− s|+ |x− y|, R) ∀ x, y ∈ B(0, R), t, s ∈ [0, T ]

for some local modulus ω.Let a(·) ∈ Uν

t for some standard reference probability space ν =(Ω,F ,F t

s ,P,WQ) and let τn be an F ts -stopping time which has finite number

of values, i.e.

τn =k

i=1

1Aiti

for some pairwise disjoint sets A1, ..., Ak such thatn

i=1

Ai = Ω and Ai ∈ F tti , i =

1, ..., k. It then follows from the proof of the first part of Theorem 2.24 (see (2.32))that

J(t, x; a (·)) =k

i=1

E1Ai

ti

te−

s

tc(X(r))drl (s,X (s) , a (s)) ds

+1AiE T

ti

e−

s


T

tc(X(r))drg (X (T )) |F t

ti

≥ E

k

i=1

1Ai

ti

te−

s


ti

tc(X(r))drV (ti, X (ti))

= E τn

te−

s


τn

tc(X(r))drV (τn, X (τn))

.

(3.177)

By Proposition 1.77, every stopping time can be approximated by stopping times τnwith finite number of values. Therefore, thanks to (1.38), (3.137), (3.139), (3.154)and (3.155) we can apply the dominated convergence theorem to obtain in the limitthat (3.177) is satisfied for every (a(·), τ) ∈ Vt. Since Vt ⊂ Vt, it thus follows thatV (t, x) = inf

a(·)∈Ut

J(t, x; a (·)) = infa(·)∈Ut

J(t, x; a (·))

≥ inf(a(·),τ)∈Vt

E τ

te−

s


τ


,

where we used Theorem 2.22 to obtain the second equality.To show the reverse inequality, let t ≤ η ≤ T , a(·) ∈ Uν

t for some standardreference probability space ν = (Ω,F ,F t

s ,P,WQ), and let X(s) = X(s; t, x, a(·)).


Let, for ω0 ∈ Ω, νω0 =Ω,Fω0 ,F

tω0,s,Pω0 ,WQ,η

, where WQ,η(s) = WQ(s) −

WQ(η), and aω0(·) = a1(·) (on [η, T ]) be from Lemma 2.26-(ii).2 We have X(·) =X(·; η, X(η), a1(·)), on [η, T ],P a.e., and (see the argument in the proof of (A3) inProposition 2.16) that it is indistinguishable from a process, still denoted by X(s),such that, for P a.e. ω0,

X(·) = Xνω0 (·; η, X(η)(ω0), aω0(·)), on [η, T ],Pω0 a.e..

Therefore, as a consequence of Theorem 2.24, applied to the reference probabilityspace νω0 , we have that, for P a.e. ω0,

V (η, X(η)(ω0)) ≤ Eω0

s

ηe−

r

ηc(X(θ))dθl (r,X (r) , aω0(·) (r)) dr

+ e−

s

ηc(X(θ))dθV (s,X (s))

. (3.178)

Set

M(s) =

s

te−

r

tc(X(θ))dθl (r,X (r) , a (r)) dr + e−

s

tc(X(θ))dθV (s,X (s)) .

Then, by (3.178) and the definition of M(s) (see also remarks at the end of Section2.2.2), for P a.e. ω0

M(η)(ω0) ≤ η

te−

r

tc(X(θ))dθl (r,X (r) , a (r)) dr

(ω0)

+e−

η

tc(X(θ))dθ

(ω0)Eω0

s

ηe−

r

ηc(X(θ))dθl (r,X (r) , aω0 (r)) dr

+ e−

s


=

η

te−

r

tc(X(θ))dθl (r,X (r) , a (r)) dr

(ω0)

+e−

η

tc(X(θ))dθ

(ω0)E

s

ηe−

r

ηc(X(θ))dθl (r,X (r) , a (r)) dr

+ e−

s


Ftη

(ω0) = E

M(s)|F t

η

(ω0) (3.179)

for every s ∈ [η, T ]. Therefore, M is a submartingale, and thus, by the OptionalSampling Theorem (Theorem 1.79), if τ is an F t

s -stopping time,

V (t, x) = M(t) ≤ EM(τ)|F t

t

= E τ

te−

s


τ


Ftt

.

Taking the expectation above (or noticing that F tt is trivial), we thus obtain

V (t, x) ≤ E τ

te−

s


τ


(3.180)for every a(·) ∈ Uν

t for any standard reference probability space ν =(Ω,F ,F t

s ,P,WQ), and every F ts -stopping time τ .

It remains to justify that (3.180) is true for every (a(·), τ) ∈ Vt. We sketchthe argument. Let a(·) ∈ Uν1

t , where ν1 =Ω1,F1,F t

1,s,P1,W 1Q

, and let τn

2In Lemma 2.26-(ii) aω0 (·) denotes the 6-upleΩ,Fω0 ,F

ηω0,s,Pω0 ,Wη , a1|[η,T ](·)

while here

we use aω0 (·) only to indicate the process.


be F t1,s-stopping times with finite number of values approximating τ . Let ν2 =

Ω2,F2,F t2,s,P2,W 2

Q

be a standard reference probability space. We proceed as

in the proof of Theorem 2.22. Let a1(·), a1(·) be as in the proof of Theorem 2.22.We denote Xν1(·) = Xν1(·; t, x, a(·)) = Xν1(·; t, x, a1(·)), Xν2(·) = Xν2(·; t, x, a1(·)).Since τn has finite number of values, we can assume that

τn =k

i=1

1Aiti

for some pairwise disjoint sets A1, ..., Ak such thatn

i=1

Ai = Ω and Ai ∈ Ft,01,ti , i =

1, ..., k. Let B1, ..., Bk ∈ B(W) be such that W 1Q(·∧ ti)−1(Bi) = Ai, i = 1, ..., k (see

Lemma 2.19). We set

τn =k

i=1

1W 2Q(·∧ti)−1(Bi)ti.

Then τn is an F t2,s-stopping time with finite number of values, and it follows that

LP1(τn ∧ ·, Xν1(·), a(·)) = LP2(τn ∧ ·, Xν2(·), a1(·)).

One can then conclude that

Eν1

τn

te−

s

tc(Xν1 (r))drl (s,Xν1 (s) , a (s)) ds+ e−

τn

tc(Xν1 (r))drV (τn, X

ν1 (τn))

= Eν2

τn

te−

s

tc(Xν2 (r))drl (s,Xν2 (s) , a1 (s)) ds+ e−

τn


ν2 (τn))

.

(3.181)

Combining (3.180) with (3.181), we thus obtain that

V (t, x) ≤ Eν1

τn

te−

s

tc(Xν1 (r))drl (s,Xν1 (s) , a (s)) ds+ e−

τ


ν1 (τn))

It remains to let n → +∞ above to conclude that (3.180) holds for every pair(a(·), τ) ∈ Vt.

Remark 3.71 The proof of existence in Theorem 3.66 works almost exactly thesame (and is in fact easier) if we use Theorem 3.70 and replace t+ by min(t+, τ),where τ is the exit time of X(·) from some ball Br0(x) for some r0 > 0 (or fromBr(x) for some r > 0). In this way one can always work with local maxima andminima in the definition of viscosity solution and avoid the requirements aboutglobal uniform continuity (and hence growth at infinity) of test functions and theirderivatives. We do not pursue this here and leave the details of such a version ofviscosity solution to the interested readers.

3.6.3. Infinite horizon problem. In this subsection we characterize thevalue function of the infinite horizon optimal control problem (3.132) as the uniquesolution of the associated HJB equation (3.133). We consider the following set ofassumptions for b : H × Λ → H,σ : H × Λ → L2(Ξ0, H), l : H × Λ → R.


There exist constants C,m ≥ 0, and a local modulus ωl such that:

|b(x, a)− b(y, a)| ≤ C|x− y| ∀x, y ∈ H, a ∈ Λ, (3.182)σ(x, a)− σ(y, a)L2(Ξ0;H) ≤ C|x− y| ∀x, y ∈ H, a ∈ Λ, (3.183)|b(x, a)| ≤ C(1 + |x|) ∀x, y ∈ H, a ∈ Λ, (3.184)σ(x, a)L2(Ξ0;H) ≤ C(1 + |x|) ∀x, y ∈ H, a ∈ Λ, (3.185)

b(x, a)− b(y, a), B(x− y) ≤ C|x− y|2−1 ∀ x, y ∈ H, a ∈ Λ, (3.186)σ(x, a)− σ(y, a)L2(Ξ0;H) ≤ C|x− y|−1 ∀ x, y ∈ H, a ∈ Λ, (3.187)|l(x, a)− l(y, a)| ≤ ωl(|x− y|, R) ∀x, y ∈ B(0, R), a ∈ Λ, (3.188)|l(x, a)− l(y, a)| ≤ ωl(|x− y|−1, R) ∀x, y ∈ B(0, R), a ∈ Λ, (3.189)|l(x, a)| ≤ C(1 + |x|m) ∀x ∈ H, a ∈ Λ. (3.190)

Proposition 3.24 suggests that in order for the value function to be well definedwe need λ to be sufficiently big. We thus impose the following hypothesis.

Hypothesis 3.72 If m > 0, the discount constant λ in the functional (3.131)satisfies λ > λ, where λ is the constant from Proposition 3.24, where C is theconstant from (3.184) and (3.185) and m is the constant appearing in (3.190). Ifm = 0, we have λ > 0.

Proposition 3.73 (Regularity of V under weak B-condition) Suppose that(3.2) hold, that b and σ are continuous, and that b σ and l satisfy (3.182), (3.184),(3.185), (3.186), (3.187), (3.189) and (3.190). Assume that Hypotheses 2.28 and3.72 hold. Then there exists a local modulus ω such that:

(i) The cost functional (3.131) satisfies

|J(x, a(·))− J(y, a(·))| ≤ ω(|x− y|−1, R), (3.191)

for all x, y ∈ B(0, R), a(·) ∈ U0.(ii) There exists a constant C such that

|J(x; a(·))|, |V (x)| ≤ C(1 + |x|m) (3.192)

for all x ∈ H, a(·) ∈ U0.(iii) The value function V defined in (3.132) satisfies

|V (x)− V (y)| ≤ ω(|x− y|−1, R) ∀ x, y ∈ B(0, R). (3.193)

Proof. Part (i): Let R > 0, x, y ∈ B(0, R), and a(·) ∈ U0. Set X(·) :=X(·; 0, x; aν(·)), Y (·) := X(·; 0, y; aν(·)).

Choose > 0. Thanks to (3.34) and Hypothesis 3.72, there exists T, alsodepending on C,m,λ, R but independent of a(·), such that

E ∞

T

e−λr|l(X(r), a(r))− l(Y (r), a(r))|dr ≤ . (3.194)


We now proceed as in the proof of Proposition 3.61. T

0e−λr

|X(r)|< 1

and |Y (r)|< 1|l(X(r), a(r))− l(Y (r), a(r))|dPdr

+

T

0e−λr

|X(r)|≥ 1

or |Y (r)|≥ 1|l(X(r), a(r))− l(Y (r), a(r))|dPdr

≤ T

0e−λr

|X(r)|< 1

and |Y (r)|< 1ωl(|X(r)− Y (r)|−1,

1

)dPdr

+

T

0e−λr

|X(r)|≥ 1

or |Y (r)|≥ 1C(2 + |X(r)|m + |Y (r)|m)dPdr

= J1 + J2. (3.195)

Thanks to (1.38), arguing as in the proof of Proposition 3.61, we have

J2 ≤ γ1(, R) (3.196)

for some local modulus γ1, independent of a(·).Let K be such ωl(s,

1 ) ≤ +Ks. Using (3.24) we obtain

J1 ≤

λ+K

T

0e−λrE|X(r)− Y (r)|−1dr ≤

λ+ C|x− y|−1 (3.197)

for some C independent of a(·).Therefore, (3.194), (3.195), (3.196) and (3.197), yield

|J(x, a(·))− J(y, a(·))| ≤ +

λ+ C|x− y|−1 + γ1(, R), (3.198)

and (3.191) follows by taking the infimum above over > 0.Estimate (3.192) follows from (3.34) and Hypothesis 3.72 and (3.193) is an

obvious consequence of (3.191).

Proposition 3.74 (Regularity of V under strong B-condition) Let (3.3)hold, let b and σ be continuous, satisfy (3.182), (3.184), (3.185) and (3.187), andlet l be continuous and satisfy (3.188), (3.190). Assume that Hypotheses 2.28 and3.72 hold. Then there exist a local modulus ω and a constant C such that (i)-(iii)of Proposition 3.73 are satisfied.

Proof. The proof is exactly the same as the proof of Proposition 3.73. Wejust have to replace the term |X(r) − Y (r)|−1 by |X(r) − Y (r)| in (3.195) and(3.197), and then use (3.29) instead of (3.24). Theorem 3.75 (Existence under weak B-condition) Let the assumptions ofProposition 3.73 be satisfied, and let, for every x,

limN→+∞

supa∈Λ

Tr

σ(x, a)Q12

σ(x, a)Q

12

∗BQN

= 0. (3.199)

Then the value function V defined in (3.132) is the unique viscosity solution of(3.133) among functions in the set

S := u : H → R : |u(x)| ≤ C1(1 + |x|k)for some C1 ≥ 0 and k ≥ 0 satisfying (3.201). (3.200)

k < λC+ 1

2C2 if λ

C+ 12C

2 ≤ 2,

Ck + 12C

2k(k − 1) < λ if λC+ 1

2C2 > 2,

k can be any positive number if C = 0.

(3.201)


(C is the constant from (3.184).)

Proof. The proof follows the lines of the proof of Theorem 3.66. Proposition3.73 and Hypothesis 3.72 guarantee that V is B-continuous and that it belongs toS. By Proposition 2.16, Theorem 2.31, and Proposition 3.73 (which ensures thatHypothesis 2.28 holds), the dynamic programming principle (2.42) is satisfied.

To show that V is a viscosity supersolution of (3.128), suppose that there exista test function ψ = ϕ+ h and a point x ∈ H such that V +ψ has a local minimumat x. Without loss of generality we can assume that h,Dh,D2h have at mostpolynomial growth at infinity, and that the minimum is global. We can also requirethat V (x) + ψ(x) = 0, so for all y we have V (y) ≥ −ψ(y).

For > 0, by the dynamic programming principle, we can find aν(·) ∈ U0 suchthat, denoting X(·) := X(·; 0, x; aν(·)),

V (x) + 2 ≥ Eν

0e−λsl(X(s), aν(s))ds+ e−λV (X())

.

Therefore we have

2 − ϕ(x)− h(x) ≥ Eν

0e−λrl(X(r), aν(r))dr − e−λ(ϕ(X()) + h(X()))

,

which, upon using (3.161), (3.162), yields

+1

Eν

0e−λr

− l(X(r), aν(r)) + ψt(X(r)) + b(X(r), aν(r)), Dψ(X(r))

+1

2Tr

σ(X(r), aν(r))Q

12

σ(X(r), aν(r))Q

12

∗D2ψ(X(r))

− λψ(X(r)) + X(r), A∗Dϕ(X(r))dr

≥ 0.

Using exactly the same arguments as these in the proof of Theorem 3.66, it followsthat there exists a modulus ρ, independent of the control aν(·), such that:

1

Eν

0λV (x)− l(x, aν(r)) + b(x, aν(r)), Dψ(x)

+1

2Tr

σ(x, aν(r))Q

12

σ(x, aν(r))Q

12

∗D2ψ(x)

+x,A∗Dϕ(x) dr

≥ −ρ().

Therefore, taking the supremum over a ∈ Λ inside the integral and then letting → 0 we obtain

λV (x) + x,A∗Dϕ(x)+ supa∈Λ

− l(x, a) + b(x, a), Dψ(x)

+1

2Tr

σ(x, a)Q

12

σ(x, a)Q

12

∗D2ψ(x)

≥ 0.

This shows that V is a viscosity supersolution of (3.133).To show that V is a viscosity subsolution we take a constant control, apply

DPP, and again argue the same as in the proof of Theorem 3.66. We leave it to thereaders.

To prove that V is the unique viscosity solution among functions in S we needto show that the hypotheses of Theorem 3.56 are satisfied. This was already donein the proof of Theorem 3.66 apart from Hypotheses 3.45 and 3.55 for γ = 1,and condition (3.106). Hypothesis 3.45 is obviously true with ν = λ. As regardsHypothesis 3.55 for γ = 1, by (3.184) and (3.185), we obtain for for all (x, r) ∈

3.7. EXISTENCE OF SOLUTIONS: FINITE DIMENSIONAL APPROXIMATIONS 201

H × R, p, q ∈ H,X, Y ∈ S(H).

|F (x, r, p+ q,X + Y )− F (x, r, p,X) |

≤ C(1 + |x|)|q|+ 1

2C2(1 + |x|)2Y ,

i.e. Hypothesis 3.55 holds with MF = C,NF = 12C

2. Condition (3.106) thus followsfrom the definition of the set S.

Theorem 3.76 (Existence under strong B-condition) Let the assumptionsof Proposition 3.74 and (3.199) be satisfied. Then the value function V defined in(3.132) is the unique viscosity solution of (3.133) among functions in S defined inTheorem 3.75.

Proof. The only difference with respect to the proof for the weak case isthat to show uniqueness we now use Theorem 3.54 instead of Theorem 3.58. Thefact that Hypothesis 3.52 is satisfied was already noticed in the proof of Theorem3.67.

When conditions (3.184) and (3.185) are replaced by σ(x, a)L2(Ξ0;H) ≤ C(1+|x|γ) and |b(x, a)| ≤ C(1 + |x|γ) for some γ ∈ [0, 1), conditions which must beimposed on a set of functions to guarantee that the value function is the uniqueviscosity solution among them, can be easily deduced from (3.104) and (3.105).

3.7. Existence of solutions: Finite dimensional approximations

We have shown in Section 3.6 that value functions of stochastic optimal controlproblems are viscosity solutions of their associated HJB equations. This gives adirect method of establishing existence of viscosity solutions for a large class ofequations where we have an explicit representation formula for a solution. How-ever many interesting equations cannot be linked to a stochastic optimal controlproblem. The best examples are Isaacs equations which are associated to zero-sum, two-player, stochastic differential games. For Isaacs equations, one way orshowing existence of viscosity solutions is by proving directly that the associated(upper or lower) value of the game is the solution. Such results can be found in[192, 361, 363]. This method however runs into technical difficulties as the proofof the dynamic programming principle is very complicated. In this section we willpresent a more general method of showing existence of viscosity solutions based onfinite dimensional approximations. This method can be thought of as a Galerkintype approximation for PDE in infinitely many variables. It was first introducedin [103] for first order equations and later generalized to second order equations in[421, 422]. We will present the proofs for the initial value problems .

Let A be a linear, densely defined, maximal dissipative operator in H. Let Bbe a bounded, strictly positive, self-adjoint, compact operator on H such that A∗Bis bounded. For N > 1 let HN be the finite dimensional space spanned by theeigenvectors of B corresponding to the eigenvalues which are greater than or equalto 1/N . Let PN , QN be defined as in Section 3.2. We notice that B commutes withPN and QN , and PN , QN are now also orthogonal projections in H.

We need to change slightly the structure conditions on the Hamiltonian F .

Hypothesis 3.77 There exists a modulus ω such that

F

t, x,

B(x− y)

, X

− F

t, y,

B(x− y)

, Y

≥ −ω

|x− y|

1 +

|x− y|−1


for all (t, x, y) ∈ (0, T ) × H × H, > 0, and X,Y ∈ S(H), X = PNXPN , Y =PNY PN for some N and such that (3.54) holds.

For a bounded, strictly positive, self-adjoint, operator C on H we will use thenotation

|x|C := |C1/2x|.Hypothesis 3.78 Let C be a bounded, strictly positive, self-adjoint, operator onH . We say that Hypothesis 3.78-C is satisfied if there exists a modulus ω1 suchthat

F (t, x, c1C(x− y), X)− F (t, y, c1C(x− y), Y )

≥ −ω1 (|x− y|C (1 + (c1 + c2 + c3)|x− y|C))for all (t, x, y) ∈ (0, T ) ×H ×H, and X,Y ∈ S(H), X = PNXPN , Y = PNY PN

for some N and such that

−c2

I 00 I

≤

X 00 −Y

≤ c3

C −C−C C

, (3.202)

for some c1, c2, c3 ≥ 0.

Hypothesis 3.79 There exists h ∈ C2(H), radial, nondecreasing, nonnegative,h(x) → ∞ as |x| → ∞, Dh, D2h are bounded, and

F (t, x, p+ αDh(x), X + αD2h(x)) ≥ F (t, x, p,X)− σ(α, p+ X) (3.203)

∀x, p,X, ∀α ≥ 0, where σ is a local modulus.

Hypotheses 3.77, 3.78, and 3.79 will be sometimes applied to Hamiltonians Fdefined on finite dimensional spaces, i.e. when F : (0, T )×HN0×HN0×S(HN0) → Rfor some N0. In such cases it will be understood that N in Hypotheses 3.77, 3.78will always be equal to N0 and that every X ∈ S(HN0) is naturally extended to anoperator in S(H) by taking PN0XPN0 .

We first show continuity estimates for viscosity solutions of finite dimensionalproblems.

Lemma 3.80 Let δ > 0, l > 0, and let ω be a modulus. Then there exist anondecreasing, concave, C2 function ϕδ on [0,+∞) such that ϕδ(0) < δ and

ω(|ϕδ (r)|r2 + ϕ

δ(r)r + r) ≤ ϕδ(r) for 0 ≤ r ≤ l. (3.204)

Proof. For ∈ (0, l], r ≥ 0, 0 < γ ≤ 1, thanks to the subadditivity of ω, wehave

ω(r) ≤ ω() +ω()

r ≤ ω() +

ω()

(1 + l)1−γrγ . (3.205)

Let be such that ω() < δ/2. Define

gγ(r) = ω() + 2ω()

(1 + l)1−γrγ .

An elementary calculation and (3.205) give

gγ(r)−ω(|gγ (r)|r2+gγ(r)r+r) ≥ ω()

(1+l)1−γrγ

1− 2γ(2− γ)

ω()

(1 + l)

≥ 0

if γ is small enough. We choose such γ0 and set

ϕδ(r) = gγ0(r + r0),

where 0 < r0 < 1 is such that gγ0(r0) < δ. The function ϕδ has the requiredproperties.


Proposition 3.81 Let C be a bounded, strictly positive, self-adjoint, operatoron Rk. Let u ∈ USC([0, T )× Rk), v ∈ LSC([0, T )× Rk) be respectively a viscositysubsolution and a viscosity supersolution of

ut + F (t, x,Du,D2u) = 0 for t ∈ (0, T ), x ∈ Rk,u(0, x) = ψ(x) for x ∈ Rk,

(3.206)

where F : (0, T )×Rk×Rk×S(Rk) → R is continuous, degenerate elliptic (Hypothesis3.46) and satisfies Hypotheses 3.78 (with H = HN = Rk) and 3.79, and ψ ∈UCb(Rk).

(i) If u ,−v ≤ M , then there is a modulus of continuity m, depending only onM,T,ω1, and a modulus of continuity of ψ in the | · |C norm, such that

u(t, x)− v(t, y) ≤ m(|x− y|C) (3.207)

for all t ∈ [0, T ) and x, y ∈ Rk.(ii) If

supx∈Rk,t∈(0,T )

|F (t, x, 0, 0)| = K < +∞, (3.208)

then there exists a unique bounded viscosity solution u of (3.206). The norm u0only depends on ψ0 and K.

Proof. (i) Let m1 be a modulus of continuity of ψ in the | · |C norm. Givenµ > 0, set

u1(t, x) = u(t, x)− µ

T − t(3.209)

v1(t, x) = v(t, x) +µ

T − t. (3.210)

Let κ = 3(T+1)(1+2C). Lemma 3.80 applied with the modulus m2(r) = m1(r)+κω(r) + (2M + 1)r and l = 2 gives us for every δ > 0 a function ϕδ ∈ C2([0,∞)),nondecreasing, concave, such that

ϕδ(0) < δ, ϕδ(1) ≥ 2M + 1 (3.211)

andϕδ(r)−m2(|ϕ

δ (r)|r2 + ϕδ(r)r + r) ≥ 0 (3.212)

for 0 ≤ r ≤ 2.We are going to show that for every δ > 0

u1(t, x)− v1(t, y) ≤ ϕδ(|x− y|C)(1 + t)

for t ∈ [0, T ] and |x− y|C < 1 = ∆. Let for γ > 0

ϕ(t, x, y) = ϕδ(|x− y|2C + γ)12 (1 + t).

Suppose that

sup(x,y)∈∆ , t∈[0,T ]

(u1(t, x)− v1(t, y)− ϕ(t, x, y)) > 0

(if not we are done). Then, for small α > 0, using h from Hypothesis 3.79,

sup(x,y)∈∆,t∈[0,T ]

(u1(t, x)− v1(t, y)− ϕ(t, x, y)− αh(x)− αh(y)) > 0

and is attained at a point (t, x, y). Moreover (3.211) and (3.212) imply that (x, y) ∈∆ and 0 < t < T .

We compute

Dxϕ(t, x, y) = ϕδ(|x− y|2C + γ)

12

C(x− y)

(|x− y|2C + γ)12

(t+ 1), (3.213)


D2xxϕ(t, x, y) = ϕδ

((|x− y|2C + γ)12 )

C(x− y)⊗ C(x− y)

|x− y|2C + γ(t+ 1)

+ ϕδ((|x− y|2C + γ)

12 )

C

(|x− y|2C + γ)12

(t+ 1) (3.214)

− ϕδ((|x− y|2C + γ)

12 )

C(x− y)⊗ C(x− y)

(|x− y|2C + γ)32

(t+ 1).

We may rewrite (3.214) as D2xxϕ(x, y) = B1 + B2 + B3, where B1, B2, B3 are the

three terms appearing in (3.214). Since ϕδ is nondecreasing and concave, B2 ≥ 0and B1, B3 ≤ 0. Using this notation we have

D2ϕ(t, x, y) =

B2 −B2

−B2 B2

+

B1 +B3 −B1 −B3

−B1 −B3 B1 +B3

. (3.215)

If we denote the two matrices in (3.215) by D1 and −D2 respectively, we obtainD2ϕ(t, x, y) = D = D1 −D2, where D1, D2 ≥ 0.

Applying Theorem E.11 with = 1/(D1+ D2), there exist b1, b2 ∈ R andmatrices X,Y ∈ S(Rk) such that

b1,ϕδ

(|x− y|2C + γ)

12

C(x− y)

(|x− y|2C + γ)12

(1 + t), X

∈ P2,+(u1 − αh)(t, x),

b2,ϕδ

(|x− y|2C + γ)

12

C(x− y)

(|x− y|2C + γ)12

(1 + t), Y

∈ P2,−(v1 + αh)(t, y), (3.216)

b1 − b2 = ϕδ

(|x− y|2C + γ)

12

, (3.217)

and

−2(D1+ D2)

I 00 I

≤

X 00 −Y

≤ D +1

D1+ D2D2 ≤ 2D1,

where in the last line we used D2 ≤ (D1 + D2)(D1 + D2). Computing thenorms, we thus have obtained

−2C(1 + T )

2ϕδ

(|x− y|2C + γ)

12

+3ϕδ

(|x− y|2C + γ)

12

(|x− y|2C + γ)12

I 00 I

≤

X 00 −Y

≤

2ϕδ(|x− y|2C + γ)

12

(|x− y|2C + γ)12

(1 + T )

C −C−C C

(3.218)

We set r = (|x− y|2C + γ)12 and

d = sup |ϕδ (r)|+ ϕ

δ(r) : 0 ≤ r ≤ 2 .

Using the equation, (3.216), (3.217), (3.218), and Hypothesis 3.79 we now have forγ < 1 and small α

ϕδ(r) +2µ

T 2≤ F

t, y,

(1 + t)ϕδ(r)

rC(x− y), Y

−F

t, x,

(1 + t)ϕδ(r)

rC(x− y), X

+ 2σ

α,

6d(T + 1)Cγ

12

+ dC 12 (T + 1) + 1

.


It thus follows from Hypothesis 3.78 that

ϕδ(r) +2µ

T 2

≤ ω1

|x− y|C

1 + 3(T + 1)(1 + 2C)ϕ

δ(r)

r+ 4(T + 1)C|ϕδ

(r)||x− y|C

+2σ

α,

6d(T + 1)Cγ

12

+ dC 12 (T + 1) + 1

for some local modulus σ1. Thus, since ω1 is concave, we get

ϕδ(r) +2µ

T 2≤ 3(T + 1)(1 + 2C)ω1

|ϕδ

(r)|r2 + ϕδ(r)r + r

+2σ

α,

6d(T + 1)Cγ

12

+ dC 12 (T + 1) + 1

Therefore we obtain a contradiction if we let α → 0. This implies

u1(t, x)− v1(t, y) ≤ ϕδ(|x− y|C)(1 + T ) + 2M |x− y|Cfor all x, y ∈ Rk and t ∈ [0, T ). The claim now follows by letting µ → 0.

(ii) We remark that part (i) in particular guarantees that comparison principleholds for equation (3.206). It is standard to notice that under our assumptions onecan construct a bounded viscosity subsolution u and a bounded viscosity superso-lution u such that u(0, x) = ψ(x) = u(0, x) and u ≤ u (see Proposition 3.94 fora similar construction). We can thus use Perron’s method (see Theorem E.12) toobtain a bounded viscosity solution which is unique by (i).

The above existence and uniqueness result for finite dimensional HJB equationswill be an important tool in constructing viscosity solutions of HJB equations inHilbert spaces by finite dimensional approximations. We begin with the case whenthe strong B-condition for A is satisfied.

Proposition 3.82 Let B be compact and satisfy the strong B-condition for Aas in Definition 3.10. Let u, v be respectively a viscosity subsolution and a viscositysupersolution of

ut − Ax,Du+ F (t, x,Du,D2u) = 0 for t ∈ (0, T ), x ∈ H,u(0, x) = ψ(x) for x ∈ H,

(3.219)

where F : (0, T )×H×H×S(H) → R satisfies (for U = H) Hypotheses 3.44, 3.46,3.47, 3.77, and 3.79. Let ψ ∈ UCb(H). Let u ,−v ≤ M and be such that

|u(t, x)− u(t, y)|+ |v(t, x)− v(t, y)| ≤ m(|x− y|) (3.220)

for all t ∈ [0, T ) and x, y ∈ H, for some modulus m. Assume moreover that

limt→0

ρ(t) = 0, (3.221)

whereρ(t) = sup

x∈H

(u(t, x)− ψ(etAx))+ + (v(t, x)− ψ(etAx))−

.

Then for every 0 < τ < T there exists a modulus mτ , depending only onτ,m,ω, ρ, T,M , the constant c0 in Definition 3.10 and the modulus of continuity ofψ, such that

u(t, x)− v(t, y) ≤ mτ (|x− y|) for all x, y ∈ H, t ∈ [τ, T ). (3.222)


Proof. We will first show that there exists constants C > 0, dependingonly on ,m, c0,ω such that

lim→0

C = 0, (3.223)

and for every 0 < τ < T ,sup

t∈[τ,T )a,C

(t) = a,C(τ), (3.224)

wherea,C(t) = sup

x,y∈H

u(t, x)− v(t, y)− |x− y|2−1

2− Ct

.

For µ > 0,α > 0,β > 0 we consider the function

Ψ(t, s, x, y) = u(t, x)− µ

T − t− v(s, y)− µ

T − s− |x− y|2−1

2

− αh(x)− αh(y)− (t− s)2

2β− Ct.

where h is the function from Hypothesis 3.79. Since B is compact, B-upper semicontinuity is equivalent to weak sequential upper semicontinuity, so Ψ attains amaximum at some point (t, s, x, y). Moreover as always we have

limβ→0

(t− s)2

2β= 0 for fixed ,α. (3.225)

Therefore, using weak sequential upper semicontinuity of the above function, it iseasy to see that if supt∈[τ,T ) a,C(t) > a,C(τ), then for small µ > 0,α > 0,β > 0,we must have τ < t, s < T .

We can now argue as in the proof of Theorem 3.50 (from (3.90) to (3.93)) toobtain that for N > 2 there exist XN , YN ∈ S(H) satisfying (3.54) and such that

t− s

β+

µ

(T − t)2+ C −

x, A∗

B(x− y)

+ F

t, x,

B(x− y)

+ αDh(x), XN +

2BQN

+ αD2h(x)

≤ 0 (3.226)

andt− s

β− µ

(T − s)2+

y, A∗

B(x− y)

+ F

s, y,

B(x− y)

− αDh(x), YN − 2BQN

− αD2h(x)

≥ 0. (3.227)

Since u,−v are bounded from below it is obvious that|B(x− y)|

≤ R (3.228)

for some R, possibly depending3 on u,−v. Also, since Ψ(t, s, x, x) +Ψ(t, s, y, y) ≤2Ψ(t, s, x, y), we get

|x− y|2−1

2≤ m(|x− y|). (3.229)

3In fact, using uniform continuity of u, since for every w ∈ H, |w| = 1 we have Ψ(t, s, x +w, y) ≤ Ψ(t, s, x, y), we can obtain for α < 1

|B(x− y)|

= sup|w|=1

B(x− y), w

≤ Bw,w2

+ u(t, x)− u(t, x− w)

+ α(h(x− w)− h(x)) ≤ B2

+m(1) + L,

where L is the Lipschitz constant of h.


Thus, subtracting (3.227) from (3.226), and using Hypotheses 3.44, 3.47, 3.77, 3.79,and (3.3), (3.225), (3.228), yields

C+2µ

T 2≤ c0

|x− y|2−1

− |x− y|2

+ω

|x− y|

1 +

|x− y|−1

+σ1(;α,β, N),

(3.230)

where limα→0 lim supβ→0 lim supN→+∞ σ1(;α,β, N) = 0. Denote

γ() = supx,y∈H

c0

|x− y|2−1

− |x− y|2

+ ω

|x− y|

1 +

|x− y|−1

.

using (3.229) we have

γ() ≤ supr≥0

2c0m(r)− r2

+ ω

r

1 +

(2m(r))12

12

. (3.231)

This expression can be estimated from above by

C1

1 + r +

r

12

+r

32

12

− r2

, (3.232)

where C1 only depends on ω,m and c0. It is easily seen that (3.232) is positive onlyif r ≤ C2

12 for ≤ 1, where C2 only depends on C1. But then it easily follows from

(3.231) that lim→0 γ() = 0. Thus, if C = C := 2γ(), we obtain a contradictionafter we take lim supα→0 lim supβ→0 lim supN→+∞ in (3.230). Hence (3.224) mustbe true with this choice of C.

A consequence of (3.224) is that for all t ∈ [τ, T ), x, y ∈ H

u(t, x)−v(t, y)− |x− y|2−1

2−Ct ≤ sup

x,y∈H

u(τ, x)− v(τ, y)− |x− y|2−1

2

. (3.233)

Let mψ be the modulus of continuity of ψ. Then, by (3.20), (3.223) and (3.221),we obtain from (3.233)

u(t, x)− v(t, y) ≤ |x− y|2−1

2+ Ct+ sup

x,y∈H

u(τ, x)− v(τ, y)− |x− y|2−1

2

≤ |x− y|2−1

2+ CT + 2ρ(τ) + sup

x,y∈H

|ψ(eτAx)− ψ(eτAy)|− |x− y|2−1

2

≤ |x− y|2−1

2+ CT + 2ρ(τ) + sup

x,y∈H

mψ

ec0T |x− y|−1

2τ12

− |x− y|2−1

2

≤ |x− y|2−1

2+ ρτ () (3.234)

where ρτ depends only on C, T, τ,mψ, ρ, and lim→0 ρτ () = 0. Thus for every > 0

u(t, x)− v(t, y) ≤ min

2M |x− y|−1,

|x− y|2−1

2+ ρτ ()

which implies (3.222). Proposition 3.82 in particular implies comparison principle for bounded and

uniformly continuous viscosity sub- and super-solutions of (3.219). However wewant to mention that comparison also holds without the requirement of uniformcontinuity of u and v with almost the same proof.

We will be using the following operators to approximate the operator A. ForN ≥ 1 we define

AN = (PNA∗PN )∗.


The AN are bounded, dissipative, operators in H and it is easy to see that

ANPN = AN = PNAN , (3.235)

and thus it follows thatetANPN = PNetAN . (3.236)

Moreover we have−A∗

NB + c0B ≥ PN . (3.237)We alert the readers that in the lemma below we will use xN to denote a

sequence in H, not PNx as we have done in previous sections.

Lemma 3.83 Let B be a positive, self-adjoint, compact operator satisfying thestrong B-condition as in Definition 3.10. Then:

(i) Let x, xN ∈ D(A∗), xN → x, and A∗xN → A∗x. Then A∗NxN → A∗x.

(ii) For every x ∈ H,T > 0

etANx → etAx (3.238)

uniformly on [0, T ].

Proof. (i) We know from Lemma 3.17(ii) that the operator S = −A∗B +c0B is invertible, S−1 ∈ L(H), and D(A∗) = D(B−1). We have

A∗ = −SB−1 + c0I, A∗N = −PNSB−1PN + c0PN .

Since B−1 = S−1(−A∗ + c0I) we thus obtain

B−1xN → B−1x.

Therefore

A∗xN −A∗Nx = −QNSB−1xN − PNSQNB−1xN + c0QNxN → 0

since PN converges strongly to I and QN converges strongly to 0. This proves theclaim.

(ii) We notice that etA∗

N and etAN are semigroups of contractions. Using (3.3)we have

|etAx|2−1 + 2

t

0esAx, SesAxds− 2c0

t

0|esAx|2−1ds = |x|2−1,

|etANx|2−1 + 2

t

0esANx, SNesANxds− 2c0

t

0|esANx|2−1ds = |x|2−1,(3.239)

where

SN = −A∗NB + c0B = −PNA∗PNB + c0B = PN (−A∗B + c0B)PN + c0QNB.

By Trotter-Kato theorem (see Theorem B.46), for every x ∈ H, etA∗

Nx → etA∗

xuniformly on [0, T ]. Thus, taking adjoints, it follows that

etANx etAx for every x ∈ H, t ≥ 0. (3.240)

Since B is compact, this implies

etANx → etAx in H−1. (3.241)

Thus, passing to the limit as N → +∞ in (3.239) and using (3.241), we obtain t

0esANx, SNesANxds →

t

0esAx, SesAxds,

which, upon observing that QNB → 0 and PNx → x, yields t

0esANx, SesANxds →

t

0esAx, SesAxds. (3.242)


Denote y = esAx, yN = esANx. Then

0 ≤ |y − yN |2 ≤ y − yN , S(y − yN ) = y, Sy − yN , Sy − y, SyN + yN , SyN .(3.243)

Using (3.240) it thus follows that

0 ≤ lim infN→+∞

yN , SyN − y, Sy.

This, together with Fatou’s lemma and (3.242), implies that

limN→+∞

esANx, SesANx = esAx, SesAx

for a.e. s. We then get from (3.243) that

limN→+∞

|esAx− esANx|2 = 0 for a.e. s. (3.244)

The uniform convergence on [0, T ] follows from standard arguments.

Theorem 3.84 Let B be compact and satisfy the strong B-condition for A as inDefinition 3.10. Let F : (0, T )×H×H×S(H) → R satisfy (for U = H) Hypotheses3.44, 3.46, Hypothesis 3.47 with B = I, and Hypotheses 3.77, 3.78-I, and 3.79. Letψ ∈ UCb(H) and let for every R > 0

FR := sup|F (t, x, p,X)| : t ∈ (0, T ), x ∈ H, |p|+ X ≤ R < +∞. (3.245)

Then there exists a unique bounded viscosity solution u ∈ UCxb ([0, T ) × H) ∩

UCxb ([τ, T )×H−1) for 0 < τ < T , of

ut − Ax,Du+ F (t, x,Du,D2u) = 0 for t ∈ (0, T ), x ∈ H,u(0, x) = ψ(x) for x ∈ H,

(3.246)

satisfyinglimt→0

supx∈H

|u(t, x)− ψ(etAx)| = 0. (3.247)

Moreover, there is a modulus ρ such that

|u(t, x)− u(s, e(t−s)Ax)| ≤ ρ(t− s) for all 0 ≤ s ≤ t < T, x ∈ H. (3.248)

Proof. We consider two approximating equations.

(uN )t − ANx,DuN + F (t, PNx, PNDuN , PND2uNPN ) = 0 in (0, T )×HuN (0, x) = ψ(PNx) in H,

(3.249)and

(vN )t − ANx,DvN + F (t, x,DvN , PND2vNPN ) = 0 in (0, T )×HN

vN (0, x) = ψ(x) in HN .(3.250)

We notice that, since AN is dissipative, the Hamiltonian FN : (0, T )×HN ×HN ×S(HN ) → R defined by

FN (t, x, p,X) = ANx, p+ F (t, x, p, PNXPN )

satisfies all the assumptions of Proposition 3.81 with C = I, uniformly in N . There-fore, by Proposition 3.81, there is a unique bounded viscosity solution vN of (3.250),M ≥ 0, and a modulus m such that for all N

vN0 ≤ M|vN (t, x)− vN (t, y)| ≤ m(|x− y|) for all t ∈ [0, T ), x, y ∈ HN

(3.251)

and so (3.251) is also satisfied by uN on H.We remark that the monotonicity of AN guarantees that vN is also a viscosity

solution in the sense of Definition 3.34 on the finite dimensional Hilbert space HN .


We now extend vN to H by setting uN (x) = vN (PNx). We claim that uN isa viscosity solution of (3.249). Again, since all the terms are bounded and AN ismonotone it is enough to show it in the classical sense of (the parabolic counterpartof) Definition 3.40. To prove that uN is a viscosity subsolution, suppose thatuN (t, x) − ϕ(t, x) has a maximum at (t, x) for a smooth test function ϕ. ThenvN (t, z) − ϕ(t, z + QN x) has a maximum at (t, PN x) in (0, T ) × HN . Therefore,using the fact that vN is a subsolution of (3.250), we get

ϕt(t, x)− ANPN x, PNDϕ(t, x)+ F (t, PN x, PNDϕ(t, x), PND2ϕ(t, x)PN ) ≤ 0

and the claim follows by (3.235). The supersolution case is done similarly.We now show that there is a modulus ρ, depending only on m and the function

FR, such that|vN (t, x)− vN (s, e−(t−s)ANx)| ≤ ρ(t− s) (3.252)

for x ∈ HN , 0 ≤ s ≤ t < T . Because of (3.251) it is enough to show (3.252) fors = 0 since the estimate can be reapplied at any later time. To do this we beginwith ψ ∈ C1,1

b (H). We denote the Lipschitz constant of Dψ by LDψ. We use thefact that w(t, x) = ψ(etANx) is a classical (and viscosity) solution of

wt − ANx,Dw = 0 in (0, T )×HN , u(0, x) = ψ(x) in HN ,

which implies that

w + tFLDψ+Dψ0, w − tFLDψ+Dψ0

are respectively a viscosity super- and a subsolution of (3.250). Comparison thengives

|vN (t, x)− ψ(etANx)| ≤ tFLDψ+Dψ0(3.253)

For ψ ∈ UCb(H) we can approximate it by its inf-sup convolutions ψ ∈ C1,1b (H)

(see Proposition D.21). This approximation is such that c = ψ − ψ0 andK = LDψ

+Dψ0 only depend on the modulus of continuity of ψ, and moreoverlim→0 c = 0. Let vN be the viscosity solution of (3.250) with initial condition ψ.It follows from comparison guaranteed by Proposition 3.81 that

vN − vN0 ≤ ψ − ψ0 = c.

Using this and (3.253) we thus have

|vN (t, x)− ψ(etANx)| ≤ |vN (t, x)− vN (t, x)|+ |vN (t, x)− ψ(etANx)|

+ |ψ(etANx)− ψ(etANx)| ≤ 2ψ − ψ0 + tFL

Dψ+Dψ

0

= 2c + tK. (3.254)

Therefore|vN (t, x)− ψ(etANx)| ≤ ρ(t) = inf

>02c + tK

which completes the proof of (3.252). We also conclude, by (3.236), that for 0 ≤s ≤ t < T, x ∈ H,

|uN (t, x)− uN (s, e(t−s)ANx)| = |vN (t, PNx)− vN (s, PNe(t−s)ANx)|= |vN (t, PNx)− vN (s, e(t−s)ANPNx)| ≤ ρ(t). (3.255)

We will now to show that for every 0 < τ < T there exists a modulus mτ , suchthat

|uN (t, x)− uN (t, y)| ≤ mτ (|x− y|−1) for all x, y ∈ H, t ∈ [τ, T ). (3.256)

We notice that (3.237) implies that B restricted to HN (i.e. BN = BPN ) satisfiesthe strong condition for AN on HN with the same constant c0. Therefore (3.256)follows from (3.251), (3.252) and Proposition 3.82 applied on spaces H = HN ,since all assumptions are independent of N . (In fact we do not need the full force


of Proposition 3.82 since we deal with bounded equations on finite dimensionalspaces.)

Since B also satisfies the weak B-condition for AN with constant c0, we noticethat, by (3.19), for every N , |etANx − x|−1 ≤ C(R)

√t for |x| ≤ R, where C(R) is

independent of N . Thus for 0 < τ ≤ s < t < T,R > 0, using (3.255), (3.256), weobtain for |x| < R

|uN (t, x)−uN (s, x)| ≤ |uN (t, x)−uN (s, e(t−s)ANx)|+|uN (s, e(t−s)ANx)−uN (s, x)|≤ ρ(|t−s|)+mτ (|e(t−s)ANx−x| ≤ ρ(|t−s|)+mτ (C(R)

|t− s|) =: ρτ,R(|t−s|).

Combining it with (3.256) we have

|uN (t, x)−uN (s, x)| ≤ mτ (|x−y|−1)+ρτ,R(|t−s|), N ≥ 1, τ ≤ t, s < T, |x|, |y| ≤ R.(3.257)

Therefore (extending uN to t = T ), the family uN is equicontinuous in thetopology of R×H−1 on sets [τ, T ]× |x| ≤ R for τ > 0. But since B is compactsuch sets are compact in R×H−1. Therefore, by the Arzela-Ascoli theorem thereis a subsequence of uN , still denoted by uN , and a function u, such that uN → uuniformly on bounded subsets of [τ, T ]×H for τ > 0. Obviously u satisfies (3.248),(3.251), (3.256) and (3.257).The conclusion that u is a viscosity solution of (3.246)will follow from Theorem 3.41 (reformulated for the initial value problem), Lemma3.83(i) and Lemma 3.85 below applied with F (X) := F (t, x, p,X) for some fixed(t, x, p) ∈ (0, T )×H ×H. Uniqueness is a consequence of Proposition 3.82.

Lemma 3.85 If F : S(H) → R is locally uniformly continuous and satisfiesHypothesis 3.46 and Hypothesis 3.47 with B = I, then for every X ∈ S(H)

F (PNXPN ) → F (X) as N → ∞.

Proof. For every > 0 we have

PN (X − X2)PN −X+ 1

QN ≤ X ≤ PN (X + X2)PN +

X+ 1

QN .

Therefore, Hypotheses 3.46 and 3.47 imply

F (PN (X + X2)PN )− σ1(N, ) ≤ F (X) ≤ F (PN (X − X2)PN ) + σ1(N, ),

where σ1 is a local modulus. Using uniform continuity of F we thus obtain

F (PNXPN )− σ1(N, )− σ2() ≤ F (X) ≤ F (PNXPN ) + σ1(N, ) + σ2(),

for some modulus σ2. Thus

|F (X)− F (PNXPN )| ≤ σ1(N, ) + σ2() → σ2() as N → +∞.

and the claim follows thanks to the arbitrariness of .

We now study the case when B satisfies the weak B-condition for A, i.e. when−A∗B + c0B ≥ 0. In this case we do not have an analogue of Lemma 3.83 so wewill have to add another layer of approximations of A. We will first replace A byits Yosida approximation Aλ and then approximate Aλ by Aλ,N = PNAλPN . Theoperators Aλ and Aλ,N are bounded and dissipative. We first notice that B alsosatisfies a weak B-condition for Aλ and Aλ,N with a different constant. Indeed,since for every y ∈ D(A),

(1− λc0)By, y ≤ B(I − λA)y, y,taking y = (I − λA)−1x, we get

(1− λc0)|B12 (I − λA)−1x|2 ≤ |B 1

2x||B 12 (I − λA)−1x|,


which yields

|B 12 (I − λA)−1x| ≤ |B 1

2x|1− λc0

.

It thus follows that for every x ∈ H,

B(I − λA)−1x, x ≤ 1

1− λc0Bx, x.

Therefore,

−BAλ +c0

1− λc0B =

1

λ

1

1− λc0B −B(I − λA)−1

≥ 0

and we conclude thatA∗

λB +c0

1− λc0B ≥ 0. (3.258)

Thus B satisfies the weak B-condition for Aλ with constant 2c0 for λ < 1/(2c0).Obviously (3.258) is also satisfied if Aλ is replaced by Aλ,N .

Theorem 3.86 Let B be compact and satisfy the weak B-condition for A as inDefinition 3.9. Let F : (0, T )×H×H×S(H) → R satisfy (for U = H) Hypotheses3.44, 3.46, Hypothesis 3.47 with B = I, and Hypotheses 3.78-B, and 3.79. Letψ ∈ UCb(H−1) and let

sup|F (t, x, 0, 0)| : t ∈ (0, T ), x ∈ H = K < +∞. (3.259)

Then there exists a unique bounded viscosity solution u ∈ UCxb ([0, T ) × H−1)] of

(3.246). Moreover, for every R > 0, there is a modulus ρR such that

|u(t, x)− u(s, x)| ≤ ρR(|t− s|) for all 0 ≤ s, t < T, |x| ≤ R. (3.260)

Proof. We first solve, for N > 2, 0 < λ < 1/(2c0), the equations

(vλ,N )t + Aλ,Nx,Dvλ,N + F (t, x,Dvλ,N , PND2vλ,NPN ) = 0 in (0, T )×HN

vλ,N (0, x) = ψ(x) in HN .(3.261)

To do this we notice that, since Aλ,N is dissipative and the weak B condition holdswith constant 1/(2c0), the Hamiltonian Fλ,N : (0, T ) × HN × HN × S(HN ) → Rdefined by

Fλ,N (t, x, p,X) = Aλ,Nx, p+ F (t, x, p, PNXPN )

satisfies all the assumptions of Proposition 3.81 with C = B, uniformly in N .(Again we identify X ∈ S(HN ) with PNXPN ∈ S(H).) In fact we have

Fλ,N (t, x, c1B(x− y), X)− Fλ,N (t, y, c1B(x− y), Y )

≥ −ω1 (|x− y|−1 (1 + (c1 + c2 + c3)|x− y|−1))−c12c0

|x− y|2−1

in Hypothesis 3.78-B now. Therefore, there exists a unique viscosity solution ofvλ,N of (3.261), M ≥ 0, and a modulus m such that for all λ, N

vλ,N0 ≤ M|vλ,N (t, x)− vλ,N (t, y)| ≤ m(|x− y|−1) for all t ∈ [0, T ), x, y ∈ HN

(3.262)As in the proof of Theorem 3.84, the functions uλ,N (t, x) = vλ,N (t, PNx) are vis-cosity solutions of

(uλ,N )t − Aλ,Nx,Duλ,N + F (t, PNx, PNDuλ,N , PND2uλ,NPN ) = 0uλ,N (0, x) = ψ(PNx) in H,

(3.263)and they also satisfy (3.262).


We will now show that for every R > 0 there exists a modulus ρR such that forall N > 2, 0 < λ < 1/(2c0),

|vλ,N (t, x)− vλ,N (s, x)| ≤ ρR(|t− s|) for 0 ≤ t, s < T, |x| ≤ R. (3.264)

It is obvious that the function

w(t, x) = Kt+M

is a classical and viscosity supersolution of (3.261) for every λ, N . For every >0, x ∈ H there exists C, depending only on m, such that

ψ(y) ≤ ψ(x) + + C|x− y|2−1.

Set

R =

KT + 2M

12

.

We notice that

ψx,(y) = ψ(x) + + C|x− y|2−1 + |y|2 > KT +M for |y| ≥ R.

Now let |x| ≤ R. Since A∗λ,NB ≤ A∗B, we have

|Aλ,Ny, 2CB(y − x)| ≤ 2CA∗BR(R +R) for |y| ≤ R.

Thus if we set

FR, = sup|F (t, x, p,X| : t ∈ (0, T ), |y| ≤ R,

|p|+ X ≤ 2[(R + 1) + CB(R+R + 1)]+ 2CA∗BR(R +R),

it is easy to see that for every , N the function

ηx,(t, y) = FR,t+ ψx,(y)

is a viscosity supersolution of (3.261) in [0, T ) × |y| < R. Therefore, for every,λ, N , the function

wx, = minw, ηx,is a bounded viscosity supersolution of (3.261) in [0, T ) ×HN . By comparison wehave vλ,N ≤ wx,. In particular

vλ,N (t, x)− ψ(x) ≤ wx,(t, x)− ψ(x) ≤ + R2 + FR,t.

Taking infimum over > 0 we obtain a modulus ρR such that

vλ,N (t, x)− ψ(x) ≤ ρR(t) for t ≥ 0, |x| ≤ R.

Similar construction for subsolutions provides the same bound from below. Sincethe construction only depended on M and m in (3.262) it can be applied for anystarting point 0 ≤ s < T which yields (3.264) which is obviously also true for uλ,N .

We can finish as in the proof of Theorem 3.84. By (3.262) and (3.264), thefamily uλ,N is equibounded and equicontinuous in the topology of R × H−1 onbounded sets of [0, T ] × H and thus by the Arzela-Ascoli theorem, for every λthere is a subsequence of uλ,N , still denoted by uλ,N , and a function uλ, such thatuλ,N → uλ uniformly on bounded subsets of [0, T ] × H. Obviously uλ satisfies(3.262) and (3.264). The fact that uλ is a viscosity solution of (3.246) with Areplaced by Aλ is standard and follows from Theorem 3.41 and Lemma 3.85 sinceall the terms are bounded. We then again us Arzela-Ascoli theorem to obtain that,up to a subsequence, uλ converges uniformly on bounded subsets of [0, T ]×H to afunction u which satisfies (3.262) and (3.264). Using again Theorem 3.41, Lemma3.85, and a well known analogue of Lemma 3.83(i) for Yosida approximations wefinally conclude that u is a viscosity solution of (3.246).


The proof of uniqueness is similar to the proof of Proposition 3.82 and it willbe omitted. Alternatively it can be deduced from the proof of Theorem 3.50 wherewe now have to first let δ → 0 and then → 0 there.

3.8. Singular perturbations

Passing to limits with viscosity solutions for equations in infinite dimensionalspaces was discussed in Section 3.4. Despite its ease, some finite dimensional tech-niques cannot be applied due to the lack of local compactness, and we need toknow a priori that solutions converge locally uniformly. When A is more coercivea version of the method of half-relaxed limits will be discussed in Section 3.9. Inthis section we look at a classical “vanishing viscosity” limit in which one tries toestablish convergence of viscosity solutions of singularly perturbed equations. Suchproblems arise for instance in large deviation considerations and we will focus onequations having such origins.

Suppose we have a sequence of SDE

dXn(s) = (AXn(s) + b(s,Xn(s)))ds+1√nσ(s,Xn(s))dWQ(s) for s > t,

X(t) = x ∈ H(3.265)

in a real, separable Hilbert space H, where A is a linear, densely defined, maximaldissipative operator in H, Q ∈ L+(H) and WQ is a Q-Wiener process defined onsome reference probability space. We want to investigate large deviation principlefor the processes Xn. One of the key components in the study of large deviationsis establishing the existence of the so called Laplace limit, i.e.

limn→+∞

1

nlogE

e−ng(Xn(T )) : Xn(t) = x

for a given continuous and bounded function g, where T > t. Defining

vn(t, x) := − 1

nlogE

e−ng(Xn(T ))

,

by formally applying Ito’s formula the function vn should be a viscosity solution ofthe second order equation

(vn)t +12nTr

(σ(t, x)Q1/2)(σ(t, x)Q1/2)∗D2vn

− 1

2 |(σ(t, x)Q12 )∗Dvn|2

+Ax+ b(t, x), Dvn = 0,

vn(T, x) = g(x) in (0, T )×H.(3.266)

Sending n → +∞ in (3.266) we obtain the limiting first order PDE

vt + Ax+ b(t, x), Dv − 12 |(σ(t, x)Q

12 )∗Dv|2 = 0,

v(T, x) = g(x) in (0, T )×H.(3.267)

This is the HJB equation associated to the deterministic optimal control problemcharacterized by the state equation

ddsX(s) = AX(s) + b(s,X(s)) + σ(s,X(s))Q

12 z(s) s > t,

X(t) = x,(3.268)

where we minimize the cost functional

J(t, x; z(·)) = T

t

1

2|z(s)|2ds+ g(X(T )) (3.269)

3.8. SINGULAR PERTURBATIONS 215

over all controls z(·) ∈ L2(t, T ;H). The value function of the problem shouldbe the unique viscosity solution of (3.267). Thus we can show the existence ofthe Laplace limit and identify it if we can prove that solutions vn of the PDE(3.266) converge to the viscosity solution v of the limiting PDE (3.267). This is aclassical singular perturbation limit problem which can be solved using the theoryof viscosity solutions presented in this book. The details of the above program(which is based on a general PDE approach to large deviations developed in [184],see also [180, 181, 182]) and further study of this large deviation problem are in[425]. Here we will only show how the convergence of the vn can be establishedusing the techniques from the proof of comparison principle. We also point out thatequations (3.266) and (3.267) have a quadratic gradient term which makes themmore difficult. In particular they do not satisfy the assumptions of Section 3.5.

Let T > 0. Let B be an operator satisfying the weak B-condition (3.2) for A.We make the following assumptions.

Hypothesis 3.87 The functions b : [0, T ]×H → H, σ : [0, T ]×H → L2(Ξ0;H)are uniformly continuous on bounded sets and there exist constants L,M such that

|b(t, x)− b(t, y)| ≤ L|x− y|, t ∈ [0, T ], x, y ∈ H, (3.270)

b(t, x)− b(t, y), B(x− y) ≤ L|x− y|2−1, t ∈ [0, T ], x, y ∈ H, (3.271)

σ(t, x)− σ(t, y)L2(Ξ0;H) ≤ L|x− y|−1, t ∈ [0, T ], x, y ∈ H, (3.272)

σ(t, x)L2(Ξ0;H) ≤ M, t ∈ [0, T ], x ∈ H. (3.273)The function g : H → R is bounded and

|g(x)− g(y)| ≤ L|x− y|−1, x, y ∈ H. (3.274)

It was shown in [425] that the functions vn are unique viscosity solutions of(3.266). The assumptions in [425] were slightly different from Hypothesis 3.87 andsome additional restrictions on test functions were placed to deal with exponen-tial moments, however the proof of existence follows the standard arguments andthe test function restrictions can be circumvented by localization using Itô’s for-mulas with stopping times and (for instance) Theorem 3.70. The uniqueness partis more difficult. Moreover it was shown in [425] that the value function v of thedeterministic control problem satisfies

|v(t, x)− v(s, y)| ≤ C1|x− y|−1 + C2(max|x|, |y|)|t− s| 12

for all x, y ∈ H and t, s ∈ [0, T ].The following theorem addresses the convergence problem. It is a general state-

ment about a singular perturbation problem. The theorem could be stated for moregeneral HJB equations, however the main difficulty here is the quadratic gradientterm.

Theorem 3.88 Let Hypothesis 3.87 hold. Let vn be a bounded viscosity solutionof (3.266), and v be a bounded viscosity solution of (3.267) such that

limt→T

|vn(t, x)− g(x)|+ |v(t, x)− g(x)| = 0, uniformly on bounded sets (3.275)

and|v(t, x)− v(t, y)| ≤ L|x− y|−1, 0 ≤ t ≤ T, x, y ∈ H. (3.276)

Then there exists a constant C independent of n such that

vn − v0 ≤ C√n. (3.277)


Proof. Set

un := v +C√n(T − t+ 1).

Then un is a viscosity solution of

(un)t + Ax+ b(t, x), Dun −1

2|(σ(t, x)Q 1

2 )∗Dun|2 = − C√n. (3.278)

We will show that there exists C independent of n such that vn ≤ un. Ifvn ≤ un, then for µ, δ,β > 0,m ∈ N, there exist pm, qm ∈ H, am, bm ∈ R such that|pm|, |qm|, |am|, |bm| ≤ 1/m, and

Ψ(t, s, x, y) := vn(t, x)− un(s, y)−µ

t− µ

s−

√n

2|x− y|2−1 − δ(|x|2 + |y|2)

− (t− s)2

2β+ Bpm, x+ Bqm, y+ amt+ bms (3.279)

has a global maximum over (0, T ]×H × (0, T ]×H at some points t, s, x, y, whereΨ(t, s, x, y) ≥ ηn > 0 for small µ, δ and large m. Similarly to the proof of Theorem3.50 we have

lim supβ→0

lim supm→∞

(t− s)2

2β= 0 for fixed µ, , δ, (3.280)

lim supδ→0

lim supβ→0

lim supm→∞

δ(|x|2 + |y|2) = 0 for fixed µ. (3.281)

Since Ψ(t, s, x, x) ≤ Ψ(t, s, x, y), it follows from (3.276)√n

2|x− y|2−1 ≤ un(s, x)− un(s, y) + δ|x|2 + Bqm, y − x

≤L+

B1/2m

|x− y|−1 + δ|x|2.

Therefore

lim supδ→0

lim supβ→0

lim supm→∞

|x− y|−1 ≤ 2L√n. (3.282)

If either s or t is equal to T , we thus obtain from (3.275), (3.276), (3.280), (3.281)and (3.282)

ηn ≤ lim supδ→0

lim supβ→0

lim supm→∞

Ψ(t, s, x, y)

≤ lim supδ→0

lim supβ→0

lim supm→∞

L|x− y|−1 −

C√n

≤ 2L2 − C√

n.

Thus if C ≥ 2L2 we must have 0 < t, s < T .We now use that vn is a viscosity subsolution of (3.266) to obtain

− µ

t2− am +

t− s

β+

1

2nTr

(σ(t, x)Q1/2)(σ(t, x)Q1/2)∗(

√nB + 2δI)

− 1

2|(σ(t, x)Q 1

2 )∗(√nB(x− y) + 2δx−Bpm)|2

+ x, A∗[√nB(x− y)−Bpm]+ b(t, x),

√nB(x− y) + 2δx−Bpm ≥ 0.

(3.283)

3.8. SINGULAR PERTURBATIONS 217

Moreover, since un is a viscosity supersolution of (3.278), we get

µ

s2+ bm +

t− s

β− 1

2|(σ(s, y)Q 1

2 )∗(√nB(x− y)− 2δy +Bqm)|2

+ y, A∗[√nB(x− y) +Bqm]+ b(s, y),

√nB(x− y)− 2δy +Bqm

≤ − C√n. (3.284)

Subtracting (3.284) from (3.283) and using (3.2), (3.270), (3.271), (3.272), (3.273),(3.280), (3.281) and (3.282) give us

2µ

T 2≤ n

2|(σ(t, y)Q 1

2 )∗B(x− y)|2 − n

2|(σ(t, x)Q 1

2 )∗B(x− y)|2

+1

2√nM2B+ c1

√n|x− y|2−1 −

C√n+ γ(δ,β,m), (3.285)

where c1 is some constant depending only on L and c0 in (3.2), and γ is a functionsuch that lim supδ→0 lim supβ→0 lim supm→∞ γ(δ,β,m) = 0. Now

|(σ(t, y)Q 12 )∗B(x− y)|2 − |(σ(t, x)Q 1

2 )∗B(x− y)|2

= Tr(σ(t, y)Q1/2)(σ(t, y)Q1/2)∗B(x− y)⊗B(x− y)

− Tr(σ(t, x)Q1/2)(σ(t, x)Q1/2)∗B(x− y)⊗B(x− y)

≤ c2|x− y|3−1,

where c2 is some constant depending only on L,M, B1/2. Plugging this inequalityinto (3.285) and invoking (3.282) we thus obtain

2µ

T 2≤ lim sup

δ→0lim supβ→0

lim supm→∞

c1√n|x− y|2−1 + c2n|x− y|3−1

+

1

2√nM2B − C√

n

≤ 1√n

4L2c1 + 8L3c2 +

1

2M2B

− C√

n.

This yields a contradiction if C ≥ 4L2c1 + 8L3c2 +12M

2B. Thus we must have

vn ≤ v +C√n(T − t+ 1) ≤ v +

C(T + 1)√n

.

Similar arguments give us

v − C√n(T − t+ 1) ≤ vn

and thus the result follows. The rate of convergence provided by Theorem 3.88 is the same as the rate for

finite dimensional problems.

Remark 3.89 It is obvious from the proof that Theorem 3.88 remains the sameif the term b(t, x), Du in (3.266) and (3.267) is replaced by a general HamiltonianF (t, x,Du), where F : [0, T ]×H×H → R is uniformly continuous on bounded setsand for instance satisfies

|F (t, x, p)− F (t, x, q)| ≤ C|p− q|(1 + |x|),

F

t, x,

B(x− y)

− F

t, y,

B(x− y)

≤ C|x− y|2−1,

for all t ∈ [0, T ], x, y, p, q ∈ H. Such equations arise in risk sensitive optimal controlproblems. We refer to [84, 85, 276, 359, 360, 362, 363, 424] for such problemsin infinite dimensional spaces and to [145, 195] for more on risk sensitive control


problems. A result similar to Theorem 3.88 has been proved in [363] for a risksensitive control problem using representation formulas and probabilistic methods.

3.9. Perron’s method and half-relaxed limits

Perron’s method is one of the main techniques for producing viscosity solu-tions of PDE in finite dimensional spaces (see [101, 269] and Appendix E.4). Itis based on the principle that the supremum of the family of all viscosity subsolu-tions which are less than or equal to a viscosity supersolution of an equation is a(possibly discontinuous) viscosity solution. Thus to construct a viscosity solution,all we need is to produce one subsolution u0 and one supersolution v0 that bothsatisfy the boundary and initial conditions and such that u0 ≤ v0. If we have acomparison theorem, the viscosity solution produced by Perron’s method can thenbe proved to be continuous. Perron’s method has a rather trivial extension to in-finite dimensional bounded equations (3.68), see [321]. Perron’s method was alsoused to prove existence of viscosity solutions using Ishii’s definitions of viscositysolutions [271, 272]. However it is not known if a version of Perron’s method canbe implemented for B-continuous viscosity solutions of (3.58) and (3.64), even ifthe equations are of first order. The reason for this is that B-continuous viscositysub-/super-solutions are semi-continious in a weaker topology and this makes theproblem difficult. However it was shown in [288] how to adapt Perron’s method toB-continuous viscosity solutions of (3.58) and (3.64) when the operator A is morecoercive. We only discuss the initial value problems

ut − Ax,Du+ F (t, x, u,Du,D2u) = 0 (t, x) ∈ (0, T )×H

u(0, x) = g(x),(3.286)

where H is a real, separable Hilbert space and A is a linear, densely defined, maxi-mal dissipative operator in H. The presentation here is based on [288] and we referto this paper for further results and more details.

In order to develop Perron’s method we need to introduce a notion of a dis-continuous viscosity solution. Let B be an operator satisfying (3.2). For a functionu we will write u∗,−1 and u∗,−1 to denote the upper- and lower-semicontinuousenvelopes of u in the | · |× | · |−1 norm, i.e.

u∗,−1(t, x) = lim supu(s, y) : s → t, |y − x|−1 → 0,

u∗,−1(t, x) = lim infu(s, y) : s → t, |y − x|−1 → 0.Observe that u∗,−1 is upper semicontinuous in the | · |× | · |−1 norm and thus, thanksto Lemma 3.6(ii), it is B-upper semicontinuous.

We assume that F : (0, T )×H ×R×H ×S(H) → R satisfies Hypotheses 3.44,3.45 and 3.46. We also impose the following coercivity condition on A.

−A∗x, x ≥ λ|x|21, for x ∈ D(A∗) (3.287)for some λ > 0.

The above implies in particular that D(A∗) ⊂ H1. Assumption (3.287) issatisfied for instance for self-adjoint invertible operators A if B = (−A)−1.

Definition 3.90 A locally bounded function u is a discontinuous viscosity sub-solution of (3.286) if u(0, y) ≤ g(y) on H, and whenever (u − h)∗,−1 − ϕ has alocal maximum in the topology of | · | × | · |−1 at a point (t, x) for a test functionψ(s, y) = ϕ(s, y) + h(s, |y|) from Definition 3.32 such that ϕ is B-continuous and

u(s, y)− h(s, |y|) → −∞ as |y| → ∞ locally uniformly in s (3.288)

3.9. PERRON’S METHOD AND HALF-RELAXED LIMITS 219

thenx ∈ H1

and

ψt(t, x) + λ|x|21hr(t, |x|)

|x| − x,A∗Dϕ(t, x)

+ F (t, x, (u− h)∗,−1(t, x) + h(t, |x|), Dψ(t, x), D2ψ(t, x)) ≤ 0,

where hr is the partial derivative of h with respect to the second variable.A locally bounded function u is a discontinuous viscosity supersolution of (3.286)if u(0, y) ≥ g(y) on H, and whenever (u + h)∗,−1 + ϕ has a local minimum in thetopology of | · |× | · |−1 at a point (t, x) for a test function ψ(s, y) = ϕ(s, y)+h(s, |y|)from Definition 3.32 such that ϕ is B-continuous and

u(s, y) + h(s, |y|) → +∞ as |y| → ∞ locally uniformly in s (3.289)then

x ∈ H1

and

− ψt(t, x)− λ|x|21hr(|x|)|x| + x,A∗Dϕ(t, x)

+ F (t, x, (u+ h)∗,−1(t, x)− h(t, |x|),−Dψ(t, x),−D2ψ(t, x)) ≥ 0.

A discontinuous viscosity solution of (3.286) is a function which is both a discon-tinuous viscosity subsolution and a discontinuous viscosity supersolution.

The maxima and minima in Definition 3.90 can be assumed to be global andstrict in the | · |× | · |−1 norm. Compared to Definition 3.34, apart from discontinuityof sub/super-solutions, the main difference here is that we require that x ∈ H1 andthe term x,A∗Dh(t, x) is not dropped entirely. We notice that if x ∈ A∗ then−x,A∗Dh(t, x) = −hr(t,|x|)

|x| x,A∗x ≥ λ|x|21hr(t,|x|)

|x| and this term is well definedand is left in the definition. If x = 0 the term |x|21

hr(t,|x|)|x| by definition is equal to 0.

We also remark that if u is B-upper semicontinuous then (u−h)∗,−1 = u−h and ifu is B-lower semicontinuous then (u+ h)∗,−1 = u+ h. Definitions of discontinuousviscosity solutions have been first used in [271, 272]. Definitions requiring thatpoints where maxima/minima occur belong to better spaces appeared in [71, 106]and have been successfully employed for some second order equations which arediscussed in this book in Sections 3.11, 3.12, 3.13 (see also [242, 244, 245]).

For simplicity we restrict ourselves to F which does not depend on u. We willoften say that u is a viscosity sub-/supersolution in an open set V . This will meanthat we disregard the initial condition and the conditions of Definition 3.90 mustbe satisfied only if (t, x) ∈ V . However all functions involved must be defined on(0, T )×H and the maxima/minima in Definition 3.90 are local in the topology of| · |× | · |−1 in the whole (0, T )×H.

Lemma 3.91 Let Hypotheses 3.44, 3.46 and condition (3.287) hold and let V bean open subset of (0, T ) ×H. Let ψ = ϕ + h be a test function such that w = −ψ(respectively, w = ψ) satisfies

wt(t, x)−x,A∗Dw(t, x)+F (t, x,Dw(t, x), D2w(t, x)) ≤ 0 x ∈ D(A∗)∩V, t ∈ (0, T )

(respectively,

wt(t, x)−x,A∗Dw(t, x)+F (t, x,Dw(t, x), D2w(t, x)) ≥ 0 x ∈ D(A∗)∩V, t ∈ (0, T ).)

Then the function w is a viscosity subsolution (respectively, supersolution) of(3.286).


Proof. We will only show the proof in the subsolution case. We notice thatw = −ψ is B-upper semicontinuous. Suppose that w(s, y) − ϕ(s, y) − h(s, |y|) hasa local maximum at (t, x) for a test function ψ = ϕ+ h. Then

wt(t, x) = ψt(t, x), −Dϕ(t, x)− hr(t, |x|)|x| x−Dϕ(t, x)− hr(t, |x|)

|x| x = 0

andD2w(t, x) ≤ D2(ϕ+ h)(t, x).

Therefore either x = 0 orhr(t, |x|)

|x| +hr(t, |x|)

|x|

x = −Dϕ(t, x)−Dϕ(t, x) ∈ D(A∗),

i.e. x ∈ D(A∗). Thus, using (3.287) and Hypothesis 3.46, we obtain

ψt(t, x) + λ|x|21hr(t,|x|)

|x| − x,A∗Dϕ(t, x)+ F (t, x,Dψ(t, x), D2ψ(t, x))

≤ wt(t, x)− hr(t,|x|)|x| x,A∗x − x,A∗Dϕ(t, x)+ F (t, x,Dw(t, x), D2w(t, x))

= wt(t, x)− x,A∗Dw(t, x)+ F (t, x,Dw(t, x), D2w(t, x)) ≤ 0

and the claim is proved.

Proposition 3.92 Let Hypotheses 3.44, 3.46 and condition (3.287) be satisfied.Let A be a family of viscosity subsolutions of (3.286) in the sense of Definition3.90. Suppose that the function

u(x) = sup w(x) : w ∈ A (3.290)

is locally bounded. Then u is a viscosity subsolution of (3.286) in the sense ofDefinition 3.90.

Proof. Suppose that (u− h)∗,−1 −ϕ has a strict in | · |× | · |−1 norm globalmaximum at a point (t, x) for a test functions ψ = ϕ + h. (We can assume that(u−h)∗,−1(s, y)−ϕ(s, y) ≤ −|y| as |y| → ∞.) Perturbed optimization (see Corollary3.26) and Definition 3.90 yield that there exist wn ∈ A, xn ∈ H1, tn, and an ∈ R,pn ∈ H, |an|+ |pn| ≤ 1/n such that

tn → t, B12xn → B

12x, xn x in H as n → ∞, (3.291)

(wn − h)∗,−1(s, y)− ϕ(s, y) + Bpn, y+ ans

has a strict in | · |× | · |−1 norm global maximum at (tn, xn), and

(wn − h)∗,−1(tn, xn) → (u− h)∗,−1(t, x) as n → ∞. (3.292)

Therefore,

ψt(tn, xn)− an + λ|xn|21hr(tn, |xn|)

|xn|− xn, A

∗(Dϕ(xn)−Bpn)

+ F (tn, xn, Dψ(tn, xn)−Bpn, D2ψ(tn, xn)) ≤ 0. (3.293)

Since the xn are bounded, using the local boundedness of F we thus obtain thateither xn → 0 = x or, up to a subsequence, |xn| > c > 0 which leads to

|xn|21 ≤ C

for some constant C which, together with (3.291), implies that x ∈ H1, andB− 1

2xn B− 12x as n → ∞. Therefore, by (3.291),

|xn − x|2 = B− 12 (xn − x), B

12 (xn − x) → 0 as n → ∞,


i.e. xn → x in H. Using this, the continuity of F , and the lower semicontinuouityof | · |1 in H, we can now pass to the lim inf as n → ∞ in (3.293) to obtain

ψt(t, x) + λ|x|21hr(t, |x|)

|x| − x,A∗Dϕ(t, x)+ F (t, x,Dψ(t, x), D2ψ(t, x)) ≤ 0

which completes the proof. Theorem 3.93 Let Hypotheses 3.44, 3.46 and condition (3.287) be satisfied. Letu0, v0 be respectively a viscosity subsolution and a viscosity supersolution of (3.286)in the sense of Definition 3.90 such that u0 ≤ v0 and u0(0, x) = v0(0, x) = g(x),x ∈ H. Then the function

u(t, x) = supv(t, x) : u0 ≤ v ≤ v0, v is a viscosity subsolutionof (3.286) in the sense of Definition 3.90 (3.294)

is a viscosity solution of (3.286) in the sense of Definition 3.90.

Proof. It follows from Proposition 3.92 that u is a viscosity subsolution.Suppose now that (u+ h)∗,−1 +ϕ has a strict in | · |× | · |−1 norm global minimumat a point (t, x) for a test function ψ = ϕ + h satisfying (3.289). First we noticethat if

(u+ h)∗,−1(t, x) = (v0 + h)∗,−1(t, x)

then (v0 + h)∗,−1 + ϕ has a global minimum at (t, x) and so we are done since v0is a viscosity supersolution. Therefore we only need to consider the case

(u+ h)∗,−1(t, x) < (v0 + h)∗,−1(t, x).

It then follows from the above inequality, the B-continuity of ϕ and the weaksequential lower semi-continuity of | · | that there is 0 > 0 such that for every R > 0there exist η0 > 0 such that

+ (u+ h)∗,−1(t, x) + ϕ(t, x)− ϕ(s, y)− h(s, |y|)< (v0 + h)∗,−1(s, y)− h(s, |y|) ≤ v0(s, y) (3.295)

for (s, y) ∈ (t− η0, t+ η0)× (BH−1(x, η0) ∩B(x,R)), 0 < < 0. Denotew(y) = + (u+ h)∗,−1(t, x) + ϕ(t, x)− ϕ(s, y)− h(s, |y|). (3.296)

By further modifying h for large values of |y| and s ∈ (t − η0, t + η0) if necessary,we can also assume that there is R0 > 0 such that

w(s, y) ≤ u(s, y)− 1 y ∈ B(x,R0), s ∈ (0, T ). (3.297)Moreover, if R0 is big enough there exist sn → t, yn ∈ B(x,R0), yn → x in H−1

such thatu(sn, yn) + h(sn, |yn|) + ϕ(sn, yn) → (u+ h)∗,−1(t, x) + ϕ(t, x)

which means that for every η > 0 there exist points (s, y) ∈ (t − η, t + η) ×(BH−1(x, η) ∩B(x,R0)) for which

u(s, y) < w(s, y). (3.298)If the condition for u being a viscosity supersolution of (3.286) is violated at

(t, x) for the test function ψ then one of the following must hold:(i) x ∈ H1.(ii) x ∈ H1 but

− ψt(t, x)− λ|x|21hr(t, |x|)

|x| + x,A∗Dϕ(t, x)

+ F (t, x,−Dψ(t, x),−D2ψ(t, x)) < −ν < 0 (3.299)


for some ν > 0.

If (i) is satisfied then we must have

lim infy→x in H−1

y∈H1

|y|1 = +∞. (3.300)

Otherwise we would have a sequence yn such that B 12 yn → B

12x and |B− 1

2 yn| ≤ C.Then for some subsequence (still denoted by yn) B− 1

2 yn z for some z ∈ H whichwould imply x ∈ H1 and z = B− 1

2x. Using the local boundedness of F , condition(3.300) now implies that for every R > 0

wt(s, y)− λ|y|21hr(s, |y|)

|y| + y,A∗Dϕ(s, y)+ F (s, y,Dw(s, y), D2w(s, y)) < −ν

2,

(3.301)for (s, y) ∈ (t− η1, t+ η1)× (BH−1(x, η1) ∩B(x,R) ∩H1) for some η1 > 0.

Suppose that (ii) is true. We will show that for every R > 0 (3.301) holds for(s, y) ∈ (t− η1, t+ η1)× (BH−1(x, η1)∩B(x,R)∩H1) for some η1 > 0. If not thereexist sequences tn → t, xn → x in H−1, |xn| ≤ R such that

wt(tn, xn)− λ|xn|21hr(tn, |xn|)

|xn|+ xn, A

∗Dϕ(tn, xn)

+ F (tn, xn, Dw(tn, xn), D2w(tn, xn)) ≥ −ν

2. (3.302)

If xn → 0 = x then letting n → +∞ in (3.302) would contradict (3.299). If xn → 0then for some subsequence (still denoted by xn) we would have hr(tn, |xn|)/|xn| ≥γ > 0 for some γ and this would imply |xn|1 ≤ C for some constant C as otherwise(3.302) would be violated. But then we must have B− 1

2xn B− 12x in H and thus

we obtain xn → x. However then (3.299), (3.302) and the lower semi-continuity of · 1 again imply

− ν

2≤ lim sup

n→∞

wt(tn, xn)− λ|xn|21

hr(tn, |xn|)|xn|

+ xn, A∗Dϕ(tn, xn)

+ F (tn, xn, Dw(tn, xn), D2w(tn, xn))

< −ν

which gives a contradiction.Thus we have proved that in both cases (i) and (ii), for every R > 0 (3.301)

holds for (s, y) ∈ (t− η1, t+ η1)× (BH−1(x, η1) ∩B(x,R) ∩H1) for some η1 > 0.

Recall now the definition of w given in (3.296). Since (u + h)∗,−1 + ϕ has aglobal minimum at (t, x), strict in | · |× | · |−1 norm, given η > 0 and small enough(depending on η) there exists a constant µη > 0 such that

w(s, y) < (u+ h)∗,−1(s, y)− h(s, |y|)− µη ≤ u(s, y)− µη (3.303)

for y ∈ BH−1(x, η), s ∈ (t− η, t+ η).Using (3.295), (3.297), (3.303), and (3.301) we can therefore conclude that there

exist numbers R, η, , µ > 0 such that

w ≤ v0 in [0, T )×H, (3.304)

w(s, y) < u(s, y)− µ for (s, y) ∈ (t− η, t+ η)× (BH−1(x, η) ∩B(x,R)), (3.305)

and such that (3.301) is satisfied for (s, y) ∈ (t − 2η, t + 2η) × (BH−1(x, 2η) ∩B(x, 2R) ∩H1).


We now claim that the function w is a viscosity subsolution of (3.286) in theinterior of (t− 2η, t+2η)× (B−1(x, 2η)∩B(x, 2R)). This follows from Lemma 3.91upon noticing that by (3.301) and (3.287) we have

wt(s, y)− y,A∗Dw(s, y)+ F (s, y,Dw(s, y), D2w(s, y))

≤ wt(s, y)− λ|y|21hr(s,|y|)

|y| + y,A∗Dϕ(s, y)+ F (s, y,Dw(s, y), D2w(s, y)) < 0

for (s, y) ∈ (t− 2η, t+ 2η)× (BH−1(x, 2η) ∩B(x, 2R) ∩D(A∗)).It remains to show that the function

u1 = max(w, u) (3.306)

is a viscosity subsolution in the sense of Definition 3.90. It follows from the definitionthat u1 is a viscosity subsolution in the interior of (t− 2η, t+ 2η)× (B−1(x, 2η) ∩B(x, 2R)). If (s, y) ∈ (t− 3/2η, t+ 3/2η)× (B−1(x, 3/2η) ∩B(x, 3/2R)) and (u1 −h)∗,−1 − ϕ has a maximum at (s, y), and

(u1 − h)∗,−1(s, y) = limn→+∞

(u1(sn, yn)− h(sn, yn)),

where |sn − s| + |yn − y|−1 → 0 and yn y, then since |y| ≤ lim infn→+∞ |yn|,we obtain (sn, yn) ∈ (t − η, t + η) × (B−1(x, η) ∩ B(x,R)) for large n. Thus by(3.305) u1(sn, yn) = u(sn, yn) which implies (u1 − h)∗,−1(s, y) = (u − h)∗,−1(s, y).Therefore the subsolution condition is satisfied for u1 at (s, y) for the test functionψ = h+ ϕ, and hence u1 is a discontinuous viscosity subsolution of (3.286).

By the definition of u1 and (3.304) we know that u0 ≤ u1 ≤ v0. Thus, by(3.294), we should have u1 ≤ u but this contradicts (3.298).

Comparison theorem in the whole space can be proved under the same as-sumptions as these of Theorem 3.50. The proof is almost exactly the same. Thereader can also check the proof of Theorem 4.1 in [288] for a proof in a simplertime independent case. Comparison theorem in particular implies that a discontin-uous viscosity solution (if it exists) is in fact B-continuous. Thus, in particular, ifcomparison theorem holds, a viscosity solution in the sense of Definition 3.90 is theusual viscosity solution in the sense of Definition 3.34 (with additional requirementthat test functions ϕ are B-continuous). However we now have a very convenientway to prove the existence of solution by Perron’s method. The remaining questionis how to construct a sub- and a supersolution u0 and v0 as in Theorem 3.93 thatin addition attain the initial condition locally uniformly so that we can later usecomparison theorem.

Proposition 3.94 Let Hypotheses 3.44, 3.46 and condition (3.287) hold andlet g be locally uniformly B-continuous and such that |g(x)| ≤ µ(1 + |x|) for x ∈H for some constant µ. Then there are a viscosity subsolution u0 and viscositysupersolution v0 of equation (3.286) in the sense of Definition 3.90 such that

limt↓0

(|u0(t, x)− g(x)|+ |v0(t, x)− g(x)|) = 0

uniformly on bounded sets of H.

Proof. We will only show how to construct v0.Define

C(r) = sup|F (t, x, p,X)| : x ∈ H, t ∈ [0, T ], |p| ≤ r, X ≤ r.

Let v(t, x) = αt + 2µ1 + |x|2. Notice that v(0, x) ≥ g(x), x ∈ H. By Lemma

3.91, v is a viscosity supersolution of (3.286) if

α+ Ft, x,Dv(t, x), D2v(t, x)

≥ 0


for all (t, x) ∈ (0, T )×H. Since Dv(t, x) and D2v(t, x) are bounded we can thereforeselect α, depending only on µ, such that the above condition is satisfied.

Let z ∈ H, > 0. We first choose a constant R = R(|z|) ≥ |z| such that(|x|− |z|)4+ ≥ 2v(t, x) for |x| ≥ R, t ∈ (0, T ). We then find M = M(|z|, ) such that

wz,(x) := g(z) + +M |x− z|2−1 + (|x|− |z|)4+ ≥ g(x)

for |x| ≤ R. Let now γ = sup|Dwz,(x)| + D2wz,(x) : |x| ≤ R. Using againLemma 3.91, in order for wz,(t, x) := βt + wz,(x) to be a viscosity supersolutionof (3.286) in the interior of (0, T )×B(0, R) we need

β + 2Mx,A∗B(x− z)+ F (t, x,Dwz,(t, x), D2wz,(t, x)) ≥ 0

in this set. This can be achieved by taking β = 2RM(R+ |z|)A∗B+ C(γ).Since wz,(t, x) > v(t, x) if t ∈ (0, T ), |x| ≥ R, it thus follows that

ωz,(t, x) := minwz,(t, x), v(t, x)is a B-lower semicontinuous viscosity supersolution of (3.286) in [0, T ) × H. It isnow clear from the construction of the ωz, and Proposition 3.92 for supersolutionsthat the function v0(t, x) := infz, ωz,(t, x) is a viscosity supersolution of (3.286)in the sense of Definition 3.90 such that limt↓0 |v0(t, x) − g(x)| = 0 uniformly onbounded sets of H.

In the last part of this section we show how the method of half-relaxed limits ofBarles-Perthame (see [101]) can be generalized to infinite dimensional spaces. Thismethod improves the general consistency result of Section 3.4. Suppose that wehave equations

ut − Anx,Du+ Fn(t, x, u,Du,D2u) = 0 (t, x) ∈ (0, T )×H, (3.307)

where Fn : [0, T ]×H ×R×H × S(H) → R, and An, n = 1, 2, ... are linear, denselydefined maximal dissipative operators in H such that D(A∗) ⊂ D(A∗

n). Let F+, F−be defined as in Theorem 3.41. We define

u+(x) = limi→∞

sup

un(y) : n ≥ i, |x− y| ≤ 1

i

,

u−(x) = limi→∞

inf

un(y) : n ≥ i, |x− y| ≤ 1

i

.

Theorem 3.95 Let the operator B satisfying (3.2) be compact. Let An be asabove, let A, An, n = 1, 2, ... satisfy (3.287), let (3.70) hold, and let for every testfunction ϕ, the family A∗

nDϕ, n = 1, 2, ... be locally uniformly bounded. Supposethat Fn, n = 1, 2, ... are continuous, locally bounded uniformly in n, and satisfyHypotheses 3.45 and 3.46. Let un be locally bounded, uniformly in n, B-uppersemicontinuous (respectively, B-lower semicontinuous) viscosity subsolutions, (re-spectively, supersolutions) of

(un)t − Anx,Dun+ Fn(t, x, un, Dun, D2un) = 0 in (0, T )×H (3.308)

in the sense of Definition 3.90. Then the function u+ (respectively, u−) is a vis-cosity subsolution (respectively, supersolution) of

(u+)t − Ax,Du++ F−(t, x, u+, Du+, D2u+) = 0 in (0, T )×H

(respectively,

(u−)t − Ax,Du−+ F+(t, x, u−, Du−, D2u−) = 0 in (0, T )×H)

in the sense of Definition 3.90.


Proof. Let (u+−h)∗,−1−ϕ have a local maximum (equal to 0) at (t, x) forsome test function ψ = ϕ+ h. In light of local uniform boundedness of the un wecan assume that the maximum is global, strict in the | · | × | · |−1 norm, and suchthat

u+(y)− h(s, |y|) → −∞, (u+ − h)∗,−1(y)− ϕ(s, y) → −∞,

andun(s, y)− h(s, |y|)− ϕ(s, y) → −∞

as |y| → +∞, uniformly in n and s ∈ (0, T ), and as s → 0 and s → T , uniformlyin n and y in bounded sets. Then there must exist a sequences tn, xn such that|tn − t|+ |xn − x|−1 → 0, |xn| ≤ C, and

u+(tn, xn)− h(tn, |xn|)− ϕ(tn, xn) ≥ − 1

n.

Therefore there exist τn, yn and in such that

uin(τn, yn)− h(τn, |yn|)− ϕ(τn, yn) ≥ − 2

n. (3.309)

Let (sn, zn) be a global maximum of

uin(s, y)− h(s, |y|)− ϕ(s, y).

It exists because of the decay of this function at infinity and around 0, T , andthe fact that, because B is compact, B-upper semicontinuity is equivalent to weaksequential upper semicontinuity. Obviously |zn| ≤ C1 and we also have

ψt(sn, zn) + λ|zn|21hr(sn, |zn|)

|zn|− zn, A∗

inDϕ(sn, zn)

+ Fin(sn, zn, uin(sn, zn), Dψ(sn, zn), D2ψ(sn, zn)) ≤ 0.

(3.310)

We can assume that sn → s. Now either zn → 0 or for a subsequence (stilldenoted by zn) |zn| ≥ c1 > 0, n = 1, 2, ..., which implies hr(sn, |zn|)/|zn| > c2 > 0,n = 1, 2, .... It then follows from the local uniform boundedness of the Fn andA∗

inDϕ, that |zn|1 ≤ C2 which implies zn z in H1 for some z ∈ H1 and thus,since B is compact, zn → z in H.

Therefore u+(s, z) ≥ lim supn→∞ uin(sn, zn) which, together with (3.309), gives

0 ≥ (u+ − h)∗,−1(s, z)− ϕ(s, z) ≥ u+(s, z)− h(s, z)− ϕ(s, z)

≥ lim supn→∞

(uin(sn, zn)− h(sn, |zn|)− ϕ(sn, zn)) ≥ 0.

Thus (s, z) = (t, x) and moreover

(u+ − h)∗,−1(t, x) + h(t, x) = lim supn→∞

uin(sn, zn).

It now remains to pass to lim infn→+∞ in (3.310) and use (3.70) to conclude theproof.

If F+ = F− and comparison holds for the limiting equation one can obtainthe convergence of the un to the unique viscosity solution of the limiting equation.Moreover, the limiting Hamiltonians F+ and F− may be of first order so the abovetheorem can be applied to singular perturbation problems discussed in Section 3.8.Other applications related to the convergence of finite dimensional approximations(like these in Section 3.7) when condition (3.287) is satisfied by the operators An

only on a family of finite dimensional spaces can be found in [288].


3.10. Infinite dimensional Black-Scholes-Barenblatt equation

In this section we show how the theory of viscosity solutions and the results ofprevious sections can be used to deal with the infinite dimensional Black-Scholes-Barenblatt equation (2.145) introduced in Section 2.6.7. We refer the reader toSection 2.6.7 for details about the financial meaning of the equation and the asso-ciated optimal control the problem.

Let H be the Sobolev space H1([0,+∞)) and let A be the maximal dissipativeoperator

D(A) := H2([0,+∞))

A(x)(σ) :=dx

dσ(σ)

The operator A generates, by Theorem B.45, a C0-semigroup of contractions etA

in H. Let B be a bounded, self-adjoint, strictly positive operator satisfying (3.2).We introduce the space

V =

x ∈ H : σ →

√σx(σ), σ →

√σdx

dσ(σ) ∈ L2(0,∞)

,

equipped with the norm

|x|2V =

∞

0(1 + σ)

x2(σ) +

dx

dσ(σ)

2dσ.

and we denote by Λ a fixed bounded and closed subset of Vd. The space H will bethe state space and Λ will be the control space.

The set of admissible controls Ut is defined as in Section 2.1.2, where W inreference probability spaces ν there are d-dimensional standard Brownian motions.

Lemma 3.96 The function b : Vd → H defined by

b(x)(σ) =d

k=1

xk(σ)

σ

0xk(µ)dµ,

is locally Lipschitz. Above x = (x1, ..., xk).

Proof. Let R > 0 and x, y ∈ Vd be such that |xk|V, |yk|V ≤ R, for k =1, ..., d. Then

|b(x)(σ)− b(y)(σ)|2 ≤d

k=1

2d

σ|xk(σ)− yk(σ)|2

σ

0x2k(µ)dµ

+ σy2k(σ)

σ

0|xk(µ)− yk(µ)|2dµ

.

Integrating we have +∞

0|b(x)(σ)− b(y)(σ)|2dσ ≤

d

k=1

2dR2

+∞

0(1 + σ)|(xk − yk)(σ)|2dσ.

Similarly, we obtain

|(b(x))(σ)− (b(y))(σ)|2 ≤ 3dd

k=1

4R2|xk(σ)− yk(σ)|2

+ σ ((yk))2(σ)

σ

0|xk(µ)− yk(µ)|2dµ

+ σ |((xk)) (σ)− ((yk)

) (σ)|2 σ

0x2k(µ)dµ

3.10. INFINITE DIMENSIONAL BLACK-SCHOLES-BARENBLATT EQUATION 227

which after integration yields +∞

0|(b(x))(σ)− (b(y))(σ)|2dσ ≤3d(4R2 + 1)

d

k=1

+∞

0|(xk − yk)(σ)|2dσ

+

+∞

0σ |((xk)

) (σ)− ((yk)) (σ)|2 dσ

.

The claim now follows easily.

The previous lemma implies in particular that for τ ∈ Ut, the process b(τ(s))is progressively measurable and bounded. Therefore the state equation for theproblem

dr(s) = (Ar(s) + b(τ(s)))ds+ τ(s) · dW (s), s ∈ (t, T ]r(t) = x,

(3.311)

is well posed in H for any reference probability space ν = (Ω,F ,F ts ,P,W ) and

any τ(·) ∈ Ut (see Theorem 1.121). We denote its unique mild solution by r(·).Our control problem consists in maximizing the cost functional

Ee−

T

tr+(s,0)dsg(r(T ))

(3.312)

over all controls τ(·) ∈ Ut. (We used r+(s, 0) to denote r+(s)(0).) This defines thevalue function

V (t, x) := supτ(·)∈Ut

Ee−

T

tr+(s,0)dsg(rT )

.

We assume the function g satisfies the following hypothesis.

Hypothesis 3.97 The function g is locally uniformly B-continuous and

|g(x)| ≤ C(1 + |x|m) for all x ∈ H,

for some C,m ≥ 0.

We can now apply the results of the previous sections to the HJB of the problem(2.145). Observe first that, if we define

c(x) = x+(0),

for x in H, then c is weakly sequentially continuous on H and so it is uniformlycontinuous in the | · |−1 norm on bounded sets of H. Moreover it is easy to see thatc has at most linear growth at infinity and for instance

x+(0) ≤ 2|x| (3.313)

so the hypotheses needed to prove the “existence” part of of Theorem 3.66 aresatisfied. It guarantees the existence of a local modulus ω such that

|V (t, x)− V (s, y)| ≤ ω|t− s|+ |x− y|−1;R

(3.314)

for all 0 ≤ t, s ≤ T , x, y ∈ B(0, R). It also ensures that V is a viscosity solution ofthe Hamilton-Jacobi-Bellman equation (the BSB equation) (2.145).

As regards the uniqueness of viscosity solutions of the BSB equation (2.145) wenotice that Hypotheses 3.44-3.46, 3.48, 3.49 with γ = 0 are satisfied. To guaranteeHypothesis 3.47 we need an additional assumption.

We suppose that Λ is a compact subset of Hd−1. It is then obvious that

supτ∈Λ

d

i=1

|QNτi|2−1 → 0 as N → ∞,


where QN is defined as in Section 3.5. This implies that Hypothesis 3.47 holds.Therefore, by Theorem 3.50, comparison holds for (2.145) and thus we have thefollowing result.

Theorem 3.98 Let Hypothesis 3.97 hold and let Λ be a bounded and closed subsetof Vd which is also a compact subset of Hd

−1. Then the value function V satisfies(3.314) and is the unique viscosity solution of the BSB equation (2.145) amongfunctions satisfying (3.76) with γ = 0 and

limt→T

|u(t, x)− g(x))| = 0 (3.315)

uniformly on bounded sets.

If g is bounded and weakly sequentially continuous, it can be shown that thevalue function can be approximated by viscosity solutions of finite dimensionalapproximations of the BSB equation (2.145). This assumption holds in many in-teresting cases, for example if g is given by (2.141). We refer to [287] for details ofthis.

3.11. HJB equation for control of the Duncan-Mortensen-Zakaiequation

This section is a continuation of Section 2.6.6 and the reader should be familiarwith it. We have seen in Section 2.6.6 how the Duncan-Mortensen-Zakai (DMZ)equation arises in control problems with partial observation. In the so called “sep-arated" problem the DMZ equation is the state equation for the unnormalizedconditional probability density of the state process with respect to the observationprocess. This gives rise to an optimal control problem for the DMZ equation whichis fully observable. In this section we discuss how the HJB techniques can be ap-plied to this problem. We first present basic results about variational solutions ofSPDE.

3.11.1. Variational solutions. In this section we make the following as-sumptions. Let V , H be real separable Hilbert spaces. We identify H with itsdual H . Suppose that V is continuously and densely embedded in H. We thenhave the continuous and dense embeddings

V ⊂ H ⊂ V

and V is also separable. We denote the norms in V , H, V by | · |V , | · |, | · |V

respectively. The inner product in H is denoted by ·, ·. The duality pairingbetween V and V are denoted by ·, ·V ,V . The duality pairing agrees with theinner product on H, i.e. for every x ∈ H, v ∈ V, x, v = x, vV ,V . The triple(V,H, V ) with the above properties is called a Gelfand triple.

Let H, Ξ be real separable Hilbert spaces, Λ be a Polish space, Q ∈ L+1 (Ξ)

and T ∈ (0,+∞). Let µ =Ω,F , Fss∈[0,T ] ,P,WQ

be a generalized reference

probability space. Let a(·) ∈ Uµ := Uµ0 (see (2.1)). We assume the following

hypothesis.

Hypothesis 3.99 The following conditions are satisfied:(i) The linear operators A(t, a) : V → V are closed with a common domain

D(A) for (t, a) ∈ [0, T ] × Λ, and for every t ∈ [0, T ], the map A :[0, T ]×Λ×V → V , A(t, a, v) = A(t, a)v, restricted to [0, t]×Λ×V , isB([0, t]) ⊗ B(Λ) ⊗ B(V )/B(V ) measurable. Moreover there exist C, γ,and β > 0 such that for all u, v ∈ V ,

|A(s, a)u, vV ,V | ≤ C|u|V |v|V , (s, a) ∈ [0, T ]× Λ, (3.316)

3.11. HJB EQUATION FOR CONTROL OF THE DUNCAN-MORTENSEN-ZAKAI EQUATION229

A(s, a)v, vV ,V ≤ −β|v|2V + γ|v|2, (s, a) ∈ [0, T ]× Λ. (3.317)(ii) The functions b : [0, T ]×V ×Λ → H and σ : [0, T ]×V ×Λ → L2(Ξ0, H)

are such that for every t ∈ [0, T ] their restrictions to[0, T ] × V × Λ are respectively B([0, t]) ⊗ B(V ) ⊗ B(Λ)/B(H) andB([0, t])⊗ B(V )⊗ B(Λ)/B(L2(Ξ0, H)) measurable.

(iii) There exists C such that for all u, v ∈ V

|b(s, v, a)|+ σ(s, v, a)L2(Ξ0,H) ≤ C(1 + |v|V ), (3.318)

|b(s, u, a)− b(s, v, a)|+ σ(s, u, a)− σ(s, v, a)L2(Ξ0,H) ≤ C|u− v|V ,(3.319)

for all (s, a) ∈ [0, T ]× Λ.

(iv) There exist C, γ1, and β1 > 0 such that for all v ∈ V ,

A(s, a)v, vV ,V + σ(s, v, a)2L2(Ξ0,H) ≤ −β1|v|2V + γ1|v|2 + C,(3.320)

for all (s, a) ∈ [0, T ]× Λ.

(v) There exists δ such that for all u, v ∈ V ,

2A(s, a)(u− v), u− vV ,V + 2b(s, u, a)− b(s, v, a), u− v+ σ(s, u, a)− σ(s, v, a)2L2(Ξ0,H) ≤ δ|u− v|2, (3.321)

for all (s, a) ∈ [0, T ]× Λ.

It is now easy to see that the maps A(s, a(s))v, b(s, v, a(s)), σ(s, v, a(s)) definedon [0, T ]× Ω× V are such that for every t ∈ [0, T ] their restrictions to [0, t]× Ω×V are respectively B([0, t]) ⊗ Ft ⊗ B(V )/B(V ), B([0, t]) ⊗ Ft ⊗ B(V )/B(H) andB([0, t]) ⊗ Ft ⊗ B(V )/B(L2(Ξ0, H)) measurable, and they satisfy (3.316)-(3.321)(with a replaced by a(s)) for a.e. (s,ω) ∈ [0, T ]× Ω.

We consider the following stochastic PDE

dX(s) = (A(s, a(s))X(s) + b(s,X(s), a(s)))ds+ σ(s,X(s), a(s)))dWQ(s)

X(0) = ξ.(3.322)

Definition 3.100 (Variational solution of (3.322)) A process X(·) ∈M2

µ(0, T ;H) is called a variational solution of (3.322) if

E T

0|X(r)|2V dr

< +∞

and for every φ ∈ V we have

X(s),φ = ξ,φ+ s

0A(r, a(r))X(r),φV ,V dr +

s

0b(r,X(r), a(r)),φdr

+

s

0σ(r,X(r), a(r))dWQ(r),φ for each s ∈ [0, T ] , P-a.e.. (3.323)

We remark that the integrand A(r, a(r))X(r) above is evaluated at a V valuedprogressively measurable equivalent version of X(·), and the process 1X(s)∈V X(s)is equivalent to the process X(s) and, by Lemma 1.17-(iii), belongs to M2

µ(0, T ;V ).Moreover, (3.323) is equivalent to the equality

X(s) = ξ +

s

0(A(r, a(r))X(r) + b(r,X(r), a(r)))dr +

s

0σ(r,X(r), a(r))dWQ(r)


as elements of V .The following result is taken from [296], Theorem I.3.1, in the version from

[220], Theorem 4.3, page 165, see also [385], Theorem 4.2.5, and [406].

Theorem 3.101 Let µ be a generalized reference probability space, ξ be F0

measurable H valued random variable such that Eµ[|ξ|2] < +∞, let Y (·) ∈M2

µ(0, T ;V), Z(·) ∈ N 2

Q(0, T ;H). We define the continuous V valued process

X(s) = ξ +

s

0Y (r)dr +

s

0Z(r)dWQ(r), s ∈ [0, T ].

If X(·) has an equivalent version X(·) ∈ M2µ(0, T ;V ), then X(·) ∈ M2

µ(0, T ;H) ∩L2(Ω;C([0, T ];H)),

E

sup0≤s≤T

|X(s)|2≤ +∞,

and the following Ito’s formula holds P-a.e.

|X(s)|2 = |ξ|2 + s

0

2Y (r), X(r)V ,V + Z(r)2L2(U0,H)

dr

+2

s

0Z(r)dWQ(r), X(r) s ∈ [0, T ].

(3.324)

Theorem 3.102 Let µ be a generalized reference probability space, ξ be F0 mea-surable H valued random variable such that Eµ[|ξ|2] < +∞, and a(·) ∈ Uµ. Then:

(i) There exists a unique variational solution of (3.322) X(·) ∈L2(Ω;C([0, T ];H)), and the energy equality holds P-a.e.

|X(s)|2 = |ξ|2 + 2

s

0A(r, a(r))X(r), X(r)V ,V dr + 2

s

0b(r,X(r), a(r)), X(r)dr

+2

s

0σ(r,X(r), a(r))dWQ(r), X(r)+

s

0σ(r,X(r), a(r))2L2(Ξ0,H)dr s ∈ [0, T ].

(3.325)

(ii) If µ1 is another generalized reference probability space, ξ1 is a Fµ10

measurable H-valued random variable such that Eµ1 [|ξ1|2] < +∞,a1(·) ∈ Uµ, and

LP1(ξ1, a1(·),WQ,1(·)) = LP(ξ, a(·),WQ(·)),

thenLP1(a1(·), X1(·)) = LP(a(·), X(·)), (3.326)

where X1(·) is the variational solution of (3.322) in µ1 with control a1(·)and initial condition ξ1.

Proof. We sketch the proof. The complete proof of the first part of thetheorem can be found in [93], pages 168–183 or [220, 296, 298, 385, 406]. Letv1, v2, ... be an orthonormal basis of H composed of elements of V . We setHn := spanv1, ..., vn, and define

Pnw :=n

k=1

w, vkV ,V vk, w ∈ V .

If w ∈ H we have

Pnw =n

k=1

w, vkvk


so Pn is an extension to V of the orthogonal projection in H onto Hn. We set

An(s,α)v := PnA(s,α)v, bn(s, v,α) := Pnb(s, v,α), σn(s, v,α) := Pnσ(s, v,α)

ξn := Pnξ.

(One can also project the Wiener process on a finite dimensional subspace but itis not necessary.) Since the above functions are Lipschitz continuous in v on Hn,standard theory guarantees that there exists a unique strong solution (in the senseof Definition 1.112) Xn(·) of

dXn(s) = (An(s, a(s))Xn(s) + bn(s,Xn(s), a(s)))ds+ σn(s,Xn(s), a(s)))dWQ(s)

Xn(0) = ξn

(3.327)which satisfies Xn(·) ∈ L2

µ(Ω;C([0, T ];H)) ∩M2µ(0, T ;V ). Moreover, Ito’s formula

gives P-a.e.

|Xn(s)|2 = |ξn|2 + 2

s

0An(r, a(r))Xn(r), Xn(r)dr

+ 2

s

0bn(r,Xn(r), a(r)), Xn(r)dr + 2

s

0σn(r,Xn(r), a(r))dWQ(r), X

n(r)

+

s

0σn(r,Xn(r), a(r))2L2(Ξ0,H)dr,

(3.328)

and using (3.328) and the assumptions one shows

E

sup0≤s≤T

|Xn(s)|2 + T

0|Xn(s)|2V ds

≤ M for all n.

Therefore, up to subsequences still denoted by Xn, bn, σn, we obtain that thereexist X(·), b(·) ∈ M2

µ(0, T ;H), X(·) ∈ M2µ(0, T ;V ), σ(·) ∈ N 2

Q(0, T ;H), and a FT

measurable random variable η ∈ L2(Ω;H) such that

Xn(·) X(·) in M2µ(0, T ;V ), Xn(T ) η in L2(Ω;H),

Xn(·) X(·), bn(r,Xn(r), a(r)) b(·) in M2µ(0, T ;H),

σn(r,Xn(r), a(r)) σ(·) in N 2Q(0, T ;H).

Obviously X(·) is an equivalent version of X(·). Passing to the limit as n → +∞one obtains that X(·) is a variational solution of the linear equation

dX(s) = (A(s, a(s))X(s) + b(s)ds+ σ(s)dWQ(s)

X(0) = ξ,

X(T ) = η, and, by Theorem 3.101, X(·) ∈ L2µ(Ω;C([0, T ];H)) and it satisfies P-a.e.

|X(s)|2 = |ξ|2 + 2

s

0A(r, a(r))X(r), X(r)V ,V dr + 2

s

0b(r), X(r)dr

+2

s

0σ(r)dWQ(r), X(r)+

s

0σ(r)2L2(U0,H)dr.

One then uses monotonicity arguments to prove that

b(r) = b(r,X(r), a(r)), σ(r) = σ(r,X(r), a(r)) dt⊗ P a.e.

and hence X(·) is a variational solution of (3.322) and (3.325) holds. More-over it also follows from these arguments (see for instance [296] for details) thatE[|Xn(T )|2] → E[|X(T )|2] and so Xn(T ) → X(T ) in L2(Ω;H). Replacing interval


[0, T ] by another interval [0, t], 0 < t < T the same arguments give Xn(t) → X(t)in L2(Ω;H) for all 0 ≤ t ≤ T .

The uniqueness of variational solution in the generalized reference probabilityspace µ follows from Theorem 3.101, elementary estimates using the assumptionson the coefficients, and Gronwall’s lemma.

Finally, if X1(·) is the variational solution in the generalized reference proba-bility space µ1 and Xn

1 (·) are the solutions of the approximating problems (3.327)in this space then we have

LP1(a1(·), Xn1 (·)) = LP(a(·), Xn(·)),

and thus (3.326) follows since Xn1 (t) → X1(t) in L2(Ω1;H) and Xn(t) → X(t) in

L2(Ω;H) for every t ∈ [0, T ].

3.11.2. Weighted Sobolev spaces. We denote the norm and the inner prod-uct in Rd by |·|Rd and ·, ·Rd respectively. Given k ∈ N we denote by Hk := Hk

Rd

the standard Sobolev space on Rd. We remind that H0 = L2Rd

, and the inner

product in H0 will be denoted by ·, ·0. Let B := (−∆+ I)−1, where D(∆) = H2.We equip Hk with the inner product

x, yk := B−k/2x,B−k/2y0.This inner product gives the norm |x|k = |B−k/2x| which is equivalent to thestandard norm in Hk given by

|α|≤k

Rd

|∂αx(ξ)|2ξ

12

.

The topological dual space of Hk is denoted by H−k. Except when explicitlystated we always identify H0 with its dual. The space H−k can be identified withthe completion of H0 under the norm

|x|−k := |Bk/2x| = Bkx, x1/20

and then B1/2 (after a natural extension) is an isometry between Hk and Hk+1,k ∈ Z. The space H−k, k ∈ N is a Hilbert space equipped with the inner product

x, y−k := Bk/2x,Bk/2y0.The duality pairing between H−k and Hk, k ∈ N is denoted by ·, ·H−k,Hk. Wehave a, bH−k,Hk =

Bk/2a,B−k/2b

0. Observe also that, for k ∈ Z, the adjoint

of the operator B1/2 : Hk → Hk+1 is B1/2 : H−k−1 → H−k.Let k = 0, 1, 2. Given a strictly positive real-valued function ρ ∈ C2(Rd), we

define the weighted Sobolev space Hkρ (Rd) (or simply Hk

ρ ) to be the completion ofC∞

c (Rd) with respect to the weighted norm

|x|k,ρ := |ρx|k.The space Hk

ρ can also be defined as the space of all measurable functions x : Rd →R such that ρ(·)x(·) ∈ Hk. We recall that the norm |x|k,ρ is equivalent to the normgiven by

|α|≤k

Rd

|∂α [ρ(ξ)x(ξ)]|2 dξ

1/2

.

We denote by Cρ the isometry Cρ : Hkρ → Hk defined as (Cρx) (ξ) = ρ (ξ)x (ξ) ,

and by C1/ρ = C−1ρ : Hk → Hk

ρ its inverse:C1/ρx

(ξ) = ρ−1 (ξ)x (ξ). We observe

that Hkρ is a Hilbert space with the inner product x, yk,ρ = Cρx,Cρyk. We


denote the topological dual space of Hkρ by H−k

ρ and, identifying H0ρ with its dual,

we haveHk

ρ ⊂ H0ρ = [H0

ρ ] ⊂

Hk

ρ

= H−k

ρ , k ≥ 0. (3.329)

We always use this identification, except when explicitly stated.The adjoint C∗

ρ of Cρ is an isometry C∗ρ : H−k −→ H−k

ρ . Observe that C∗ρ can

be identified with C1/ρ.To simplify notation we write Xk := Hk

ρ .Let Bρ := C1/ρ

(−∆+ I)−1

Cρ = C1/ρBCρ. Similarly to the case of

non-weighted spaces X−k can be identified with the completion of X0 underthe norm |x|2−k,ρ := Bk

ρx, x0,ρ = BkCρx,Cρx0 and then B1/2ρ is an isom-

etry between X−2, X−1, X0, X1 and X−1, X0, X1, X2 respectively. We remarkthat B−1

ρ = C1/ρB−1Cρ, B1/2

ρ = C1/ρB1/2Cρ, B−1/2

ρ = C1/ρB−1/2Cρ. Thus

|x|−k,ρ = |Bk/2ρ x|0,ρ and |x|k,ρ = |B−k/2

ρ x|0,ρ. The duality pairing between X−k

and Xk is denoted by ·, ·X−k,Xk. We have

a, bX−k,Xk = Bk/2ρ a,B−k/2

ρ b0,ρ = Cρa, CρbH−k,Hk.

In what follows we consider weight functions ρ of the form

ρβ (ξ) = 1 + |ξ|βRd , β > 2. (3.330)

With such choice of ρ it can be showed that Xk ⊂ Hk, and if β > d/2 thenX0 ⊂ L1

Rd

and Xk ⊂ W k,1

Rd

.

3.11.3. Optimal control of the Duncan-Mortensen-Zakai equation.We now study the optimal control problem for the DMZ equation derived in Section2.6.6. We study it in the weighted space X0 using the formalism of abstract controlproblems. Let T > 0. The control set Λ was originally a subset of Rn but we willconsider Λ to be a more general Polish space. For every 0 ≤ t ≤ T , the referenceand generalized reference probability spaces are defined by Definitions 2.7 and 1.95where WQ is now just a standard Wiener process in Rm (i.e. Ξ = Rm, Q = I) whichis denoted by W . The classes of admissible controls with respect to all referenceand generalized reference probability spaces are defined as always by Ut and U t.We remark that for a reference probability space µ =

Ω,F , F t

ss∈[t,T ] ,P,W,

P now corresponds to P in Section 2.6.6, F ts to F y1,t

s , and W to y1. Without lossof generality we will always assume that the Q-Wiener processes in the referenceprobability spaces have everywhere continuous paths.

Recall from Section 2.6.6 that for every a ∈ Λ we have the differential operatorsAa and Sk

a (k = 1, . . . ,m)

(Aax) (ξ) =d

i,j=1

∂i [ai,j(ξ, a)∂jx(ξ)] +d

i=1

∂i [bi(ξ, a)x(ξ)] , (3.331)

Skax

(ξ) =

d

i=1

dik(ξ, a)∂ix(ξ) + ek(ξ, a)x(ξ); k = 1, ...,m. (3.332)

Typically we set D(Aa) = X2, D(Ska) = X1, k = 1, ...,m, however the operators

will be considered with different domains in different Gelfand triples. Having inmind the original Hypothesis 2.46, we assume the following hypothesis.

Hypothesis 3.103(i) Λ is a compact metric space.


(ii) The coefficients

(aij)i,j=1,...,d , (bi)i=1,...,d , c, (dik)i=1,...,d; k=1,...,m , (ek)k=1,...,m : Rd × Λ → R

are continuous in (ξ, a) and, as functions of ξ are in C2b

Rd

for every

a, with their norms in C2b

Rd

bounded uniformly in a ∈ Λ. Moreover,

there exists a constant λ > 0 such thatd

i,j=1

ai,j(ξ, a)−

1

2

m

k=1

dik(ξ, a)djk(ξ, a)

zizj ≥ λ|z|2 (3.333)

for every a ∈ Λ and ξ, z ∈ Rd.(iii) The weight ρ is of the form (3.330).

For every a(·) ∈ Ut the DMZ equation is considered in X0:

dY (s) = Aa(s)Y (s)ds+m

k=1 Ska(s)Y (s)dWk(s), s > t

Y (t) = x ∈ X0.(3.334)

We consider the cost functional

J(t, x; a(·)) = E T

tl(Y (s), a(s))ds+ g(Y (T ))

and we make the following assumptions about the cost functions l and g.

Hypothesis 3.104(i) l : X0 × Λ → R and g : X0 → R are continuous and there exist C > 0

and γ < 2 such that

|l (x, a)| , |g (x)| ≤ C1 + |x|γ0,ρ

for every (x, a) ∈ X0 × Λ;(ii) for every R > 0 there exists a modulus ωR such that

|l (x, a)− l (y, a)| ≤ ωR

|x− y|0,ρ

, |g(x)− g(y)| ≤ ωR

|x− y|−1,ρ

(3.335)

for every x, y ∈ BX0 (0, R), a ∈ Λ.

The optimal control problem in the weak formulation we study consists inminimizing the cost J(t, x; a(·)) over all admissible controls a(·) ∈ Ut.

The associated HJB equation in X0 has the form

vt + infa∈Λ

12

mk=1

D2vSk

ax, Skax

ρ,0

+ Aax,Dvρ,0 + l(x, a)= 0,

in (0, T )×X0,

v(T, x) = g(x),(3.336)

and thevalue function is


J(t, x; a(·)). (3.337)

Let us now describe which restrictions may be placed on the original separatedproblem of Section 2.6.6 so that the current assumptions be satisfied.

First the law of the initial datum η of equation (2.129) must have density x inL2ρ(Rd). To guarantee that the density is also in L1(Rd) we should assume β > d/2.

The density x is polynomially decreasing when |ξ|Rd → +∞. This is of course afurther restriction with respect to assuming only x ∈ L1

Rd

but it is satisfied in

many practical cases, for instance when the starting distribution is normal. Onecan consult e.g. [34], pages 36, 204, for the use of x being Gaussian or [34], pages


82, 167, for other integrability assumptions on x (see also [468], [369] on this).Regarding the cost functional 2.134 we have (recall that we now use E in place ofE)

J(t, x; a(·)) = E T

tl1(·, a(s)), Y (s)0 ds+ g1(·), Y (T )0

= E T

t

(1/ρ2)l1(·, a(s)), Y (s)

0,ρ

ds+(1/ρ2)g1(·), Y (T )

0,ρ

= E T

tl(Y (s), a(s))ds+ g(Y (T ))

,

where we set

l (x, a) = l1 (·, a) , x0 =

1

ρ2l1 (·, a) , x

0,ρ

,

g (x) = g1 (·) , x0 =

1

ρ2g1 (·) , x

0,ρ

. (3.338)

It is easy to see that Hypothesis 3.104 is satisfied if the functions l1 : Rd ×Rn → Rand g1 : Rd → R are continuous and supa∈Λ | 1ρ l1 (·, a) |0+ | 1ρg1 (·) |0 < +∞, since inthis case the function g is weakly sequentially continuous in X0. For instance, if

l1(ξ,α) = Mξ, ξRd + Nα,αRn , g1(ξ) = Gξ, ξRd ,

where M , N and G are suitable non-negative definite matrices, Λ = BRn(0, R),Hypothesis 3.104 is satisfied if β > 2 + d/2. This is the main advantage of usingweighted spaces. When the initial density is, say, polynomially decreasing at infin-ity, we can deal with polynomially growing cost functions. This is not possible ifwe took ρ = 1. Finally we mention that in the absence of density the separatedproblem has to be studied in the space of measures.

3.11.4. Estimates for the DMZ equation.

Lemma 3.105 Let Hypothesis 3.103 hold. Then:

(i) The DMZ equation satisfies the assumptions of Hypothesis 3.99 for theGelfand triple (X1, X0, X−1) and also for (X2, X1, X0).

(ii) There exist constants λ > 0,K ≥ 0 such that for all a ∈ Λ

Aax, xX−1,X1 +1

2

m

k=1

Skax, S

kax

0,ρ

≤ −λ |x|21,ρ +K |x|20,ρ , x ∈ X1, (3.339)

Aax,B

−1ρ x

0,ρ

+1

2

m

k=1

B−1

ρ Skax, S

kax

X−1,X1

≤ −λ |x|22,ρ +K |x|21,ρ , x ∈ X2, (3.340)

Aax,BρxX−2,X2 +1

2

m

k=1

BρS

kax, S

kax

X1,X−1

≤ −λ |x|20,ρ +K |x|2−1,ρ , x ∈ X0. (3.341)

Proof. Part (i) follows from direct computations and estimates in (ii). Part(ii) is proved in [245], Lemma 3.3.


We also record for future use that

supa∈Λ,k=1,...,m

BρAaL(X0) + B1/2

ρ SaL(X0)

≤ C (3.342)

Proposition 3.106 Assume that Hypothesis 3.103 hold. Let 0 ≤ t ≤ T , let µbe a generalized reference probability space, a(·) ∈ Uµ

t and x ∈ L2 (Ω;X0) be anFt measurable random variable. Then there exists a unique variational solutionY (s) := Y (·; t, x, a(·)) ∈ L2(Ω;C([t, T ];H)) of the state equation (3.334). Moreoverwe have:

• P a.s.

|Y (s)|20,ρ = |x|20,ρ + 2

s

t

Aa(r)Y (r), Y (r)

X−1,X1 dr

+m

k=1

s

t

Ska(r)Y (r), Y (r)

0,ρdWk(r) (3.343)

+m

k=1

s

t

Ska(r)Y (r), Sk

a(r)Y (r)

0,ρdr, s ∈ [t, T ].

In particular

E |Y (s)|20,ρ = E |x|20,ρ + 2E s

t

Aa(r)Y (r), Y (r)

X−1,X1 dr

+m

k=1

E s

t

Ska(r)Y (r), Sk

a(r)Y (r)

0,ρdr. (3.344)

• There exists a constant C > 0 independent of µ, a(·) ∈ Uµt and x such

thatE |Y (s)|20,ρ ≤ E |x|20,ρ (1 + C(s− t)), (3.345)

E T

t|Y (s)|21,ρ ds ≤ CE |x|20,ρ . (3.346)

• The conclusion of Theorem 3.102-(ii) (with 0 replaced by t) is satisfied.

Proof. The results follows from Theorem 3.102.

The following proposition collects various estimates for solutions of (3.334).

Proposition 3.107 Assume that Hypothesis 3.103 hold and let 0 ≤ t ≤ T . Leta(·) ∈ Ut and x ∈ X0. Then:

(i) There exists a constant C > 0 independent of a(·) ∈ Ut and x such that

E |Y (s)|2−1,ρ ≤ |x|2−1,ρ (1 + C(s− t)), (3.347)

E T

t|Y (s)|20,ρ ds ≤ C |x|2−1,ρ . (3.348)

E |Y (s)− x|2−1,ρ ≤ C (s− t) |x|20,ρ , (3.349)

E s

t|Y (r)− x|20,ρ dr ≤ C (s− t) |x|20,ρ . (3.350)

There is a modulus σx, independent of a(·) ∈ Ut such that

E |Y (s)− x|20,ρ ≤ σx(s− t). (3.351)


(ii) If in addition x ∈ X1, then Y (s) is a strong solution and there exists aconstant C > 0 independent of a(·) ∈ Ut and x such that

E |Y (s)|21,ρ ≤ |x|21,ρ (1 + C(s− t)), (3.352)

E T

t|Y (s)|22,ρ ds ≤ C |x|21,ρ , (3.353)

E |Y (s)− x|20,ρ ≤ C |x|21,ρ (s− t), (3.354)

E s

t|Y (r)− x|21,ρ dr ≤ C(s− t) |x|21,ρ . (3.355)

There is a modulus σx, independent of a(·) ∈ Ut such that

E |Y (s)− x|21,ρ ≤ σx(s− t). (3.356)

Proof.(i). By Itô’s formula we have

E |Y (s)|2−1,ρ = EB1/2

ρ Y (s)2

0,ρ

= EB1/2

ρ x2

0,ρ+ 2E

s

t

B1/2

ρ Aa(r)Y (r), B1/2ρ Y (r)

0,ρdr

+m

k=1

E s

t

B1/2

ρ Ska(r)Y (r), B1/2

ρ Ska(r)Y (r)

0,ρdr. (3.357)

Since B1/2

ρ Aa(r)Y (r), B1/2ρ Y (r)

0,ρ=

Aa(r)Y (r), BρY (r)

X−2,X2

and B1/2

ρ Ska(r)Y (r), B1/2

ρ Ska(r)Y (r)

0,ρ=

BρS

ka(r)Y (r), Sk

a(r)Y (r)

X1,X−1

we have, thanks to (3.341),

E |Y (s)|2−1,ρ + 2λ

s

tE |Y (r)|20,ρ dr ≤ |x|2−1,ρ + 2K

s

tE |Y (r)|2−1,ρ dr.

Estimates (3.347)-(3.348) follow by applying Gronwall’s inequality.To show (3.349), we have

E |Y (s)− x|2−1,ρ = E |Y (s)|2−1,ρ + |x|2−1,ρ − 2E Y (s), Bρx0,ρwhich gives, by (3.357) and by the definition of variational solution,

E |Y (s)− x|2−1,ρ

= 2E s

t

Aa(r)Y (r), BρY (r)

X−1,X1 +

1

2

m

k=1

BρS

ka(r)Y (r), Sk

a(r)Y (r)

0,ρ

dr

−2E s

t

BρAa(r)Y (r), x

0,ρ

.

Therefore, by (3.341),

E |Y (s)− x|2−1,ρ + 2λE s

t|Y (r)|20,ρ dr

≤ 2KE s

t|Y (r)|2−1,ρ dr + 2E

s

t|x|0,ρ

BρAa(r)Y (r)0,ρ

dr

which, upon using (3.342), (3.345) and straightforward calculations, yields

E |Y (s)− x|2−1,ρ + λE s

t|Y (r)|20,ρ dr ≤ C (s− t)

|x|2−1,ρ + |x|20,ρ


for some constant C > 0. This proves (3.349). Estimate (3.350) follows uponnoticing that

E s

t|Y (r)− x|20,ρ dr ≤ 2E

s

t

|Y (r)|20,ρ + |x|20,ρ

dr ≤ C (s− t) |x|20,ρ . (3.358)

To prove (3.351), we assume by contradiction that it is not satisfied. In this casethere are an(·) ∈ Ut (which we can assume to be F t,0

s -predictable) and tn → t suchthat E|Yn(tn)− x|20,ρ → 0 as n → +∞. Because of Corollary 2.21 and Proposition3.106 we can assume that all an(·) are defined on the same reference probabilityspace. Since E |Y (tn)|20,ρ is bounded, there exists a subsequence, still denoted bytn, tn → 0 as n → +∞, and an element Y of L2(Ω;X0) such that, as n → +∞

Y (tn) Y , weakly in L2(Ω;X0)

and hence also weakly in L2(Ω;X−1). Since by (3.349)

Y (tn) → x, strongly in L2(Ω;X−1)

as n → +∞, we obtain Y = x. This, plus the fact that E |Y (tn)|20,ρ → |x|20,ρ asn → +∞ provided by (3.345), implies that Y (tn) → x, strongly in L2(Ω;X0) whichgives a contradiction.

(ii). The existence of the strong solution is known (see [295, 296, 298]) andcan be obtained similarly to Proposition 3.106 by applying it to the Gelfand triple(X2, X1, X0). One now obtains

E |Y (s)|21,ρ = E |x|21,ρ + 2E s

t

Aa(r)Y (r), Y (r)

X0,X2 dr

+m

k=1

E s

t

Ska(r)Y (r), Sk

a(r)Y (r)

1,ρdr,

which, upon using (3.340) and applying the same arguments as those in the proofof (i), yields (3.352) and (3.353). The proof of the final three estimates is analogousto the similar ones proved in (i).

3.11.5. Viscosity solutions. The definition of solution for (3.336) is similarto the general definition given in Section 3.3. However here we have unboundedfirst and second order terms so the equation is different. We also make use of thecoercivity of the operators Aa.

Definition 3.108 A function ψ is a test function if ψ = ϕ+ δ(t)|x|20,ρ, where:

(i) ϕ ∈ C1,2 ((0, T )×X0) is Bρ-lower semicontinuous, andϕt ∈ UC ((0, T )×X0) , Dϕ ∈ UC ((0, T )×X0;X2) , D2ϕ ∈BUC ((0, T )×X0;L(X−1;X1)).

(ii) δ ∈ C[0, T ] ∩ C1 (0, T ) is such that δ > 0.

Definition 3.109 A locally bounded Bρ-upper (respectively, lower) semicontin-uous function u : (0, T ] × X0 → R is a viscosity subsolution (respectively, super-solution) of (3.336) if u(T, x) ≤ g(x) (respectively, u(T, x) ≥ g(x)) on X0 and forevery test function ψ, whenever u − ψ (respectively u + ψ) has a global maximum(respectively, minimum) at (t, x) ∈ (0, T )×X0, then x ∈ X1,ρ and

ψt(t, x)+ infa∈Λ

1

2

m

k=1

D2ψ(t, x)Sk

ax, Skax

0,ρ+ Aax,Dψ(t, x)X−1,X1+f(x, a)

≥0,


respectively

− ψt(t, x) + infa∈Λ

1

2

m

k=1

−D2ψ(t, x)Sk

ax, Skax

0,ρ

+ Aax,−Dψ(t, x)X−1,X1 + f(x, a)

≤ 0. (3.359)

A function is a viscosity solution if it is both a viscosity subsolution and a viscositysupersolution.

The main difference between this and the definition in Section 3.3 is that werequire that the point x where the maximum/minimum occurs belongs to a smallersubspace X1 (of more regular functions. This is possible because of the coercivity ofthe operators Aa. In this way all terms appearing in the equation are well definedand there is no need to discard any of them. Such definitions originated for firstorder equations in [71, 106]. Compared to Definition 3.32 we put more conditionson ϕ and restricted the class of radial functions. The role of A∗ in Definition 3.32 isnow played by B−1

ρ . The radial test functions are quadratic since we only considersolutions with smaller growth rate at infinity, which is a reasonable assumption forvalue functions coming from separated problems (see (3.338)). We remark that thedefinition of viscosity solution here is different from the definition given in [245])where it was required that ϕ ∈ UC1,2 ((0, T )×X−1). Both allow to prove thesame results. We decided to change the definition to make it more in line with thepresentation of the material in the book.

If u has less than quadratic growth in x as |x|0,ρ → +∞ then the maxima andminima in the definition of solution can be assumed to be strict: if u−(ϕ+δ(t) |x|20,ρ)has a global maximum at (t, x) and λ ∈ C2([0,+∞)) is such that λ > 0, λ(r) = r4

if r ≤ 1 and λ(r) = 1 if r ≥ 2, then it is easy to see that u − (ϕ + δ(t) |x|20,ρ) −λ(|x− x|−1,ρ)− (t− t)2 has a strict global maximum at (t, x).

It is easy to see that Aax,Dϕ(t, x)X−1,X1 =BρAax,B−1

ρ Dϕ(t, x)−,ρ

andD2ϕ(t, x)Sk

ax, Skax

0,ρ

=B−1/2

ρ D2ϕ(t, x)B−1/2ρ B1/2

ρ Skax,B

1/2ρ Sk

ax

0,ρ, k =

1, ...,m. Moreover

B−1ρ Dϕ ∈ UC ((0, T )×X0;X0) , B−1/2

ρ D2ϕB−1/2ρ ∈ BUC ((0, T )×X0;L(X0))

(3.360)We also remark that Itô’s formula holds for the test functions. For the radial

part of test function this follows from (3.343) and Itô’s formula. As regards ϕ, theeasiest way to see it is to use the fact that if Y (s) := Y (·; t, x, a(·)) is the solutionof equation (3.334) on [t, T ], x ∈ X0 then Y (·) is in fact a strong solution on anyinterval [s, T ], s > t. Thus if ϕ is a test function as above and a(·) ∈ Ut then by theusual Itô’s formula we have for t < s < η

Eϕ(η, Y (η)) = Eϕ(s, Y (s)) + E η

s

ϕt (r, Y (r)) +

Aa(r)Y (r) , Dϕ (r, Y (r))

0,ρ

+1

2

m

k=1

D2ϕ (r, Y (r))Sk

a(r)Y (r) , Ska(s)Y (r)

0,ρ

dr

→ ϕ(t, x) + E η

t

ϕt (r, Y (r)) +

Aa(r)Y (r) , Dϕ (r, Y (r))

X−1,X1

+1

2

m

k=1

D2ϕ (r, Y (r))Sk

a(r)Y (r) , Ska(r)Y (r)

0,ρ

dr


as s → t using for instance (3.345) and (3.351), since by (3.342)

|ϕt (r, Y (r))|+Aa(r)Y (r) , Dϕ (r, Y (r))

X−1,X1

≤ C1 + |Y (r)|20,ρ

,

D2ϕ (r, Y (r))Sk

a(r)Y (r) , Ska(r)Y (r)

0,ρ

≤ C|Y (r)|20,ρ, k = 1, ...,m.

Itô’s formulas for test functions from [245] are proved in [364].We prove comparison principle. We use the notation of Section 3.2: en∞n=1 ⊂

X0 is now an orthonormal basis in X−1, XN = spane1, ..., eN, PN is the or-thogonal projection from X−1 onto XN , QN = I − PN , and Y N = QNX−1. Wehave an orthogonal decomposition X−1 = XN × Y N , and for x ∈ X−1 we writex = (PNx,QNx).

Theorem 3.110 Let Hypotheses 3.103 and 3.104 hold. Let u, v : X0 → R berespectively a viscosity subsolution, and a viscosity supersolution of (3.336) (asdefined in Definition 3.109). Let

lim sup|x|0,ρ→∞

u(t, x)

|x|20,ρ= 0, lim sup

|x|0,ρ→∞

−v(t, x)

|x|20,ρ= 0. (3.361)

uniformly for t ∈ [0, T ], and

(i) limt↑T (u(t, x)− g(x))+ = 0

(ii) limt↑T (v(t, x)− g(x))− = 0(3.362)

uniformly on bounded subsets of X0. Then u ≤ v.

Proof. Without loss of generality we can assume that u and −v are boundedfrom above and such that

lim|x|0,ρ→∞

u(t, x) = −∞, lim|x|0,ρ→∞

v(t, x) = +∞. (3.363)

To see this we notice that if K is the constant from (3.339) then for every η > 0

uη(t, x) = u(t, x)− ηe2K(T−t)|x|20,ρ, vη(t, x) = v(t, x) + ηe2K(T−t)|x|20,ρare respectively viscosity sub- and supersolutions of (3.336) and satisfy (3.362).This follows from (3.339) since, denoting h(t, x) = ηe2K(T−t)|x|20,ρ, we have

ht+supa∈Λ

1

2

m

k=1

D2hSk

ax, Skax

0,ρ

+ Aax,DhX−1,X1

≤ −ηKe2K(T−t)|x|20,ρ ≤ 0

on (0, T ) × X0. The functions uη,−vη satisfy (3.363). Therefore, if we can provethat uη ≤ vη for every η > 0 we recover u ≤ v by letting η → 0.

The proof basically follows the lines of the proof of Theorem 3.50. Supposethat u ≤ v. Let for µ, , δ,β > 0,

Ψ(t, s, x, y) := u(t, x)− v(s, y)− µ

t− µ

t−

|x− y|2−1,ρ

2− δ(|x|20,ρ + |y|20,ρ)−

(t− s)2

2β.

For every n ∈ N there exist pn, qn ∈ X0, an, bn ∈ R such that|pn|0,ρ, |qn|0,ρ, |an|, |bn| ≤ 1/n, and

Ψ(t, s, x, y) + Bρpn, x0,ρ + Bρqn, y0,ρ + ant+ bns

has a strict global maximum at (t, s, x, y) over (0, T ] × (0, T ] ×X0 ×X0. Arguinglike in the proof of Theorem 3.50 we obtain

lim supβ→0

lim supn→∞

(t− s)2

2β= 0 for fixed µ, , δ, (3.364)


|x|0,ρ + |y|0,ρ ≤ R for some R, independently of µ, , δ,β, n, (3.365)and

lim sup→0

lim supδ→0

lim supβ→0

lim supn→∞

|x− y|2−1,ρ

2= 0 for fixed µ. (3.366)

Therefore, it follows from (3.362), (3.363), and (3.364) that 0 < t, and s < T . Wenow fix N ∈ N. Defining

u1(t, x) = u(t, x)− µ

t− BρQN (x− y), x0,ρ

−

|QN (x− x)|2−1,ρ

+|QN (x− y)|2−1,ρ

2− δ|x|20,ρ + ant+ Bpn, x0,ρ

and

v1(s, y) = v(s, y) +µ

s− BρQN (x− y), y0,ρ

+

|QN (y − y)|2−1,ρ

+ δ|y|20,ρ − bns− Bqn, y0,ρ.

we see that

u1(t, x)− v1(s, y)−1

2|PN (x− y)|2−1,ρ −

1

2β|t− s|2

has a strict global maximum at (t, s, x, y). It now follows from Corollary 3.29 andthe proof of Theorem 3.27 that for every ν > 1 there exist test functions ϕi, andψi, i = 1, 2, ..., such that

u(t, x)− ϕi(t, x)

has a global maximum at some point (ti, xi),

v(s, y)− ψi(s, y)

has a global minimum at some point (si, yi), and

ti, xi, u(ti, xi), Dϕi(ti, xi), D

2ϕi(ti, xi) k→∞−−−−→

t, x, u(t, x),

Bρ(xN − yN )

, LN

in R×X0 × R×X2 × L(X−1, X1), (3.367)

si, yi, v(si, yi), Dψi(si, yi), D

2ψi(si, yi) k→∞−−−−→

s, y, v(s, y),

Bρ(xN − yN )

,MN

in R×X0 × R×X2 × L(X−1, X1), (3.368)

where LN = P ∗NLNPN , MN = P ∗

NMNPN and

LN 00 −MN

≤ ν

BρPN −BρPN

−BρPN BρPN

. (3.369)

Using the definition of viscosity subsolution we thus obtain

infa∈Λ

1

2

m

k=1

(D2ϕi(ti, xi) +2

BρQN + 2δI)Sk

axi, Skaxi0,ρ

+Aaxi, Dϕi(ti, xi) +BρQN (x− y)

+

2BρQN (xi − x)

+ 2δxi −BρpnX−1,X1

+f(xi, a)

− an + (ϕi)t (ti, xi) ≥

µ

T 2(3.370)

(We remark that the function µt can be modified around 0 so that it is part of a

test function.)


We now pass to the limit in (3.370) as i → ∞. We notice that by (3.339), forevery a ∈ Λ

m

k=1

Skaxi, S

kaxi0 + 2Aaxi, xiX−1,X1 ≤ 2δK|xi|20,ρ → 2δK|x|20,ρ

as i → ∞. (In fact one can prove xi x in X1.) Moreover we observe thatby (3.360), B−1/2

ρ D2ϕi(ti, xi)B−1/2ρ → B−1/2

ρ LNB−1/2ρ in L(X0). Using these,

(3.367), and (3.342), i.e. that BρAaL(X0), B12ρ Sk

aL(X0) ≤ C independently ofa ∈ Λ, 1 ≤ k ≤ m, we obtain upon passing to lim supi→∞ in (3.370) that

− an +t− s

β+ inf

a∈Λ

1

2

m

k=1

LN +2

BρQN )Sk

a x, Ska x0,ρ

+ Aax,Bρ(x− y)

−BρpnX−1,X1 + f(x, a)

+ 2δK|x|20,ρ ≥ µ

T 2. (3.371)

We obtain similarly for the supersolution v

bn +t− s

β+ inf

a∈Λ

1

2

m

k=1

MN − 2

BρQN )Sk

a y, Ska y0,ρ

+ Aay,Bρ(x− y)

+BρqnX−1,X1 + f(y, a)

− 2δK|y|20,ρ ≤ − µ

T 2. (3.372)

By Hypothesis 3.103 the closures of the sets Ska x : a ∈ Λ, 1 ≤ k ≤ m and

Ska y : a ∈ Λ, 1 ≤ k ≤ m are compact in X0, and hence in X−1. This yields

sup|B1/2ρ QNSk

a x|0,ρ + |B1/2ρ QNSk

a y|0,ρ : a ∈ Λ, 1 ≤ k ≤ m → 0 (3.373)

as N → ∞. Moreover, (3.369) implies that

LNSka x, S

ka x0,ρ − MNSk

a y, Ska y0,ρ ≤ ν

2BρS

ka(x− y), Sk

a(x− y)0,ρ. (3.374)

Therefore, subtracting (3.371) from (3.372) and using (3.335), (3.373), and (3.374),we have

infa∈Λ

− ν

2

m

k=1

BρSka(x− y), Sk

a(x− y)0,ρ −1

Aa(x− y), Bρ(x− y)X−1,X1

+an + bn − ωR(|x− y|0)− 2δK(|x|20,ρ + |y|20,ρ)− σ(1/N, n) ≤ − 2µ

T 2

for some local modulus σ. Now, if ν is is close to 1, it follows from (3.341) that

an + bn +λ

2|x− y|20,ρ −

K

|x− y|2−1,ρ

−ωR(|x− y|0,ρ)− 2δK(|x|20,ρ + |y|20,ρ)− σ(1/N, n) ≤ − 2µ

T 2. (3.375)

Since ωR is a modulus we have

lim→0

infr≥0

λ

2r2 − ωR(r)

= 0. (3.376)

Therefore we obtain a contradiction in (3.375) after sending N → ∞, n → ∞,β → 0, δ → 0, → 0 in the above order, and using (3.365), (3.366), and (3.376).


3.11.6. Value function and existence of solutions. In this subsection weshow that the value function V defined by (3.337) is the unique viscosity solutionof (3.336).

Proposition 3.111 Assume that Hypotheses 3.103 and 3.104 are satisfied.Then for every R > 0 there exists a modulus σR such that

|V (t, x)−V (s, y)| ≤ σR(|t−s|+|x−y|−1,ρ), t, s ∈ [0, T ], |x|0,ρ, |y|0,ρ ≤ R, (3.377)

and there is C > 0 such that

|V (t, x)| ≤ C1 + |x|γ0,ρ

, t ∈ [0, T ], x ∈ X0. (3.378)

Moreover the Dynamic Programming Principle (2.25) is satisfied.

Proof. We only sketch the proof since it is very similar to the proof ofProposition 3.61. Using similar arguments it follows from (3.345), (3.347), (3.348),linearity of the DMZ equation (3.334), and Hypothesis 3.104 that (3.378) holds andthere are moduli σ1

R such that

|J(t, x; a(·))−J(t, y; a(·))| ≤ σ1R(|x−y|−1,ρ), t ∈ [0, T ], |x|0,ρ, |y|0,ρ ≤ R, a(·) ∈ Ut,

(3.379)and thus the same inequality is satisfied by V . We now claim that the DynamicProgramming Principle holds. To do this we need to check that the assumptionsof Hypothesis 2.12 are satisfied. Parts (A0) and (A2) follow from the definition ofvariational solution, properties of stochastic integrals and standard manipulations.Part (A1) was proved in Proposition 3.106 (i.e. Theorem 3.102-(ii)). Part (A3) canbe proved similarly to the proof of Proposition 2.16. It thus follows from Theorem2.24 that for every x ∈ X0, 0 ≤ t ≤ η ≤ T ,


E η

tl(Y (s), a(s))ds+ V (η, Y (η))

. (3.380)

Using (3.380) we again argue as in the proof of Proposition 3.61 using (3.345),(3.349), (3.378), and (3.379) to obtain that there exist moduli σ2

R such that

|V (t, x)− V (s, x)| ≤ σ2R(|t− s|), t, s ∈ [0, T ], |x|0,ρ ≤ R. (3.381)

Obviously (3.379) and (3.381) produce (3.377).

Theorem 3.112 Assume that Hypotheses 3.103 and 3.104 are true. Then thevalue function V is the unique viscosity solution of the HJB equation (3.336) amongfunctions satisfying

lim sup|x|0,ρ→∞

|u(t, x)||x|20,ρ

= 0 uniformly for t ∈ [0, T ],

andlimt↑T

|u(t, x)− g(x)| = 0 uniformly on bounded subsets of X0.

Proof. The uniqueness is a consequence of Theorem 3.110 and Proposition3.111. Therefore it remains to show that V is a viscosity solution of (3.336).

We only argue about the supersolution property as the subsolution part iseasier. Suppose that V + (ϕ + δ(t) |x|20,ρ) has a global minimum at (t0, x0) ∈(0, T )×X0.

We need to prove that x0 ∈ X1. For every (t, x) ∈ (0, T )×X0

V (t, x)− V (t0, x0) ≥ −(ϕ(t, x)− ϕ(t0, x0))−δ(t) |x|20,ρ − δ(t0) |x0|20,ρ

. (3.382)


By the dynamic programming principle for every > 0 there exists and a(·) ∈ Ut0

such that, writing Y (s) for Y (s; t0, x0, a(·)) , we have

V (t0, x0) + 2 > E t0+

t0

l (Y (s) , a(s)) ds+ V (t0 + , Y (t0 + ))

.

In light of Corollary 2.21 and Proposition 3.106, without loss of generality we canassume that all a(·) are defined on the same reference probability space. We have,by (3.382),

2 − E t0+

t0

l (Y (s) , a(s)) ds ≥ E [V (t0 + , Y (t0 + ))− V (t0, x0)]

≥ −E [ϕ (t0 + , Y (t0 + ))− ϕ (t0, x0)]−Eδ(t0 + ) |Y (t0 + ε)|20,ρ − δ(t0) |x0|20,ρ

and, by Itô’s formula

− E1

t0+

t0

l (Y (s) ,α(s)) ds

≥ −E1

t0+

t0

ϕt (s, Y (s)) +

Aa(s)Y (s) , Dϕ (s, Y (s))

X−1,X1

+1

2

m

k=1

D2ϕ (s, Y (s))S

ka(s)Y (s) , Sk

a(s)Y (s)

0,ρ

ds (3.383)

−E1

t0+

t0

δ (s) |Y (s)|20,ρ + 2δ(s)

Aa(s)Y (s) , Y (s)

X−1,X1

+1

2

m

k=1

Ska(s)Y (s) , S

ka(s)Y (s)

0,ρ

ds.

By (3.339) we have

−2δ(s)

Aa(s)Y (s) , Y (s)

X−1,X1 +

1

2

m

k=1

Ska(s)Y (s) , S

ka(s)Y (s)

0,ρ

≥ 2δ(s)λ |Y (s)|21,ρ −K |Y (s)|20,ρ

.

Moreover

|ϕt (s, Y (s))|+Aa(s)Y (s) , Dϕ (s, Y (s))

X−1,X1

≤ C1(1 + |Y (s)|20,ρ ,

m

k=1

D2ϕ (s, Y (s))S

ka(s)Y (s) , S

ka(s)Y (s)

0,ρ

≤ C2 |Y (s)|20

Therefore, using Hypothesis 3.104, (3.345), and δ(s) ≥ γ > 0 for s close to t0 forsome γ > 0, we obtain

2λγE1

t0+

t0

|Y (s)|21,ρ ds ≤ C3

1 + E1

t0+

t0

|Y (s)|20,ρ ds≤ C4.

Take now = 1/n and set Yn (s) := Ys; t0, x0, a1/n

. The above inequality yields

n

t0+1/n

t0

E |Yn (s)|21 ds ≤ C5

so that, along a sequence tn ∈ (t0, t0 + 1/n)

E |Yn (tn)|21,ρ ≤ C5

3.12. HJB EQUATION FOR A BOUNDARY CONTROL PROBLEM 245

and thus, along a subsequence, still denoted by tn, we have

Yn (tn) Y

weakly in L2 (Ω;X1) for some Y ∈ L2 (Ω;X1). This also clearly implies weakconvergence in L2 (Ω;X0). However, by (3.351), Yn (tn) → x0 strongly (and weakly)in L2 (Ω;X0). Thus Y = x0 ∈ X1.

Having established that x0 ∈ X1, we now go back to (3.383), use the propertiesof test functions and estimates of Proposition 3.107 to obtain

≥ −E1

t0+

t0

ϕt (t0, x0) +

Aa(s)x0, Dϕ (t0, x0)

X−1,X1

+1

2

m

k=1

D2ϕ (t0, x0)S

ka(s)x0, S

ka(s)x0

0,ρ

ds

−E1

t0+

t0

δ (t0) |x0|20,ρ + 2δ(t0)

Aa(s)x0, x0

X−1,X1

+1

2

m

k=1

Ska(s)x0, S

ka(s)x0

0,ρ

− l (x0,α(s))

ds− γ(),

≥ E1

t0+

t0

− ψt(t, x) + inf

a∈Λ

1

2

m

k=1

−D2ψ(t, x)Sk

ax, Skax

0,ρ

+ Aax,−Dψ(t, x)X−1,X1 + f(x, a)

ds− γ(),

where lim→0 γ() = 0. It remains to let → 0. We refer the reader to the proof ofTheorem 5.4 in [245] for more details.

3.12. HJB equation for a boundary control problem

In this section we discuss how the theory of viscosity solutions can be applied tosolve HJB equations coming from stochastic boundary control problems discussedin Section 2.6.2. We will only consider time independent problems. Suppose that His a real, separable Hilbert space and A is an operator in H satisfying the followinghypothesis.

Hypothesis 3.113 A : D(A) ⊂ H → H is a (densely defined) self-adjointoperator, there exists a > 0 such that Ax, x ≤ −a|x|2 for all x ∈ D(A) and A−1

is compact.

Hypothesis 3.113 implies in particular that A is the infinitesimal generator ofan analytic semigroup with compact resolvent satisfying etA ≤ e−at for all t ≥ 0and that there is an orthonormal basis of H composed of eigenvectors of A such thatthe corresponding sequence of (positive) eigenvalues diverges to +∞ as n → ∞.Moreover the fractional powers (−A)γ , γ > 0 are well defined, and if γ ∈ (0, 1] andα ∈ (0, γ), a well known interpolation inequality (see e.g. [375], p. 73-74) gives usthat for every σ > 0 there exists Cσ > 0 such that

|(−A)αx| ≤ σ|(−A)γx|+ Cσ|x|, for every x ∈ D((−A)γ). (3.384)

The HJB equations introduced in Section 2.6.2 (see (2.105)) have the form

λv − Ax,Dv+ F (x,Dv,D2v) = 0, x ∈ H, (3.385)

where F : Z ⊂ H × H × S(H) → R, λ > 0. In particular F (x, p,X) may onlybe defined if p ∈ D((−A)β) for some β > 0 and may be undefined if X is not oftrace class. The unboundedness in the first order terms comes from the boundary


control term rewritten as a distributed control terms, and the unboundedness inthe second derivative terms comes from noise with non-nuclear covariance in thecontrol problem. Such second order unboundedness has not been discussed so farin this book and indeed it is not easy to handle by the viscosity solution methods.Here we suggest one way to do it. The idea is the following. To deal with theunboundedness in the first and second derivatives we introduce a change of variablesx = (−A)

β

2 y,β > 0. Then the function u(y) := v((−A)β

2 y) should formally solve

λu− Ay,Dv+ F ((−A)β

2 y, (−A)−β

2 Dv, (−A)−β

2 D2v(−A)−β

2 ) = 0. (3.386)

This equation contains less unbounded terms and is easier to handle in spite of theadditional difficulty created by the presence of the new unbounded term A

β

2 y. Wewill define a viscosity solution of (3.385) to be a function v such that u(·) def

= v(Aβ

2 ·)is a viscosity solution of (3.386). We will make this idea rigorous in the nextsection. The definition is meaningful, indeed, when (3.385) comes from a stochasticboundary control problem, v and u can be respectively characterized as the valuefunctions of their control problems.

3.12.1. Definition of viscosity solution. We first consider the followingHJB equation

λu− Ay,Du+G(y,Du,D2u) = 0, y ∈ H, (3.387)

where G : D((−A)β

2 )×D((−A)β

2 )× S(H) → R.

Definition 3.114 We say that a function ψ is a test function if ψ(x) = ϕ(x) +δ|x|2, where δ > 0 and

(i) ϕ ∈ C2(H) and is weakly sequentially lower semicontinuous on H.(ii) Dϕ ∈ UC(H,H)∩UC

D((−A)

12−), D((−A)

12 )

for some = (ϕ) >

0.(iii) D2ϕ ∈ BUC(H,S(H)).

Definition 3.115 We say that a function w : H → R is a viscosity subsolutionof (3.387) if w is weakly sequentially upper semicontinuous on H, and wheneverw − ψ has a local maximum at x for a test function ψ, then

x ∈ D((−A)12 )

and

λw(x) + (−A)12x, (−A)

12Dϕ(x)+ 2δ|(−A)

12x|2 +G

x,Dψ(x), D2ψ(x)

≤ 0.

We say that w is a viscosity supersolution of (3.387) if w is weakly sequentiallylower semicontinuous on H, and whenever w + ψ has a local minimum at x for atest function ψ, then

x ∈ D((−A)12 )

and

λw(x)− (−A)12 y, (−A)

12Dϕ(x) − 2δ|(−A)

12x|2 +G

x,−Dψ(x),−D2ψ(x)

≥ 0.

We say that w is a viscosity solution of (3.387) if it is both a viscosity subsolutionand a supersolution.

Suppose now that F from (3.385) is such that F : H × D((−A)β) ×(−A)−

β

2 S(H)(−A)−β

2 → R. We define

GF (z, p, S)def= F

(−A)

β

2 z, (−A)−β

2 p, (−A)−β

2 S(−A)−β

2

, (3.388)


Definition 3.116 A bounded continuous function v : H → R is said to be aviscosity solution of equation (3.385) if the function

u(y)def= v((−A)

β

2 y)

is a viscosity solution of the equation

λu− Ay,Du+GF (y,Du,D2u) = 0, y ∈ H. (3.389)

Similarly we define a viscosity subsolution and a supersolution of (3.385).

We remark that the function v is uniquely determined once u has been charac-terized on D((−A)

β

2 ).

3.12.2. Comparison and existence theorem. For γ > 0 we denote byH−γ the completion of H in the norm |x|−γ = |(−A)−

γ

2 x|, and for 0 < γ ≤ 2, andD((−A)

γ

2 ) is equipped with the norm |x|γ = |(−A)γ

2 x|. For N > 2 let HN be finitedimensional subspaces of H generated by eigenvectors of (−A)−1 correspondingto the eigenvalues which are greater than or equal to 1/N . Denote by PN theorthogonal projection in H−1 onto HN , QN = I−PN , and H⊥

N = QNH. PN and QN

are also orthogonal projections in H. We then have an orthogonal decompositionH = HN ×H⊥

N and we will write x = (xN , x⊥N ) = (PNx,QNx).

We assume:

Hypothesis 3.117(i) There exists β ∈ (0, 1) such that the function G : D((−A)

β

2 ) ×D((−A)

β

2 ) × S(H) → R is uniformly continuous (in the topologyof D((−A)

β

2 ) × D((−A)β

2 ) × S(H)) on bounded sets of D((−A)β

2 ) ×D((−A)

β

2 )× S(H).

(ii) G(y, p, S1) ≤ G(y, p, S2) if S1 ≥ S2, for all y, p ∈ D((−A)β

2 ).

(iii) There exists a modulus ρ such that

|G(y, p, S1)−G(y, q, S2)|

≤ ρ(1 + |(−A)

β

2 y|)|(−A)β

2 (p− q)|+ (1 + |(−A)β

2 y|2)S1 − S2

for all y, p, q ∈ D((−A)β

2 ) and S1, S2 ∈ S(H).

(iv) There exist 0 < η < 1 − β and a modulus ω such that, for all N > 2, > 0,

G

x,

(−A)−η(x− y)

, Z

−G

y,

(−A)−η(x− y)

, Y

≥ −ω

|(−A)

β

2 (x− y)|1 +

|(−A)β

2 (x− y)|

for all x, y ∈ D((−A)β

2 ) and Z, Y ∈ S(H), Z = PNZPN , Y = PNY PN

such that

Z 00 −Y

≤ 3

(−A)−ηPN −(−A)−ηPN

−(−A)−ηPN (−A)−ηPN

(3.390)


(v) Let η be from (iv). For every R < +∞, |λ| ≤ R, p, x ∈ D((−A)β

2 )

sup

|G(x, p, S + λQN )−G(x, p, S)| : S ≤ R,S = PNSPN

→ 0.

(3.391)as N → ∞.

Some of the conditions of Hypothesis 3.117 can be weakened. By the propertiesof moduli, Hypothesis 3.117-(iii) guarantees that there exists a constant C suchthat, for every y, p, S,

|G(y, p, S)| ≤ C1 + (1 + |(−A)

β

2 y|)|(−A)β

2 p|+ (1 + |(−A)β

2 y|2)S+|G(y, 0, 0)|,

(3.392)and conditions (i), (iv) of Hypothesis 3.117 imply that there is C1 such that

|G(y, 0, 0)| ≤ C1

1 +

(−A)β

2 y. (3.393)

Theorem 3.118 Let Hypotheses 3.113 and 3.117 be satisfied. Then:Comparison: Let u,−v ≤ M for some constant M . If u is a viscosity subsolutionof (3.387) and v is a viscosity supersolution of (3.387) then u ≤ v on H. Moreover,if u is a viscosity solution then

|u(x)− u(y)| ≤ m(|(−A)−η

2 (x− y)|) (3.394)

for all x, y ∈ H and some modulus m, where η is the constant in (iv).Existence: If

supx∈D(A

β

2 )

|G(x, 0, 0)| = K < ∞, (3.395)

then there exists a unique viscosity solution u ∈ BUC(H−η) of (3.387).

Proof. Comparison. Let , δ > 0. We set

Φ(x, y) = u(x)− v(y)− |(−A)−η

2 (x− y)|22

− δ

2|x|2 − δ

2|y|2

Since u−v is bounded from above and weakly sequentially upper-semicontinuous inH ×H, Φ must attain its maximum at some point (x, y) ∈ D((−A)

12 )×D((−A)

12 )

(which can be assumed to be strict by subtracting for instance µ(|(−A)−1(x−x)|2+|(−A)−1(y − y)|2) and then letting µ → 0). Moreover, arguing similarly as in theproof of Theorem 3.56, we have

limδ→0

δ|x|2 + δ|y|2

= 0 for every fixed > 0, (3.396)

lim→0

lim supδ→0

|(−A)−

η

2 (x− y)|2

= 0. (3.397)

Then (see the proof of Theorem 3.50), defining

u1(x) = u(x)− x,QN (−A)−ηQN (x− y)

+QN (−A)−ηQN (x− y), x− y

2ε

− |(−A)−η

2 QN (x− x)|2

− δ

2|x|2,

v1(y) = v(y)− y,QN (−A)−ηQN (x− y)ε

+|(−A)−

η

2 QN (y − y)|2

+δ

2|y|2,

it follows that the function

Φ(x, y) def= u1(x)− v1(y)−

|(−A)−η

2 PN (x− y)|22

(3.398)


always satisfies Φ ≤ Φ and Φ attains a strict global maximum at (x, y), whereΦ(x, y) = Φ(x, y). Using Corollary 3.28 (with B = (−A)−η) we thus obtain func-tions ϕn,−ψn satisfying conditions (i)− (iii) of Definition 3.114 such that

u1(x)− ϕn(x)

has a global maximum at some point xn,

v1(y) + ψn(y)

has a global minimum at some point yn, and

xn, u1(x

n), Dϕn(xn), D2ϕn(x

n) n→∞−−−−→

x, u1(x),

(−A)−ηPN (x− y)

, ZN

in H × R×D(−A)× L(H;H), (3.399)

yn, v1(y

n),−Dψn(yn),−D2ψn(y

n) n→∞−−−−→

y, v(y),

(−A)−ηPN (x− y)

, YN

in H × R×D(−A)× L(H;H), (3.400)

for some ZN , YN ∈ S(H) such that ZN = PNZNPN , YN = PNYNPN , they satisfy(3.390) and ZN+ YN ≤ C for some constant C.

Therefore, by the definition of viscosity subsolution, xn ∈ D((−A)12 ) and

λu(xn) +

(−A)

12xn, (−A)

12Dϕn(x

n) +(−A)

12−ηQN (x− y)

+2(−A)

12−ηQN (xn − x)

+ δ|(−A)

12xn|2

+G

xn, Dϕn(x

n) +(−A)−ηQN (x− y)

+

2(−A)−ηQN (xn − x)

+ δxn,

D2ϕn(xn) +

2(−A)−ηQN

+ δI

≤ 0. (3.401)

Thus, using (3.399), (3.392), (3.393), and (3.384), it follows from (3.401) that|(−A)

12xn| are bounded independently of n which implies, thanks to (3.399) that

(−A)12xn (−A)

12 x as n → +∞. (3.402)

Since (−A)β−12 and (−A)−

η

2 are compact we conclude that, as n → +∞,

(−A)β

2 xn = (−A)β−12 ((−A)

12xn) → (−A)

β

2 x and (−A)1−η

2 xn → (−A)1−η

2 x.(3.403)

Using (3.399), (3.402), (3.403), and the weak sequential lower semicontinuity of thenorm we thus obtain

(−A)

1−η

2 x,(−A)

1−η

2 (x− y)

+ δ|(−A)

12 x|2

≤ lim infn→∞

(−A)

12xn, (−A)

12Dϕn(x

n) +(−A)

12−ηQN (x− y)

+2(−A)

12−ηQN (xn − x)

+ δ|(−A)

12xn|2


and then letting n → ∞ in (3.401) yields

λu(x) +

(−A)

1−η

2 x,(−A)

1−η

2 (x− y)

+ δ|(−A)

12 x|2

+G

x,

(−A)−η(x− y)

+ δx, ZN +

2(−A)−ηQN

+ δI

≤ 0.

(3.404)Using Hypothesis 3.117-(iii) we have

G

x,

(−A)−η(x− y)

, ZN +

2(−A)−ηQN

− ρ

cδ(1 + |(−A)

β

2 x|2)

≤ G

x,

(−A)−η(x− y)

+ δx, ZN +

2(−A)−ηQN

+ δI

(3.405)for some constant c > 0. Now, given τ > 0, let Kτ be such that ρ(s) ≤ τ +Kτs.Applying (3.384) with α = β/2 and γ = 1/2 we obtain

ρcδ(1 + |(−A)

β

2 x|2)≤ δ|(−A)

12 x|2 + δCτ |x|2 + τ +Kτdδ

for some constant Cτ > 0 independent of δ and . It then follows from (3.396) that

lim supδ→0

ρcδ(1 + |(−A)

β

2 x|2)− δ|(−A)

12 x|2

≤ 0. (3.406)

Using (3.405), (3.406) and (3.391) in (3.404) we thus obtain

λu(x) +

(−A)

1−η

2 x,(−A)

1−η

2 (x− y)

+G

x,

(−A)−η(x− y)

, ZN

≤ ω1(, δ;N) + ω2(; δ),

(3.407)

where limN→∞ ω1(, δ;N) = 0, limδ→0 ω2(; δ) = 0 . Similarly we obtain

λv(y) +

(−A)

1−η

2 y,(−A)

1−η

2 (x− y)

+G

y,

(−A)−η(x− y)

, YN

≥ −ω1(, δ;N)− ω2(; δ).

(3.408)

We subtract (3.408) from (3.407), use Hypothesis 3.117-(iv), and let N → +∞ toconclude

λ(u(x)− v(y)) ≤ ω

|(−A)

β

2 (x− y)|1 +

|(−A)β

2 (x− y)|

− |(−A)1−η

2 (x− y)|2

+ 2ω2(; δ). (3.409)

By (3.384), for every σ > 0

|(−A)β

2 (x− y)| ≤ σ|(−A)1−η

2 (x− y)|+ Cσ|(−A)−η

2 (x− y)|. (3.410)

Since for every α > 0, ω(s) ≤ α/2 +Kαs, if σ is sufficiently small, we obtain afterelementary calculations

λ(u(x)− v(y)) ≤ α+ Kα|(−A)−

η

2 (x− y)|2

+ 2ω2(; δ). (3.411)

By (3.397) this implies

lim sup→0

lim supδ→0

(u(x)− v(y)) ≤ α

λ


for all α > 0, which gives u ≤ v in H, since for all x ∈ H we have

Φ(x, x) ≤ Φ(x, y) ≤ u(x)− v(y).

If u is a solution, we can set u = v in the proof to obtain that for all x, y ∈ H

u(x)− u(y)− |(−A)−η

2 (x− y)|22

= limδ→0

Φ(x, y) ≤ lim supδ→0

(u(x)− u(y)) ≤ ρ1()

for some modulus ρ1 in light of (3.411). This proves (3.394).Existence. Existence of a viscosity solution will be proved by the method of

finite dimensional approximations similar to this of Section 3.7. We consider forN > 2 the approximating equations

λuN − Ax,DuN +G(x,DuN , D2uN ) = 0 in HN . (3.412)

We notice that for every γ > 0, (−A)γx = PN (−A)γx, (−A)−γx = PN (−A)−γxfor x ∈ HN , and thus (3.412) satisfies Hypotheses 3.113 and 3.117 with constantsand moduli independent of N . Since u(x) = −K/λ is a viscosity subsolutionand u(x) = K/λ is a viscosity supersolution of (3.412), it follows from the finitedimensional Perron’s method that (3.412) has a (unique) bounded viscosity solutionuN such that uN0 ≤ K/λ.

We will prove that there exists a modulus ση independent of N such that

|uN (x)− uN (y)| ≤ ση(|x− y|−η)

for all x, y ∈ HN . To do this we adapt the technique of Section 3.7.For every > 0 let K be such that ω(r) ≤ λ/2 +Kr. For L > K/λ + 1 we

set

ψL(r) = 2Lr12L .

The function ψL ∈ C2(0,∞), is increasing, concave, ψL(r) ≥ 1 for 0 < r ≤ 1,

ψL(0) = 0, ψL(1) > 2(K/λ+ 1), and

ψL(r) > L (ψL(r)r + r) for 0 ≤ r ≤ 1. (3.413)

We will show that, for every > 0 there exists L = L such that

uN (x)− uN (y) ≤ ψL(|(−A)−η

2 (x− y)|) + for every x, y ∈ HN . (3.414)

Set ∆ =(x, y) ∈ H ×H : |(−A)−

η

2 (x− y)| < 1. It is clear from the properties

of ψL, that for (x, y) ∈ ∆ (3.414) is always satisfied independently of L. Assumenow by contradiction that (3.414) is false. Then, for any L > K

λ + 1 we have, forsmall δ > 0,

sup(x,y)∈HN×HN

uN (x)− uN (y)− ψL(|(−A)−

η

2 (x− y)|)− − δ

2|x|2 − δ

2|y|2

> 0

(3.415)and is attained at (x, y) ∈ ∆ such that x = y. Denote s = |(−A)−

η

2 (x− y)|.Repeating the arguments from the proof of Proposition 3.81 that led to (3.218)

and then the arguments from the just finished proof of comparison we obtain thatthere exist Z, Y ∈ S(HN ) such that

Z 00 −Y

≤ 2ψ

L(s)

s

(−A)−ηPN −(−A)−ηPN

−(−A)−ηPN (−A)−ηPN


and

λ(uN (x)− uN (y)) ≤ −ψL(s)

s|(−A)

1−η

2 (x− y)|2 +G(y,ψL(s)

s(−A)−η(x− y), Y )

−G(x,ψL(s)

s(−A)−η(x− y), Z) + ρ(L; δ)

≤ −ψL(s)

s|(−A)

1−η

2 (x− y)|2 + λ

4

+K

|(−A)

β

2 (x− y)|1 +

ψL(s)

s|(−A)

β

2 (x− y)|

+ ρ(L; δ),

where limδ→0 ρ(L; δ) = 0. Therefore using (3.410) with sufficiently small σ it followsthat

λ(uN (x)− uN (y)) ≤ λ

2+ Cε(ψ

L(s)s+ s) + ρ(L; δ),

where C only depends on K and the interpolation constant but not on L. ChoosingL = C/λ, using (3.413), and letting δ → 0 we arrive at

uN (x)− uN (y) ≤

2+ ψL(s).

This is a contradiction since we obviously have by (3.415)

ψL(s) + ε ≤ uN (x)− uN (y).

Hence we obtain the existence of the required modulus of continuity ση.Now set vN (x) = uN (PNx). Since (−A)−

η

2 is compact we are in a position toapply Arzela-Ascoli theorem to find a subsequence (still denoted by vN ) converginguniformly on bounded sets of H to a function u that obviously satisfies the sameestimates as uN ’s. It remains to show that u solves the limiting equation (3.387).To this end let u − ψ have a maximum at x (which we may assume to be strict)for some test function ψ(x) = ϕ(x) + δ|x|2. It follows from the local uniformconvergence of the vN and the strictness of the maximum at x that there exists asequence xN = PN xN → x as N → ∞ such that, for every x ∈ HN ,

vN (x)− ϕ(x)− δ|x|2 ≤ vN (xN )− ϕ(xN )− δ|xN |2.Therefore, since APN = PNA,

λuN (xN ) + (−A)12 xN , (−A)

12Dϕ(xN )+ 2δ|(−A)

12 xN |2

+GxN , PNDϕ(xN ) + 2δxN , PN (D2ϕ(xN ) + 2δI)PN

≤ 0.

(3.416)

Since ϕ is a test function we have

|(−A)12Dϕ(xN )| ≤ C1 + C2|(−A)

12−xN | (3.417)

for some independent constants C1, C2. Also, by (3.384), (3.392), (3.395) and(3.417),

|GxN , PNDϕ(xN ) + 2δxN , PN (D2ϕ(xN ) + 2δI)PN

|

≤ C3

1 + |(−A)

β

2 xN |2 + |(−A)12−xN |2

≤ C4 +

δ

2|(−A)

12 xN |2.

Using this, (3.417) and (3.384), we therefore obtain from (3.416) that

|(−A)12 xN | ≤ C5

for some constant C5 independent of N . Thus (−A)12 xN (−A)

12 x (so x ∈

D((−A)12 )) and hence

(−A)β

2 xN → (−A)β

2 x, and (−A)12Dϕ(xN ) → (−A)

12Dϕ(x).


These convergences and Lemma 3.85 allow us to pass to the limit in (3.416) asN → ∞ to conclude thatλu(x)+(−A)

12 x, (−A)

12Dϕ(x)+2δ|(−A)

12 x|2+G

x, Dϕ(x) + δx, D2ϕ(x) + δI

≤ 0.

The proof of the supersolution property is analogous. 3.12.3. Stochastic control problem. We present an application of the re-

sults of the previous section to an abstract infinite horizon stochastic optimal controlproblem which includes the class of problems discussed in Section 2.6.2 and maycome from a boundary control problem of Dirichlet type with distributed controls.We take the usual setup. Let H,Ξ be real separable Hilbert spaces, and Q ∈ L+(Ξ).Let Λ = Λ1× Λ2, where Λ1,Λ2 are real separable Hilbert spaces, and Λ2 is a closedbounded subset of Λ2. We set R := supa2∈Λ2

|a2|Λ2 . Given a reference probabilityspace ν = (Ω,F , Fss≥0,P,WQ) we have the set of admissible controls

Uν :=a(·) = (a1(·), a2(·)) : [0,+∞)× Ω → Λ :

a1(·), a2(·) are Fs − progressively measurable, (3.418)

and we denote U :=

ν Uν to be the set of all admissible controls.We control the state given by the SDE

dX(t) =AX(t) + b(X(t), a1(t)) + (−A)βCa2(t)

dt

+σ(X(t), a1(t))dWQ(t), t > 0

X(0) = x0 ∈ H,

(3.419)

i.e.

X(t) = etAx0 +

t

0e(t−s)Ab(X(s), a1(s))ds+ (−A)β

t

0e(t−s)ACa2(s)ds

+

t

0(−A)

β

2 e(t−s)A(−A)−β

2 σ(X(s), a1(s))dWQ(s),

(3.420)and try to minimize the cost functional

J(x0; a(·)) = E +∞

0e−λtl (X(t;x0, a(·)), a(t)) dt, (3.421)

over all admissible controls a(·) ∈ U . We denote by v the value function for thisproblem. We assume that A satisfies Hypothesis 3.113 and λ > 0. We also makethe following assumptions.

Hypothesis 3.119(i) The function b is continuous from H × Λ1 to H and there exists a

constant c0 > 0 such that|b(x, a1)| ≤ c0(1 + |x|) for all x ∈ H, a1 ∈ Λ1,

|b(x1, a1)− b(x2, a1)| ≤ c0|x1 − x2| for all x1, x2 ∈ H, a1 ∈ Λ1.

(ii) C ∈ L(Λ2, H) and β ∈34 , 1

.

(iii) σ : H × Λ1 → L(Ξ0, H), the map (−A)−β

2 σ : H × Λ1 → L2(Ξ0, H) iscontinuous and moreover there exists a constant K1 > 0 such that

(−A)−β

2 σ(x, a1)L2(Ξ0,H) ≤ K1(1 + |x|)for all x ∈ H, a1 ∈ Λ1,

(−A)−β

2 [σ(x1, a1)− σ(x2, a1)]L2(Ξ0,H) ≤ K1|x1 − x2|for all x1, x2 ∈ H, a1 ∈ Λ1.


(iv) For all x ∈ H

limN→+∞

supa1∈Λ1

QN (−A)−β

2 σ(x, a1)L2(Ξ0,H) = 0.

(v) l ∈ C(H × Λ) and

|l(x, a)| ≤ Cl , for all (x, a) ∈ H × Λ,

|l(x1, a)− l(x2, a)| ≤ ωl(|x1 − x2|), for all a ∈ Λ, x1, x2 ∈ H,

for some positive constant Cl and modulus ωl.

Remark 3.120(1) Hypotheses 3.119-(iii),(iv) are satisfied if we assume, for example, that

there exists a constant K2 > 0 such that

σ(x, a1)L(Ξ0,H) ≤ K2(1 + |x|)

for all x ∈ H, a1 ∈ Λ1

σ(x1, a1)− σ(x2, a1)L(Ξ0,H) ≤ K2|x1 − x2|

for all x1, x2 ∈ H, a1 ∈ Λ1, and if the operator (−A)−β is trace class.

(2) Hypothesis 3.119-(iv) is satisfied if, for instance, for every x ∈ H thereexists η ∈ (0,β/2) such that (−A)−ησ(x, a1) is bounded in L2(Ξ0, H)independently of a1 ∈ Λ1.

It is a consequence of Theorem 1.135 that for every generalized reference prob-ability space µ = (Ω,F , Fss≥0,P,WQ), T > 0, a(·) ∈ Uµ, and ξ ∈ L2(Ω,F0,P),equation (3.419) with X(0) = ξ has a unique mild solution in Hµ

2 (0, T ;H) withcontinuous trajectories. Following the strategy described in Remark 2.17 and usingProposition 1.136 one can then argue that all the assumptions needed to prove DPPfor the problem are satisfied. However we will not look directly into this, since weneed to study the transformed HJB equation 3.389 and the optimal control problemassociated with it.

Let us first look how the state equation is transformed by the changeof variables. If X(·;x0, a(·)) satisfies (3.419) then Y (·) = Y (·; y0, a(·)) :=

(−A)−β

2 X(·;x0, a(·)) satisfies the equation

dY (t) =AY (t) + (−A)−

β

2 b((−A)β

2 Y (t), a1(t)) + (−A)β

2 Ca2(t)dt

+(−A)−β

2 σ((−A)β

2 Y (t), a1(t))dWQ(t)

Y (0) = y0 = (−A)−β

2 x0 ∈ H,

(3.422)

which is understood in its mild form

Y (t) = etAy0 +

t

0e(t−s)A(−A)−

β

2 b((−A)β

2 Y (s), a1(s))ds

+ (−A)β

2

t

0 e(t−s)ACa2(s)ds +

t

0e(t−s)A(−A)−

β

2 σ((−A)β

2 Y (s), a1(s))dWQ(s).

(3.423)


We are now minimizing the cost functional

J(y0; a(·)) = E +∞

0e−λtl

(−A)

β

2 Y (t; y0, a(·)), a(t)dt, (3.424)

over all admissible controls and we denote by u the value function for this problem.The HJB equation associated with this new control problem is of the form (3.389)with G : D((−A)

β

2 )×D((−A)β

2 )× S(H) → R given by

G(y, q, S)

= supa∈Λ

−1

2Tr

((−A)−

β

2 σ((−A)β

2 y, a1)Q12 )((−A)−

β

2 σ((−A)β

2 y, a1)Q12 )∗S

−b((−A)

β

2 y, a1), (−A)−β

2 q−Ca2, (−A)

β

2 p− l((−A)

β

2 y, a)

.

(3.425)We will see that the value functions v and u are linked by the relation v(x) =

u((−A)−β

2 x) for x ∈ H. Thus u should correspond to an HJB equation (3.385)with a Hamiltonian F : H × D((−A)β) × (−A)−

β

2 S(H)(−A)−β

2 → R such thatG = GF , where GF is given by (3.388). An easy calculation shows that this is trueif

F (x, p, S) = supa∈Λ

−1

2Tr

(σ(x, a1)Q

12 )∗S(σ(x, a1)Q

12 )

− b(x, a1), q −Ca2, (−A)βp

− l(x, a)

. (3.426)

This is just the formal Hamiltonian corresponding to the original control problem,however notice that the second order terms in F are written in a slightly differentform since we do not know that σ(x, a1)Q

12 ∈ L2(U,H). We remark that if either

(σ(x, a1)Q12 )∗S or S(σ(x, a1)Q

12 ) is trace class then (see Appendix B.3)

Tr(σ(x, a1)Q

12 )∗S(σ(x, a1)Q

12 )= Tr

(σ(x, a1)Q

12 )(σ(x, a1)Q

12 )∗S

.

Proposition 3.121 Assume that Hypotheses 3.113 and 3.119 hold. Let µ =(Ω,F , Fss≥0,P,WQ) be a generalized reference probability space, T > 0, a(·) ∈Uµ, and ξ ∈ L2(Ω,F0,P). Then equation (3.422) with Y (0) = ξ has a unique mildsolution Y (·) = Y (· ; ξ, a(·)) in M2

µ(0, T ;D((−A)β

2 )) with continuous trajectories inH.

Proof. The proof of existence and uniqueness will follow from the contrac-tion mapping principle. Assume first that ξ ∈ Lp(Ω,F0,P), 2 ≤ p < 2/β. ForZ ∈ Mp

µ(0, T ;D((−A)β

2 )) we define a map K on Mpµ(0, T ;D((−A)

β

2 ) by

K[Z](t) = etAξ +

t

0e(t−s)A(−A)−

β

2 b((−A)β

2 Z(s), a1(s))ds

+ (−A)β

2

t

0e(t−s)ACa2(s)ds+

t

0e(t−s)A(−A)−

β

2 σ((−A)β

2 Z(s), a1(s))dWQ(s).

Thus, thanks to Hypothesis 3.119(i) and (B.15), for suitable constants C1, C2 > 0,

|(−A)β

2 K[Z](t)| ≤ |(−A)β

2 etAξ|+ C1

t

0e−a(t−s)[1 + |(−A)

β

2 Z(s)|]ds

+C2R

t

0

e−a(t−s)

(t− s)βds+

t

0(−A)

β

2 e(t−s)A(−A)−β

2 σ((−A)β

2 Z(s), a1(s))dWQ(s)

.


Then, taking the expectation of the p-th power of the terms of this last inequalityand using (1.106) and Hypothesis 3.119(iii) we get

E(−A)

β

2 K[Z](t)p≤ C3

1

tpβ

2

E|ξ|p + 1

+

t

0E|(−A)

β

2 Z(s)|pds+ t

0

1

(t− s)pβ

2

[1 + E|(−A)β

2 Z(s)|p]ds.

(3.427)

Therefore

|K[Z]|pMp

µ(0,T ;D((−A)β

2 ))=

T

0E(−A)

β

2 K[Z](t)pdt

≤ C4(T )

E|ξ|p + 1 +

T

0

t

0

1 +

1

(t− s)pβ

2

E|(−A)

β

2 Z(s)|pds

≤ C4(T )

E|ξ|p + 1 +

T

0E|(−A)

β

2 Z(s)|p T

s

1 +

1

(t− s)pβ

2

dtds

≤ C5(T )

E|ξ|p + 1 + |Z|p

Mp

µ(0,T ;D((−A)β

2 ))

. (3.428)

We now prove that K is a contraction on Mpµ(0, T ;D((−A)

β

2 )) if T is sufficientlysmall. Let Z1(·), Z2(·) ∈ Mp

µ(0, T ;D((−A)β

2 )). Then, arguing as above, we have

E(−A)

β

2 (K[Z1](t)−K[Z2](t))p

≤ C6

t

0

1 +

1

(t− s)pβ

2

E(−A)

β

2 (Z1(s)− Z2(s))pds (3.429)

for some constant C6 > 0, which implies

|K[Z1]−K[Z2]|Mp

µ(0,T ;D((−A)β

2 ))≤ C7

T + T 1− pβ

2

12 |Z1 − Z2|

Mp

µ(0,T ;D((−A)β

2 )).

Thus K is a contraction on Mpµ(0, T ;D((−A)

β

2 )) for small T > 0. The existence ofa unique solution in Mp

µ(0, T ;D((−A)β

2 )) for any T > 0 follows by repeating theargument a finite number of times. The second and third terms of the right handside of (3.423) have continuous trajectories by Lemma 1.110 whereas the stochasticintegral there has continuous trajectories if p > 2 by Proposition 1.111. To provethat the solution has continuous trajectories if ξ ∈ L2(Ω,F0,P) we argue as in theproof of Theorem 1.135. We approximate ξ by random variables

ξn =

ξ if |ξ| ≤ n0 if |ξ| > n.

The solutions Y (·; ξn, a(·)) have continuous trajectories in H, P-a.s. and since thesolutions are obtained by fixed point, one can show that Y (·; ξ, a(·)) = Y (·; ξn, a(·)),P-a.s. on ω : |ξ(ω)| ≤ n.

Proposition 3.122 Assume that Hypotheses 3.113 and 3.119 hold. Let T > 0,y0 ∈ H. Then there exists a constant C(T, |y0|) ≥ 0 such that, for all t ∈ (0, T ] anda(·) ∈ U ,

E|(−A)β

2 Y (t; y0, a(·))|2 ≤ C(T, |y0|)1

tβ. (3.430)


Moreover, for every γ ∈ (0, 1−β), there exists a constant Cγ(T, |y0|) ≥ 0 such that,for all t ∈ (0, T ] and a(·) ∈ U ,

t

0

1

(t− s)β+γE|(−A)

β

2 Y (s; y0, a(·))|2ds ≤ Cγ(T, |y0|)1

tβ. (3.431)

Proof. Estimate (3.427) for p = 2 applied to the solution Y (·) =Y (·; y0, a(·)) implies

E(−A)

β

2 Y (t)2≤ C(T )

1

tβ(|y0|2 + 1) +

t

0

1

(t− s)βE|(−A)

β

2 Y (s)|2ds.

Estimate (3.430) thus follows from Proposition D.25. Now t

0

1

(t− s)β+γE|(−A)

β

2 Y (s)|2ds ≤ C(T, |y0|) t

0

1

(t− s)β+γ

1

sβds ≤ Cγ(T, |y0|)

1

tβ

since t

0

1

(t− s)β+γ

1

sβds = t1−(2β+γ)

1

0

1

(1− s)β+γ

1

sβds

and this last integral is bounded and 1− (2β + γ) > −β.

Theorem 3.123 Assume that Hypotheses 3.113, 3.119 hold. Then the valuefunction v is the unique BUC(H−η) viscosity solution (for every η ∈ (0, 1)) of theHJB equation (3.385) with the Hamiltonian F given by (3.426). Moreover, thedynamic programming principle holds for u and v, i.e. for x ∈ H and all T > 0,

v(x) = infa(·)∈U

E T

0e−λtl(X(t;x, a(·)), a(t))dt+ e−λT v(X(T ;x, a(·)))

and

u(y) = infa(·)∈U

E T

0e−λtl((−A)

β

2 Y (t; y, a(·)), a(t))dt+ e−λTu(Y (T ; y, a(·))).

We will prove this theorem by the approximation argument used in the proofof existence of Theorem 3.118. We consider for N ≥ 1 the following SDE approxi-mating the state equation (3.422).

dYN (t) =PNAYN (t) + (−A)−

β

2 PNb((−A)β

2 Y (t), a1(t)) + (−A)β

2 PNCa2(t)dt

+(−A)−β

2 PNσ((−A)β

2 Y (t), a1(t))dWQ(t)YN (0) = PNy0 ∈ HN .

(3.432)These are finite dimensional SDE (even though the noise is infinite dimensional) inthe spaces HN which have unique strong solutions YN (·) = YN (·; y0, a(·)) and goodcontinuous dependence estimates like these in Sections 1.4.3 and 3.1.2 with respectto the norm in HN . The solutions YN (·) can be also written in the mild form

YN (t) = etAPNy0 +

t

0e(t−s)A(−A)−

β

2 PNb((−A)β

2 YN (s), a1(s))ds

+ (−A)β

2

t

0e(t−s)APNCa2(s)ds

+

t

0e(t−s)A(−A)−

β

2 PNσ((−A)β

2 YN (s), a1(s))dWQ(s). (3.433)


Lemma 3.124 Let Hypotheses 3.113, 3.119 hold, y0 ∈ H, and T > 0. Then

limN→+∞

supa(·)∈U

|YN (·; y0, a(·))− Y (·; y0, a(·))|M2(0,T ;D((−A)

β

2 ))= 0.

Proof. We denote Y (·) = Y (·; y0, a(·)), YN (·) = YN (·; y0, a(·)) and fix γ ∈(0, 1 − β). Recall that PN , QN commute with −A, its fractional powers and etA.We have

(−A)β

2 (YN (t)− Y (t)) = −(−A)β

2 etAQNy0

−QN (−A)−γ

2

t

0(−A)

γ

2 e(t−s)Ab((−A)β

2 Y (s), a1(s))ds

−QN (−A)−γ

2

t

0(−A)β+

γ

2 e(t−s)ACa2(s)ds

−QN (−A)−γ

2

t

0(−A)

β+γ

2 e(t−s)A(−A)β

2 σ((−A)β

2 Y (s), a1(s))dW (s)

+

t

0e(t−s)APN [b((−A)

β

2 YN (s), a1(s))− b((−A)β

2 Y (s), a1(s))]ds

+

t

0(−A)

β

2 e(t−s)A(−A)β

2 PN [σ((−A)β

2 YN (s), a1(s))−σ((−A)β

2 Y (s),α1(s))]dWQ(s),

which yields, for a suitable Cγ(T ) > 0,

E|(−A)β

2 (YN (t)− Y (t))|2

≤ Cγ(T )

1

tβ|QNy0|2 + QNA− γ

2 21 +

t

0

1 +

1

(t− s)β+γ

E|(−A)

β

2 Y (s)|2ds

+

t

0

1 +

1

(t− s)β

E|(−A)

β

2 (YN (s)− Y (s))|2ds.

Since A−γ/2 is compact, QNA−γ/2 → 0 as N → +∞, and by using (3.431) wethus deduce

E|(−A)β

2 (YN (t)− Y (t))|2 ≤ Cγ,T,y0(N)

1 +

1

tβ

+ Cγ,T

t

0

1

(t− s)βE|(−A)

β

2 (YN (s)− Y (s))|2ds,

where Cγ,T,y0(N) → 0 as N → +∞. Using Proposition D.25 we thus obtain

E|(−A)β

2 (YN (t)− Y (t))|2 ≤ Cγ,T,y0(N)M1

tβ(3.434)

for some constant M independent of N . This implies the claim.

Proof of Theorem 3.123. We notice that under our assumptions equa-tion (3.389) with G given by (3.425) has a unique viscosity solution in BUC(H−η)for every η ∈ (0, 1 − β). To verify that the value function u is the solution weconsider the approximating problems

λuN − Ax,DuN +G(x,DuN , D2uN ) = 0 in HN . (3.435)

Equation (3.435) is the one used in the proof of Theorem 3.118 and it is easy to seethat it is the equation in HN corresponding to the control problem with evolutiongiven by (3.432). Therefore, by the results of Section 3.6.3, the function

uN (y0) = infa(·)∈U

E +∞

0e−λtl((−A)

β

2 YN (t; y0, a(·)), a(t))dt (3.436)


belongs to BUC(HN ), it satisfies the dynamic programming principle, i.e. for everyy0 ∈ HN , T ≥ 0

uN (y0) = infa(·)∈U

E T

0e−λtl((−A)

β

2 YN (t; y0, a(·)), a(t))dt

+ e−λTuN (YN (T ; y0, a(·))), (3.437)

and uN is the unique viscosity solution of (3.435) in BUC(HN ).Since for every y0 ∈ H, YN (t; y0, a(·)) = YN (t;PNy0, a(·)), extending uN to

H by putting uN (y) = uN (PNy) we obtain (3.436) and (3.435) for every y0 ∈ H.Moreover, from the proof of existence of Theorem 3.118, we know that for everyη ∈ (0, 1− β) and N ≥ 1

uN0 ≤ Cl

λ, |uN (x)− uN (y)| ≤ ση(|x− y|−η) (3.438)

for some modulus ση and uN → u uniformly on bounded sets, where u is the uniqueviscosity solution of (3.389) in BUC(H−η), η ∈ (0, 1− β).

We need to show that u = u. We will prove that uN converges pointwise to uas N → ∞. Let y0 ∈ H. For every T > 0,

|uN (y0)− u(y0)|

≤ supa(·)∈U

T

0e−λtEωl(|(−A)

β

2 (YN (t; y0, a(·))− Y (t; y0, a(·)))|)dt+ 2Cle−λT

λ.

Let > 0 and T > 0 be such that such that 2Cle−λT/λ ≤ . If ωl(s) ≤ +Ks, s ≥0, we obtain by Cauchy-Schwarz inequality

T

0e−λtEωl(|(−A)

β

2 (YN (t; y0, a(·))− Y (t; y0, a(·)))|)dt

≤

λ+

K

λ|YN (·; y0, a(·))− Y (·; y0, a(·))|

M2(0,T;D((−A)β

2 ))

for all N ≥ 1 and all a(·) ∈ U . The conclusion thus follows by letting N → +∞and using Lemma 3.124 since is arbitrary.

It remains to show the dynamic programming principle for u. By (3.437), wehaveuN (y0)− inf

a(·)∈UE T

0e−λtl((−A)

β

2 Y (t; y0, a(·)), a(t))dt+ e−λTu(Y (T ; y0, a(·)))

≤ supa(·)∈U

E T

0e−λtωl(|(−A)

β

2 (YN (t; y0, a(·))− Y (t; y0, a(·)))|)dt

+e−λT supa(·)∈U

E|uN (YN (T ; y0, a(·)))− u(y(T ; y0, a(·)))|.

The first term of the right-hand side converges to 0 when N goes to infinity by thesame argument as this used in the previous paragraph. For the second term, weproceed as follows:

E|uN (YN (T ; y0, a(·)))− u(Y (T ; y0, a(·)))|≤ E|uN (YN (T ; y0, a(·)))− uN (Y (T ; y0, a(·)))|

+ E|uN (Y (T ; y0, a(·)))− u(Y (T ; y0, a(·)))|.


The first term of the right-hand side converges to 0 uniformly in a(·) ∈ U when Ngoes to infinity by (3.434) and (3.438). It remains to prove that

supa(·)∈U

E|uN (Y (T ; y0, a(·)))− u(Y (T ; y0, a(·)))|

goes to 0 when N goes to infinity. By Proposition 3.122, estimate (3.430),E|Y (T ; y0, a(·))|2 is bounded by a constant C(T, |y0|) > 0 which does not dependon a(·) ∈ U . Hence, for all R > 0,

P |Y (T ; y0, a(·))| > R ≤ C(T, |y0|)R2

.

Let > 0 and choose R > 0 sufficiently large so that this probability be smallerthan . Then

supa(·)∈U

E|uN (Y (T ; y0, a(·)))− u(Y (T ; y0, a(·)))| ≤2Cl

λ+ sup

|y|≤R

|uN (y)− u(y)|.

We conclude by letting N → +∞ since was arbitrary.Finally we observe that for x0 ∈ H and y0 = (−A)−

β

2 x0, the mild solutionsX(·) of (3.420) and Y (·) of (3.423) are related by Y (·) = (−A)−

β

2 X(·). Therefore wehave v(x0) = u((−A)−

β

2 x0). Thus the dynamic programming principle also holdsfor v, and by definition v is the unique viscosity solution in BUC(H−η), η ∈ (0, 1)of the HJB equation (3.385) with F given by (3.426).

Remark 3.125 We would obtain the same results if instead of the change ofvariables y = (−A)−

β

2 x we applied the change of variables y = (−A)−γ

2 x for β ≤γ < 1. This may be beneficial for a boundary control problem with the Neumannboundary condition, where we have β < 1/2, as it may help make the second orderterms satisfy Hypothesis 3.119-(iii)(iv), which would then have (−A)−

β

2 replacedby (−A)−

γ

2 there (see also the example below).

We now discuss a specific example of a stochastic boundary control problemwith Dirichlet boundary conditions. It is more general than the one from Section2.6.2 since it also contains distributed controls and allows multiplicative noise Goodexamples of deterministic boundary control problems can be found in [309]. Theresults of this section would apply to suitable stochastic perturbations of examplesbelonging to the “first abstract class” in [309].

Let O ⊂ RN be an open, connected and bounded set with smooth boundary.Consider, as in Section 2.6.2, the following stochastic controlled PDE

∂x

∂t(t, ξ) = ∆ξx(t, ξ) + f1 (x(t, ξ),α1(t, ξ))

+f2 (x(t, ξ),α1(t, ξ)) WQ(t, ξ) in (0,∞)×O

x(0, ξ) = x0(ξ) on O

x(t, ξ) = α2(t, ξ) on (0,∞)× ∂O,(3.439)

where WQ is a Q-Wiener process, Q ∈ L+(L2(O)), x0 ∈ L2(O), and f1, f2 : R2 →R. We take H = L2(O), Λ1 = L2(O) and Λ2 to be the closed ball centered at0 with radius R in L2(∂O), and assume that the control a(t) = (a1(t), a2(t)) :=(α1(t, ·),α2(t, ·)) belongs to U as defined in (3.418). As it was discussed in Section2.6.2, (3.439) can be rewritten as an abstract stochastic evolution equation (3.419)-(3.420), where A is the Laplace operator with zero Dirichlet boundary conditions,


C is the the Dirichlet operator, and

b(x, a1)(ξ) = f1(x(ξ), a1(ξ)), [σ(x, a1)y](ξ) = f2(x(ξ), a1(ξ))y(ξ).

Suppose that f1, f2 satisfy for i = 1, 2

|fi(r, s)| ≤ c1(1 + |r|) for all r, s ∈ R,

|fi(r1, s)− fi(r2, s)| ≤ c1|r1 − r2| for all r1, r2, s ∈ R.

It is then easy to see that b satisfies Hypothesis 3.119-(i). As regards B, supposethat Q = I and N = 1. Let ek be the orthonormal basis of eigenvectors of A.In this case −Aek = ck2ek where c > 0. Moreover the ek are bounded in L∞(O),uniformly in k. Therefore we obtain

(−A)−β

2 σ(x, a1)2L2(H) =+∞

k=1

|(−A)−β

2 σ(x, a1)ek|2

= c−β+∞

k=1

+∞

h=1

h−2βσ(x, a1)ek, eh2 = c−β+∞

h=1

h−2β |σ(x, a1)eh|2

= c−β+∞

h=1

h−2β

O|f2(x(ξ), a1(ξ))eh(ξ)|2dξ ≤ C1(1 + |x|2),

where we used that σ(x, a1)ek, eh = ek,σ(x, a1)eh to justify the third equalityabove, and the fact that β > 1/2. However the above computation does not workfor for N ≥ 2, where stronger assumptions either on f2 or Q need to be imposed. Italso does not work in the Neumann case when β < 1/2, even if f2 ≡ 1, and this iswhy it may be beneficial to use the change of variables y = (−A)−

γ

2 x for β ≤ γ < 1as discussed in Remark 3.125. The other conditions of Hypothesis 3.119-(iii),(iv)are checked similarly. If f2 is constant (and thus so is B), Hypothesis 3.119-(iii),(iv)holds if we assume that there is an orthonormal basis ek of H such that

Aek = −λkek, Qek = βkek, k ∈ N,

where λk is a sequence of positive numbers increasing to +∞ while βk is abounded sequence of nonnegative real numbers and

∞

k=1

βk

λβk

< +∞.

Since for the Laplace operator A we have λk ≈ k2N as k → +∞, this condition is

fulfilled if for some > 0, βk ≤ Ck2βN

−1−. When Q is invertible this is possibleonly for N = 1. However Q can have finite rank.

If the original cost functional was given by

E +∞

0e−λt

Of3(x(t, ξ),α1(t, ξ))dξdt,

where λ > 0 and f3 : R2 → R, then the cost functional for the abstract evolutionsystem (3.419)-(3.420) is given by (3.421), where

l(x, a1) :=

Of3(x(ξ), a1(ξ))dξ.

If f3 ∈ BUC(R2) then l satisfies Hypothesis 3.119-(v). The original cost functionalcan be more general and depend explicitly on the boundary control α2.


3.13. HJB equations for control of stochastic Navier-Stokes equations

In this section we present another special class of equations which can be studiedby viscosity solution methods, which however require modification of the generaldefinition of viscosity solution from Section 3.3 and the techniques of Sections 3.5and 3.6. We will study the second order HJB equations that arise in problemsof optimal control of stochastic Navier-Stokes equations. No much is known aboutequations of this type. Kolmogorov equations for stochastic Navier-Stokes equationshave been studied by Komech and Vishik (see [443] and the references therein) andmore recently in [25, 26, 188, 401] for two-dimensional stochastic Navier-Stokesequations and by Da Prato and Debussche [117] for the three-dimensional case.Only existence of strict and mild solutions has been proved in [117]. A semilinearequation associated to a special optimal control problem has been investigated byDa Prato and Debussche in [115] from the point of view of mild solutions. Someof these results have been generalized to the three dimensional case in [328]. Themild solution approach of [115] is discussed in Section 4.9. The viscosity solutionapproach is more general in a sense that it can handle more complicated costfunctionals and applies to stochastic optimal control problems with the associatedHJB equations that are fully nonlinear in the gradient variable and the noise. Onthe other hand the covariance operator of the Wiener process here must be of traceclass and thus the viscosity solution approach cannot cover non-degenerate casesstudied in [115], where regular solutions were obtained and a formula for optimalfeedback was derived.

We will consider an optimal control problem for the 2-dimensional stochasticNavier-Stokes equations with periodic boundary conditions in the setting of an ab-stract stochastic evolution equation for the velocity vector field discussed in Section2.6.5. Let O = [0, L]× [0, L], and let ν > 0. We define the spaces

V =

x ∈ H1

p

O;R2

, div x = 0,

Ox = 0

,

H = the closure of V in L2O;R2

,

where for an integer k ≥ 1, Hkp

O;R2

is the space of R2 valued functions x

that are in Hkloc

R2;R2

and such that x(y + Lei) = x(y) for every y ∈ R2 and

i = 1, 2. We will denote by ·, ·, and | · | respectively the inner product and thenorm in L2

O;R2

. The space H inherits the same inner product and norm, and

V has the norm inherited from H1p

O;R2

. Let PH be the orthogonal projection in

L2O;R2

onto H. Define Ax = PH∆x with the domain D (A) = H2

p

O;R2

∩V ,

and B(x, y) = PH [(x · ∇)y] for x, y ∈ V . The operator A is maximal dissipative,self-adjoint, and (−A)−1 is compact. For γ = 1, 2 we denote Vγ := D((−A)

γ

2 ),equipped with the norm

|x|γ := |(−A)γ

2 x|. (3.440)The space V1 coincides with V . Recall that

O|curl x(ξ)|2dξ =

O|∇x(ξ)|2dξ, for x ∈ V.

Hence |x|1-norm is equivalent to

O|curl x(ξ)|2dξ

1/2

.

The dual space V of V can be identified with the space V−1, which is the completionof H with respect to the norm

|x|−1 := |(−A)−12x|.

3.13. HJB EQUATIONS FOR CONTROL OF STOCHASTIC NAVIER-STOKES EQUATIONS263

The duality is then given by

x, yV ,V = (−A)−12x, (−A)

12 y.

Let T > 0, Λ be a complete separable metric space. For every 0 ≤ t < T ,reference probability space ν = (Ω,F ,F t

s ,P,WQ), where WQ ia an H-valued Q-Wiener process with Q ∈ L+

1 (H), and a(·) ∈ Uµt , the abstract controlled stochastic

Navier-Stokes (SNS) equations describe the evolution of the velocity vector fieldX : [t, T ]×O × Ω → R2 that satisfies the stochastic evolution equation

dX(s) = (AX(s)−B (X(s), X(s)) + f (s, a(s))) ds+ dWQ(s) in (t, T ]×H,

X(t) = x ∈ H,(3.441)

where f : [0, T ] × Λ → V . (We remark that without loss of generality we set theviscosity coefficient in front of A to be 1.) The optimal control problem consists inthe minimization, over all controls a(·) ∈ Ut, of a cost functional

J(t, x; a(·)) = E T


.

The value functionv(t, x) = inf

a(·)∈Ut

J(t, x; a(·)), (3.442)

and the associated Hamilton-Jacobi-Bellman equation is

ut +1

2Tr

QD2u

+ Ax−B(x, x), Du+ F (t, x,Du) = 0,

u(T, x) = g(x) for (t, x) ∈ (0, T )×H,

(3.443)

where the Hamiltonian function F is defined byF (t, x, p) := inf

a∈Λf(t, a), p+ l(t, x, a) . (3.444)

It is convenient to introduce the trilinear form b(·, ·, ·) : V × V × V → R. It isdefined as

b (x, y, z) =

Oz(ξ) · (x(ξ) ·∇ξ)y(ξ) dξ = B(x, y), z.

It is a continuous operator on V × V × V but it can also be extended to a contin-uous map in different topologies, for instance it is also continuous on V × V2 ×H(see [436] and (3.449) below.) The incompressibility condition gives the standardorthogonality relations

b (x, y, z) = −b (x, z, y) , b (x, y, y) = 0. (3.445)Also, because of the periodic boundary conditions (see for instance [436]),

b (x, x,Ax) = 0 for x ∈ V2. (3.446)We will be using the following inequalities. If x, y, z ∈ V then

|b (x, y, z)| ≤ C |x|1/2 |x|1/21 |y|1 |z|1/2 |z|1/21 , (3.447)

which gives when z = x

|b (x, y, x)| ≤ C |x| |x|1 |y|1 . (3.448)Also, if x ∈ V , y ∈ V2, z ∈ H, then

|b (x, y, z)| ≤ C |x|1 |y|2 |z| . (3.449)We will assume the following hypothesis throughout the rest of this section

Hypothesis 3.126


(i) (−A)12Q

12 ∈ L2(H).

(ii) The function f : [0, T ]× Λ → V is continuous and there is R ≥ 0 suchthat

|f(t, a)|1 ≤ R for all t ∈ [0, T ], a ∈ Λ. (3.450)

We remark that Hypothesis 3.126-(i) is equivalent to the requirement thatTr(Q1) < +∞, where Q1 := (−A)

12Q(−A)

12 . By this we mean that (−A)

12Q(−A)

12

is densely defined and it extends to a bounded operator, still denoted by Q1, be-longing to L+

1 (H).

3.13.1. Estimates for controlled SNS equations. We will be using thenotions of variational and strong solutions of the SNS equations (3.441). Thedefinition of a variational solution is the same as this in Section 3.11.1, howeversince the generic equation (3.322) there is slightly different from (3.441), we repeatthe definition below.

Definition 3.127 Let 0 ≤ t < T . Let µ =Ω,F , F t

ss∈[t,T ] ,P,WQ

be a

generalized reference probability space, let Hypothesis 3.126 be satisfied. Let ξ bean Ft measurable H-valued random variable such that Eµ[|ξ|2] < +∞, and leta(·) ∈ Uµ

t .• A process X(·) ∈ M2

µ(t, T ;H) is called a variational solution of (3.441)with initial condition X(t) = ξ if

E T

t|X(r)|2V dr

< +∞

and for every φ ∈ V we have

X(s),φ = ξ,φ+ s

tAX(r)−B(X(r), X(r)) + f(r, a(r)),φV ,V dr

+

s

tdWQ(r),φ for each s ∈ [t, T ] , P-a.e..

• A process X(·) ∈ M2µ(t, T ;H) is called a strong solution of (3.441) with

initial condition X(t) = ξ if

E T

t|X(r)|2V2

dr

< +∞

and we have

X(s) = ξ +

s

t(AX(r)−B(X(r), X(r)) + f (r, a(r)))dr +

s

tdWQ(r)

for each s ∈ [t, T ],P-a.e..

Proposition 3.128 Let 0 ≤ t < T and p ≥ 2. Let µ =Ω,F , F t

ss∈[t,T ] ,P,WQ

be a generalized reference probability space, and let Hy-

pothesis 3.126 be satisfied. Let ξ be an F tt measurable H-valued random variable

such that Eµ[|ξ|p] < +∞, and let a(·) ∈ Uµt . Then:

(i) There exists a unique variational solution X(·) = X(·; t, ξ, a(·)) of(3.441) with initial condition X(t) = ξ. The solution has continuoustrajectories and satisfies for t ≤ s ≤ T

E|X(s)|p + E s

t|X(τ)|21|X(τ)|p−2dτ ≤ E|ξ|p + C(p,R,Q)(s− t). (3.451)


and

E

supt≤s≤T

|X(s)|p≤ C(p, T,R,Q) (1 + E|ξ|p) . (3.452)

(ii) If E|ξ|p1 < +∞, then the variational solution X(·) = X(·; t, ξ, a(·)) is astrong solution with trajectories continuous in V . Moreover we have fort ≤ s ≤ T

E|X(s)|p1 + E s

t|X(τ)|22|X(τ)|p−2

1 dτ ≤ E|ξ|p1 + C(p,R,Q1)(s− t) (3.453)

and

E

supt≤s≤T

|X(s)|p1≤ C(p, T,R,Q1) (1 + E|ξ|p1) . (3.454)

(iii) If µ1 is another generalized reference probability space, ξ1 is a Ft,µ1t

measurable H-valued random variable such that Eµ1 [|ξ1|p] < +∞,a1(·) ∈ Uµ1

t , and

LP1(ξ1, a1(·),WQ,1(·)) = LP(ξ, a(·),WQ(·)),then

LP1(a1(·), X1(·)) = LP(a(·), X(·)), (3.455)where X1(·) = X1(·; t, ξ1, a1(·)) is the variational solution of (3.441) inµ1 with control a1(·) and initial condition ξ1.

Proof. (i) The general strategy of the poof of part (i) is similar to the proofof Theorem 3.102. More precisely, part (i) is proved in [342], Proposition 3.3 (seealso [93] for a similar proof and [127, 443] for related results and estimates). Wesketch the main points of the proof since we will need them to explain parts (ii)and (iii).

Let e1, e2, ... be the orthonormal basis of H composed of eigenvectors of A,Hn := spane1, ..., en, and Pn be the orthogonal projection in H onto Hn. Inthis case Pn extends to the orthogonal projection in V onto Hn. Also we havePnA = APn. Let Xn(·) be the unique strong solution of

dXn(s) = (PnAXn(s)− PnB(Xn(s), Xn(s)) + Pnf(s, a(s)))ds+ PndWQ(s)

Xn(t) = Pnξ(3.456)

We first assume that p ≥ 8. If follows using Itô’s formula (see also [342]) thatwe have

E

supt≤s≤T

|Xn(s)|p + T

t|Xn(s)|2V (1 + |Xn(s)|p−2)ds

≤ M for all n.

It can also be deduced from the estimates for Xn(·) obtained from Itô’s formula,and a martingale inequality, that the norms of Xn(·) in L8(Ω;L4((t, T ) × O)) arebounded uniformly in n. Therefore, there exists a process X(·) ∈ M2

µ(t, T ;V ) ∩Lp(Ω;L∞(t, T ;H)), and a FT measurable random variable η ∈ L2(Ω;H) such that(up to a subsequence and identifying X(·) with its versions)

Xn(·) X(·) in M2µ(t, T ;V ), Xn(T ) η in L2(Ω;H),

Xn(·) → X(·) weak star in Lp(Ω;L∞(t, T ;H)).

We can also assume that X(·) ∈ L8(Ω;L4((t, T ) × O)). Passing to the limit asn → +∞ we obtain that there is a process F0(·) ∈ M2

µ(t, T ;V) such that, up to a

subsequence,PnAXn(·)− PnB(Xn(·), Xn(·)) F0(·) in M2

µ(t, T ;V),


X(·) is a variational solution of

dX(s) = (F0(s) + f(s, a(s)))ds+ dWQ(s)

X(t) = ξ,

X(T ) = η, and, by Theorem 3.101, X(·) ∈ Lp(Ω;C([t, T ];H)) and P-a.e.

|X(s)|2 = |ξ|2 + 2

s

tF0(r) + f(r, a(r)), X(r)V ,V dr

+2

s

tdWQ(r), X(r)+Tr(Q)(s− t).

One then uses an argument based on the monotonicity of the operator −Ax+B(x, x)on balls in L4(O) (see [342]) to show that F0(·) = AX(·)−B(X(·), X(·)), i.e. X(·)is a variational solution of (3.441), and thus using (3.445), we have

|X(s)|2 = |ξ|2 − 2

s

t|X(r)|21dr + 2

s

tf(r, a(r)), X(r)dr

+2

s

tdWQ(r), X(r)+Tr(Q)(s− t).

(3.457)

(The monotonicity argument uses the fact that X(·) ∈ L8(Ω;L4((t, T )×O)).) Wealso have a similar identity as (3.457) for |Xn(s)|2. Taking expectation in both ofthem for s = T , passing to the limit as n → +∞, and recalling that Xn(T ) X(T ),we deduce E|Xn(T )|2 → E|X(T )|2, which gives E|Xn(T )−X(T )|2 → 0 as n → +∞.Replacing T by s ∈ (t, T ), the same arguments give E|Xn(s)−X(s)|2 → 0.

Estimates (3.451) and (3.452) can now be proved by applying Itô’s formula tothe function ϕ(r) = rp/2 and using identity (3.457).

Uniqueness of variational solutions for any p ≥ 2 follows from Proposition3.129-(i).

Let now 2 ≤ p < 8. For n ≥ 1 we denote Ωn := ω ∈ Ω : |ξ(ω)| ≤ n andξn := ξ1Ωn . Then X(·; t, ξn, a(·)) = X(·; t, ξm, a(·)) on Ωn if n ≤ m, and estimates(3.451) and (3.452) are true for the processes X(·; t, ξn, a(·)), n ≥ 1. Therefore,the process X(s) := limn→+∞ X(·; t, ξn, a(·)) is well defined, (3.451) and (3.452)for X(·; t, ξn, a(·)) follow from Fatou’s lemma, and it is easy to see that X(·) is avariational solution of (3.441).

(ii) Let 2p ≥ 8 and Xn(·) be the processes from part (i). It follows from Itô’sformula, Hypothesis 3.126 and (3.446), that there is M ≥ 0 such that the processesXn(·) satisfy in this case

E

supt≤s≤T

|Xn(s)|p1 + T

t|Xn(s)|22(1 + |Xn(s)|p−2

1 )ds

≤ M for all n.

Therefore by passing to a weak limit we obtain that the variational solution frompart (i) satisfies X(·) ∈ Lp(Ω;L∞(t, T ;V )) ∩ M2

µ(t, T ;V2), and thus it is a strongsolution. Moreover (−A)

12 (AX(r)−B(X(r), X(r)) + f(r, a(r))) ∈ M2

µ(t, T ;V−1),the process

(−A)12X(s) = (−A)

12 ξ

+

s

t(−A)

12 (AX(r)−B(X(r), X(r)) + f(r, a(r))) dr +

s

t(−A)

12 dWQ(r)


is a continuous process with values in V−1, and (−A)12X(·) ∈ M2

µ(t, T ;V ). Thus,by Theorem 3.101, X(·) ∈ Lp(Ω;C([t, T ];V )) and P-a.e.

|X(s)|21 = |ξ|21 − 2

s

t

|AX(r)|2 + f(r, a(r)), AX(r)

dr

−2

s

tdWQ(r), AX(r)+Tr(Q1)(s− t).

(3.458)

Estimates (3.453) and (3.454) now follow by standard arguments applying Itô’sformula to the function ϕ(r) = rp/2 and using identity (3.458). For 2 ≤ p < 8 weproceed as in the proof of part (i).

(iii) Similarly to the proof of part (ii) of Theorem 3.102, if p ≥ 8 and X1(·) isthe variational solution in the generalized reference probability space µ1 and Xn

1 (·)are the solutions of the approximating problems (3.456) in this space, then

LP1(a1(·), Xn1 (·)) = LP(a(·), Xn(·)),

and thus (3.455) follows since Xn1 (s) → X1(s) in L2(Ω1;H) and Xn(s) → X(s) in

L2(Ω;H) for every s ∈ [t, T ]. For 2 ≤ p < 8 we have

LP1(a1(·), X1(·; t, ξn1 , a1(·))) = LP(a(·), X(·; t, ξn, a(·))),

which gives the claim in the limit as n → +∞.

Without loss of generality we will always assume from now on that the Q-Wiener processes in the reference probability spaces have everywhere continuouspaths.

Proposition 3.129 Let 0 ≤ t < T and p ≥ 2. Let ν =Ω,F , F t

ss∈[t,T ] ,P,WQ

be a reference probability space, and let Hypothesis

3.126 be satisfied. Let ξ, η be F tt measurable H-valued random variables such that

Eν [|ξ|p + |η|p] < +∞, and let a(·) ∈ Uνt . Then:

(i) There exists a constant C independent of t, ξ, η, a(·) and µ, such thata.s. on Ω

|X(s)− Y (s)|2 + s

t|X(τ)− Y (τ)|21dτ ≤ |ξ − η|2exp

s

tC|X(τ)|21dτ

(3.459)

for all s ∈ [t, T ], where X(·) = X(·; t, ξ, a(·)), Y (·) = Y (·; t, η, a(·)) aresolutions of (3.441) with initial conditions X(t) = ξ and Y (t) = η.

(ii) If |x|1 ≤ R1 then there exists a constant C = C(p, T,R,R1, Q) suchthat

E|X(s)− x|p ≤ C(p, T,R,R1, Q)(s− t), (3.460)

where X(·) = X(·; t, x, a(·)).(iii) For every initial condition x ∈ V there exists a modulus ω, independent

of the reference probability spaces ν and controls a(·) ∈ Uνt , such that

E|X(s)− x|21 ≤ ωx(s− t), (3.461)

where X(·) = X(·; t, x, a(·)).Proof. (i) Denote Z(s) = X(s)− Y (s). Then Z(·) satisfies

Z(s) = ξ − η +

s

tAZ(τ)dτ +

s

t[B(Y (τ), Y (τ))−B(X(τ), X(τ))]dτ.


Hence using (3.445) and (3.448) we obtain

|Z(s)|2 = |ξ − η|2 − 2

s

t|Z(τ)|21dτ −

s

tb(Z(τ), X(τ), Z(τ))dτ

≤ |ξ − η|2 − 2ν

s

t|Z(τ)|21dτ +

s

tC|Z(τ)|1|X(τ)|1|Z(τ)|dτ

≤ |ξ − η|2 − ν

s

t|Z(τ)|21dτ + C

s

t|X(τ)|21|Z(τ)|2dτ. (3.462)

Here we have used Young’s inequality. Then it follows from Gronwall’s lemma that

|Z(s)|2 ≤ |ξ − η|2exp s

tC|X(τ)|21dτ P a.s.

Plugging this back into (3.462) yields (3.459) with another constant C.(ii) Denote Y (s) = X(s)− x. Then

Y (s) =

s

t(AX(τ)−B (X(τ), X(τ)) + f (τ, a(τ))) dτ +

s

tdW (τ).

Therefore applying Itô’s formula, taking expectation, and using (3.445) and theCauchy-Schwartz inequality, we obtain

E|Y (s)|p ≤ E s

tpAX(τ)−B (X(τ), X(τ)) + f (τ, a(τ))), Y (τ)|Y (τ)|p−2dτ

+ E s

t

p(p− 1)

2tr(Q)|Y (τ)|p−2dτ

≤ −p

2E s

t|X(τ)|21|Y (τ)|p−2dτ + CpE

s

t|x|21|Y (τ)|p−2dτ

+ C(p,R,R1, Q)E s

t(|Y (τ)|p−1 + |Y (τ)|p−2)dτ

+ pE s

t|b(X(τ), X(τ), x)||Y (τ)|p−2dτ.

Since

|b(X(τ), X(τ), x)| ≤ C|X(τ)|1|X(τ)||x|1 ≤ 1

2|X(τ)|21 +

C2

2|X(τ)|2|x|21,

plugging this into the previous inequality and using (3.452) finally yields

E|Y (s)|p ≤ C(p,R,R1, Q)E s

t(|Y (τ)|p−1 + |Y (τ)|p−2 + |X(τ)|2|Y (τ)|p−2)dτ

≤ C(p, T,R,R1, Q)(s− t).

(iii) If (3.461) is not satisfied then there are > 0, an(·) ∈ Ut (which we canassume to be F t,0

s -predictable) and sn → t such that E|Xn(sn) − x|21 ≥ for alln ≥ 1, where Xn(·) = X(·; t, x, an(·). By Corollary 2.21 and Proposition 3.128-(iii),we can assume that all an(·) are defined on the same reference probability space.

However it follows from (3.460) and (3.453) that, up to a subsequence, we have

Xn(sn) → x strongly in L2(Ω;H) and weakly in L2(Ω;V ).

Since the weak sequential convergence in L2(Ω;V ) implies

|x|21 ≤ lim infn→∞

E|X(sn)|21,

this, together with (3.453), implies |x|21 = limn→∞ E|X(sn)|21. Therefore X(sn) → xstrongly in L2(Ω;V ), contrary to our assumption.


3.13.2. Value function. In this section we show continuity properties of thevalue function of the stochastic optimal control problem and the Dynamic Pro-gramming Principle.

To minimize non-essential technical difficulties we will assume that the runningcost function l is independent of t. The case of l depending on t is a straightforwardextension of the methods presented here. The continuity of the value function is notentirely trivial since continuous dependence estimates in the mean for solutions ofthe stochastic Navier-Stokes equation depend on exponential moments of solutions(3.459) and these seem to be bounded only for a short time (see [443, CorollaryXI.3.1], also [425]). We make the following assumptions about the cost functions land g.

Hypothesis 3.130 The functions l : V ×Λ → R, and g : H → R are continuousand there exist k ≥ 0 and for every r > 0 a modulus σr such that

|l(x, a)|, |g(x)| ≤ C(1 + |x|k1) for all x ∈ V, a ∈ Λ, (3.463)

|l(x, a)− l(y, a)| ≤ σr (|x− y|1) if |x|1, |y|1 ≤ r, a ∈ Λ (3.464)

|g(x)− g(y)| ≤ σr (|x− y|) if |x|1, |y|1 ≤ r. (3.465)

Proposition 3.131 Let Hypotheses 3.126 and 3.130 be satisfied. Then:(i) For every r > 0 there exists a modulus ωr such that for every t ∈

[0, T ], a (·) ∈ Ut

|J(t, x; a(·))− J(t, y; a(·))| ≤ ωr (|x− y|) if |x|1, |y|1 ≤ r. (3.466)

(ii) The value function v satisfies the Dynamic Programming Principle, i.e.for every 0 ≤ t ≤ η ≤ T and x ∈ V ,

v(t, x) = infa(·)∈Ut

E η

tl(X(s; t, x, a(·)), a(s))ds+ v(η, X(η; t, x, a(·)))

. (3.467)

(iii) For every r > 0 there exists a modulus ωr such that

|v(t1, x)− v(t2, y)| ≤ ωr (|t1 − t2|+ |x− y|) (3.468)

for all t1, t2 ∈ [0, T ] and |x|1, |y|1 ≤ r, and there exists C ≥ 0 such that

|v(t, x)| ≤ C(1 + |x|k1) (3.469)

for all t ∈ [0, T ] and x ∈ V .

Proof. (i) Let x, y ∈ V, t ∈ [0, T ] and a (·) ∈ Ut. For every m > 0 let Dm

be a constant such that σm(s) ≤ 1m +Dms. Denote X(s) = X(s; t, x, a(·)), Y (s) =

Y (s; t, y, a(·)), and Am = ω ∈ Ω : maxt≤s≤T |X(s)|1 ≤ m, Bm = ω ∈ Ω :maxt≤s≤T |Y (s)|1 ≤ m. Then, using (3.454), (3.459), (3.463) and (3.464), weobtain

E T

t|l(X(s), a(s))− l(Y (s), a(s))|ds ≤ T

m+ E

T

tDm|X(s)− Y (s)|1 1Am∩Bm

ds

+E T

tC(2 + |X(s)|k1 + |Y (s)|k1)1Ω\(Am∩Bm)ds

≤ T

m+Dm|x− y|E

T

texp

C

s

t|X(τ)|21dτ

1Am

ds

+

T

tC2 + (E|X(s)|2k1 )

12 + (E|Y (s)|2k1 )

12

(P(Ω \Am))

12 + (P(Ω \Bm))

12

ds

≤ T

m+DmT |x− y|eCTm2

+ C1(p, T,R,Q1)(1 + |x|k1 + |y|k1)1 + |x|1 + |y|1

m.


Applying the same process to estimate |g(X(T )) − g(Y (T ))| we therefore obtainthat for every r,m > 0 there exist constants cm, dr such that for every t ∈ [0, T ]and a (·) ∈ Ut

|J(t, x; a(·))− J(t, y; a(·))| ≤ drm

+ cm|x− y| if |x|1, |y|1 ≤ r.

Estimate (3.466) now follows by taking the infimum over all m > 0.(ii) We need to show that the problem satisfies the assumptions of Hypothesis

2.12. However here the statement of the DPP is restricted to points in V butthe filtrations are still generated by Q-Wiener processes with values in H. Thus toproceed with the proof of the DPP described in Section 2.3 it is enough to assume inHypothesis 2.12 that the random variable ξ there satisfies Eµ|ξ|21 < +∞. We recallthat in this case we have strong solutions so conditions (A0) and (A2) follow fromthe definition of solution and standard arguments. In particular (A0) follows from(3.459) and (A2) from an obvious generalization of (3.459). Condition (A1) followsfrom Proposition 3.128-(iii). We point out that if LP1(X1(·; t1, x, a1(·)), a1(·)) =LP2(X2(·; t1, x, a2(·)), a2(·)) as processes with values in H × Λ then, by Lemma1.17-(i), they have the same laws as processes with values in V × Λ. The proof ofcondition (A3) starts as the proof of it in Proposition 2.16 however the argumentsare now obvious since here the stochastic integral is just Wt1(s).

(iii) We notice that (3.466) implies

|v(t, x)− v(t, y)| ≤ ωr (|x− y|) (3.470)

for all t ∈ [0, T ] and |x|1, |y|1 ≤ r. Moreover (3.469) is a direct consequence of(3.454) and (3.463).

Let now 0 ≤ t1 < t2 ≤ T, x ∈ V, |x|1 ≤ r. We will denote X(s) =X (s; t1, x, a(·)). Using (3.454), (3.460), (3.467), (3.469), (3.470), we obtain form > r,

|v(t1, x)− v(t2, x)| ≤ supa(·)∈Ut1

E t2

t1

(1 + |X(s)|k1)ds

+ supa(·)∈Ut1

E|v(t2, X(t2))− v(t2, x)|

≤ C(R, T,Q1, r)(t2 − t1) + supa(·)∈Ut1

EC(1 + |X(t2)|k1 + |x|k1)1|X(t2)|1>m

+ supa(·)∈Ut1

Eσm (|X(t2)− x|)

≤ C(R, T,Q1, r)(t2 − t1) + C(R, T,Q1)(1 + |x|k1)1 + |x|1

m

+σm

C(R,Q, r)(t2 − t1)

12

.

(We also used that σm above can be assumed to be concave.) The result now followsby taking the infimum over m > r.

3.13.3. Viscosity solutions and comparison theorem. Since we onlyhave continuity of the value function on [0, T ]× V , the definition of viscosity solu-tion has to be restricted to this space. From the point of view of the HJB equationit might be better to set it up in this space, however because of the associatedcontrol problem, we want to keep H as our reference space. We achieve it by aproper choice of test functions. By using a special radial function of | · |1 as testfunction we first restrict the points where maxima or minima occur in the defini-tion of viscosity sub/solution to be in (0, T )× V . Then we require that the pointswhere the maxima/minima occur belong to (0, T ) × V2. Having this property wecan interpret all terms appearing in the HJB equation. In this way we gain some


coercive terms which had to be discarded in the generic definition given in Section3.3, which are very useful in the proof of comparison principle. The definition ismeaningful as we are able to show, using properties of the Navier-Stokes equationand the coercivity of the operator −A, that the value function is a viscosity so-lution. The definition of viscosity solution here is thus similar to the one used inSections 3.11 and 3.12, however we use a radial test function of a different type. Ifdifferent continuity requirements were imposed in Hypothesis 3.130, we would havedifferent continuity properties of the value function, and then we could work witha definition of viscosity solution which resembles more the definition from Section3.11, as it was done for first order equations in [243].

Definition 3.132 A function ψ is a test function for equation (3.443) if ψ =ϕ+ δ(t)(1 + |x|21)m, where

(i) ϕ ∈ C1,2 ((0, T )×H), and is such that ϕ,ϕt, Dϕ, D2ϕ are uniformlycontinuous on [, T − ]×H for every > 0.

(ii) δ ∈ C1 ((0, T )) is such that δ > 0 on (0, T ), and m ≥ 1.

The function h(t, x) = δ(t)(1+ |x|21)m is not Fréchet differentiable in H. There-fore the terms involving Dh and D2h, in particular Ax − B(x, x), Dh(t, x) andTr(QD2h(t, x)), have to be understood properly. We define

Dh(t, x) := −δ(t)2m(1 + |x|21)m−1Ax

,

and we will writeDψ := Dϕ+Dh

even though this is a slight abuse of notation. Then, if (t, x) ∈ (0, T )×V2, Dψ(t, x)makes sense, and so does the term Ax − B(x, x), Dψ(t, x). As regards the termTr(QD2ψ(t, x)), without defining D2h(t, x), we interpret is by defining

Tr(QD2ψ(t, x)) := Tr(QD2ϕ(t, x)) + δ(t)2m(1 + |x|21)m−1Tr(Q1)

+4m(m− 1)(1 + |x|21)m−2|Q 12Ax|2

.

It will be seen in the next section that the above interpretations appear as directconsequences of Itô’s formula applied to h.

We give a definition of viscosity solution for a general equation (3.443) wherethe Hamiltonian function F is not necessarily given by (3.444). Thus we assume inthis section that F : [0, T ]× V ×H → R is any function.

Definition 3.133 A weakly sequentially upper-semicontinuous (respectively,lower-semicontinuous) function u : (0, T ] × V → R is called a viscosity sub-solution (respectively, supersolution) of (3.443) if u(T, y) ≤ h(y) (respectively,u(T, y) ≥ h(y)) for all y ∈ V and if, for every test function ψ, whenever u− ψ hasa global maximum (respectively u + ψ has a global minimum) over (0, T ) × V at(t, x), then x ∈ V2 and

ψt(t, x) +1

2Tr(QD2ψ(t, x)) + Ax−B(x, x), Dψ(t, x)+ F (t, x,Dψ(t, x)) ≥ 0

(respectively,

−ψt(t, x)−1

2Tr(QD2ψ(t, x))− Ax−B(x, x), Dψ(t, x)+ F (t, x,−Dψ(t, x)) ≤ 0.)

A function u is a viscosity solution of (3.443) if it is both a viscosity subsolutionand a viscosity supersolution of (3.443).

Hypothesis 3.134 F : [0, T ]× V ×H → R and there exist a modulus of conti-nuity ω, and moduli ωr such that for every r > 0 we have

|F (t, x, p)− F (t, y, p)| ≤ ωr (|x− y|1) + ω (|x− y|1|p|) , if |x|1, |y|1 ≤ r, (3.471)


|F (t, x, p)− F (t, x, q)| ≤ ω ((1 + |x|1)|p− q|) , (3.472)

|F (t, x, p)− F (s, x, p)| ≤ ωr(|t− s|), if |x|1, |p|1 ≤ r, (3.473)

|g(x)− g(y)| ≤ ωr(|x− y|), if |x|1, |y|1 ≤ r. (3.474)

Theorem 3.135 Let Hypothesis 3.134 hold. Let u, v : (0, T ]×V → R be, respec-tively, a viscosity subsolution, and a viscosity supersolution of (3.443). Let

u(t, x), −v(t, x), |g(x)| ≤ C(1 + |x|k1) (3.475)

for some k ≥ 0. Then u ≤ v on (0, T ]× V .

Proof. We observe that weak sequential upper-semicontinuity of u andweak sequential lower-semicontinuity of v imply that

limt↑T (u(t, x)− g(x))+ = 0limt↑T (v(t, x)− g(x))− = 0

(3.476)

uniformly on bounded subsets of V . We define for µ > 0,

uµ(t, x) = u(t, x)− µ

t, vµ(t, x) = v(t, x) +

µ

t.

Then uµ and vµ are, respectively, a viscosity subsolution, and a viscosity superso-lution of

(uµ)t +1

2Tr(QD2uµ) + Ax−B(x, x), Duµ+ F (t, x,Duµ) =

µ

T 2

and

(vµ)t +1

2Tr(QD2uµ) + Ax−B(x, x), Dvµ+ F (t, x,Dvµ) = − µ

T 2.

Let m be a number such that m ≥ 1 and 2m ≥ k + 1. For 0 < , δ,β ≤ 1, weconsider the function

Φ(t, s, x, y) = uµ(t, x)− vµ(s, y)−|x− y|2

2

−δeKµ(T−t)(1 + |x|21)m − δeKµ(T−s)(1 + |y|21)m − (t− s)2

2β

and setΦ(t, s, x, y) = −∞ if x, y ∈ V,

The constant Kµ will be chosen later. Obviously Φ → −∞ as max(|x|1, |y|1) →+∞. We claim that Φ is weakly sequentially upper-semicontinuous on (0, T ] ×(0, T ]×H ×H.

It is well known that functions x → (1+ |x|21)m, y → (1+ |y|21)m and |x−y|2 areweakly sequentially lower-semicontinuous, respectively in H and H ×H. To showthat, say,

uµ(t, x)− δeKµ(T−t)(1 + |x|21)m

is weakly sequentially upper-semicontinuous on (0, T ]×H, we suppose that this isnot the case, i.e. there exist sequences tn → t ∈ (0, T ], xn x ∈ H such that

lim supn→∞

uµ(tn, xn)− δeKµ(T−tn)(1 + |xn|21)m

> uµ(t, x)− δeKµ(T−t)(1 + |x|21)m.

If lim infn→∞ |xn|1 = +∞, this is impossible by (3.475). So there must exist asubsequence (still denoted by (tn, xn)) such that lim supn→∞ |xn|1 < +∞. But thenwe have xn x in V , which contradicts the weak sequential upper-semicontinuityof uµ.

Therefore Φ has a global maximum over (0, T ]× (0, T ]×H ×H at some point(t, s, x, y) ∈ (0, T ] × (0, T ] × V × V , where |x|1, |y|1 are bounded independently of


,β for a fixed δ. We can assume that the maximum is strict. By the definition ofviscosity solution, x, y ∈ V2. Moreover it is standard to notice that

limβ→0

(t− s)2

2β= 0 for fixed δ, , (3.477)

and

lim→0

lim supβ→0

|x− y|22

= 0 for fixed δ. (3.478)

If u v it then follows from (3.478), (3.477), (3.474) and (3.476) that for small µand δ, we have t, s < T if β and are sufficiently small.

We use the projections from the proof of Proposition 3.128. Let e1, e2, ... bethe orthonormal basis of H composed of eigenvectors of A, HN := spane1, ..., eN,PN be the orthogonal projection in H onto HN , and QN = I − PN for N ≥ 2. Wedefine

u(t, x) = uµ(t, x)−x,QN (x− y)

+

|QN (x− y)|22

− |QN (x− x)|2

− δeKµ(T−t)(1 + |x|21)m,

v1(s, y) = vµ(s, y)−y,QN (x− y)

+

|QN (y − y)|2

+ δeKµ(T−s)(1 + |y|21)m,

and we set u(t, x) = −∞, v(s, y) = +∞ if x, y ∈ V . Then u, v are respectivelyweakly sequentially upper- and lower-semicontinuous on (0, T ] × H and it is easyto see (as in the proof of Theorem 3.50) that

u(t, x)− v(s, y)− |PN (x− y)|22

− (t− s)2

2β

attains a strict global maximum over (0, T ]× (0, T ]×H×H at (t, s, x, y). Moreoverthe functions u,−v satisfy (3.55). Therefore the assumptions of Corollary 3.29 aresatisfied for B = I there. Therefore there exist functions ϕk,ψk ∈ C2((0, T ) ×H)for k = 1, 2, ... such that ϕk, (ϕk)t, Dϕk, D2ϕk,ψk, (ψk)t, Dψk, D2ψk are boundedand uniformly continuous, and such that

u(t, x)− ϕk(t, x)

has a global maximum at some point (tk, xk) ∈ (0, T )× V ,

v(s, y)− ψk(s, y)

has a global minimum at some point (sk, yk) ∈ (0, T )× V , andtk, xk, u(tk, xk), (ϕk)t(tk, xk), Dϕk(tk, xk), D

2ϕk(tk, xk)

k→∞−−−−−−−−−−−−→R×H×R×H×L(H)

t, x, u(t, x),

t− s

β,PN (x− y)

, XN

(3.479)

sk, yk, v(sk, yk), (ψk)t(sk, yk), Dψk(sk, yk), D

2ψk(sk, yk)

k→∞−−−−−−−−−−−−→R×H×R×H×L(H)

s, y, v(s, y),

t− s

β,PN (x− y)

, YN

, (3.480)

where XN = PNXNPN , YN = PNYNPN and XN ≤ YN . Moreover, since thefunctions u,−v are weakly sequentially upper-semicontinuous, it follows fromthe proof of Corollary 3.29 (see the proof of Theorem 3.27) that ϕk(t, x) =ϕk(t, PNx),ψk(s, y) = ψk(s, PNy), and thus in particular we have

Dϕk(tk, xk) →PN (x− y)

, Dψk(sk, yk) →

PN (x− y)

in H1. (3.481)


In addition it is easy to see that we must have |xk|1, |yk|1 ≤ C for some C. Thereforexk x, yk y in V . This, together with (3.479), (3.479), and the fact that(t, s, x, y) is the maximum point of Φ, implies

|xk|1 → |x|1, |yk|1 → |y|1,

which in turns givesxk → x, yk → y in V. (3.482)

By the definition of viscosity solution, we have xk, yk ∈ V2, and

− δKµeKµ(T−tk)(1 + |xk|21)m + (ϕk)t(tk, xk)

+δ

2eKµ(T−tk)

2mTr(Q1)(1 + |xk|21)m−1 + 4m(m− 1)|Q 1

2Axk|2(1 + |xk|21)m−2

+1

2Tr

QD2ϕk(tk, xk) + 2QQN

+

Axk, Dϕk(tk, xk) +

QN (x− y)

+

2QN (xk − x)

−2mδeKµ(T−tk)(1 + |xk|21)m−1Axk

− b

xk, xk, Dϕk(tk, xk) +

QN (x− y)

+

2QN (xk − x)

+ F

tk, xk, Dϕk(tk, xk) +

QN (x− y)

+

2QN (xk − x)


≥ µ

T 2. (3.483)

Above we have used (3.446) to get b (xk, xk, Axk) = 0. We now want to pass to thelimit as k → ∞. Let Cµ be a constant such that

ω(s) ≤ µ

2T 2+ Cµs.

It then follows from (3.472) thatF


QN (x− y)

+

2QN (xk − x)


−F


QN (x− y)

+

2QN (xk − x)

≤ µ

2T 2+ Cµ(1 + |xk|1)2mδeKµ(T−tk)(1 + |xk|21)m−1|Axk|.

Moreover

Cµ(1 + |xk|1)2mδeKµ(T−tk)(1 + |xk|21)m−1|Axk|

+δ

2eKµ(T−tk)

2mTr(Q1)(1 + |xk|21)m−1 + 4m(m− 1)|Q 1

2Axk|2(1 + |xk|21)m−2

≤ 2mδC2µe

Kµ(T−tk)(1 + |xk|21)m +mδeKµ(T−tk)|Axk|2(1 + |xk|21)m−1

+ δeKµ(T−tk)m(2m− 1)Tr(Q1)(1 + |xk|21)m

≤ mδeKµ(T−tk)|Axk|2(1 + |xk|21)m−1

+ δeKµ(T−tk)2mC2

µ +m(2m− 1)Tr(Q1)(1 + |xk|21)m. (3.484)


Therefore, choosing Kµ = 1+2(2mC2µ+m(2m− 1)Tr(Q1)) we obtain from (3.483)

and (3.484) that

− δ

2Kµe

Kµ(T−tk)(1 + |xk|21)m + (ϕk)t(tk, xk)

+1

2Tr

QD2ϕk(tk, xk) + 2QQN

+

Axk, Dϕk(tk, xk) +

QN (x− y)

+

2QN (xk − x)

−mδeKµ(T−tk)Axk(1 + |xk|21)m−1

− b

xk, xk, Dϕk(tk, xk) +

QN (x− y)

+

2QN (xk − x)

+ F


QN (x− y)

+

2QN (xk − x)

≥ µ

2T 2.(3.485)

Using (3.479), (3.481), (3.482), (3.471)-(3.473), and the continuity of b on V ×V × V , we obtain from (3.485) that the norms |Axk| are bounded and thereforexk x in V2. Therefore, using the above again, we can pass to the lim sup asn → ∞ in (3.485) to get

− δ

2Kµe

Kµ(T−t)(1 + |x|21)m +t− s

β+

1

2Tr (QXN + 2QQN )

+ Ax,x− y

− b

x, x,

x− y

+ F

t, x,

x− y

≥ µ

2T 2. (3.486)

Similarly, we obtainδ

2Kµe

Kµ(T−s)(1 + |y|21)m +t− s

β+

1

2Tr (QYN − 2QQN )

+Ay,x− y

− b

y, y,

x− y

+ F

s, y,

x− y

≤ − µ

2T 2. (3.487)

Combining (3.486) and (3.487), using XN ≤ YN , and then sending N → ∞ yieldsδ

2

(1 + |x|21)m + (1 + |y|21)m

+

|x− y|21

+b

x, x,

x− y

− b

y, y,

x− y

+F

t, x,

x− y

− F

s, y,

x− y

≤ − µ

T 2. (3.488)

To estimate the trinilear form terms we use (3.445), (3.448), and then (3.478)to produce

bx, x,

x− y

− b

y, y,

x− y

=1

|b (x− y, x, x− y)| ≤ C

|x|1|x− y||x− y|1 (3.489)

≤ δ

2|x|21 + Cδ

|x− y|2

|x− y|21

≤ δ

2(1 + |x|21)m + σ2(β, ; δ, µ)

|x− y|21

,

where, for fixed µ, δ, lim→0 lim supβ→0 σ2(β, ; δ, µ) = 0.Finally we need to estimate the terms containing F . We know that for µ and

δ fixed |x|1, |y|1 ≤ Rδ for some Rδ > 0. Let Dµ,δ be a constant such that

ωRδ(s) ≤ µ

4T 2+Dµ,δs,


and denote Rδ, := 2Rδ/. Then (3.471), (3.473), (3.477) and (3.478) implyF

t, x,

x− y

− F

s, y,

x− y

≤ ωRδ,(|t− s|) + ωRδ

(|x− y|1) + ω

|x− y|1

|x− y|

≤ ωRδ,(|t− s|) + 3µ

4T 2+Dµ,δ|x− y|1 + Cµ|x− y|1

|x− y|

≤ ωRδ,(|t− s|) + 3µ

4T 2+Dµ,δ|x− y|1 +

|x− y|212

+ 2C2µ|x− y|2

≤ 3µ

4T 2+ σ3(β, ; δ, µ) +Dµ,δ|x− y|1 +

|x− y|212

, (3.490)

where, for fixed µ, δ, lim→0 lim supβ→0 σ3(β, ; δ, µ) = 0.Therefore, using (3.489) and (3.490) in (3.488), we obtain1

2− σ2(β, ; δ, µ)

|x− y|21

−Dµ,δ|x− y|1 ≤ − µ

4T 2+ σ3(β, , δ;µ). (3.491)

We now notice that if and β are small, then 12 − σ2(β, ; δ, µ) >

14 and that

lim→0

infr>0

r2

4−Dµ,δr

= 0.

Therefore, it remains to take lim inf→0 lim infβ→0 in (3.491) to obtain a contradic-tion, which proves that we must have u ≤ v.

3.13.4. Existence of viscosity solutions. We go back to the HJB equation(3.443) with the Hamiltonian function F defined by (3.444) and show that thevalue function of the associated stochastic optimal control problem is its viscositysolution.

Theorem 3.136 Let Hypotheses 3.126 and 3.130 be satisfied, and let in additionf : [0, T ]×Λ → V be such that f(·, a) is uniformly continuous, uniformly for a ∈ Λ.Then the value function v defined by (3.442) is the unique viscosity solution of theHJB equation (3.443)-(3.444) within the class of viscosity solutions u satisfying

|u(t, x)| ≤ C(1 + |x|k1), (t, x) ∈ (0, T ]× V,

for some k ≥ 0.

Proof. First of all we notice that under our assumptions, the Hamilton-ian F in (3.444) satisfies Hypothesis 3.134. Moreover, by Proposition 3.131, thevalue function v satisfies (3.468), (3.469), and the Dynamic Programming Principle(3.467). In particular v is weakly sequentially continuous on (0, T ]× V . Therefore,if v is a viscosity solution of (3.443)-(3.444), the uniqueness part is a direct conse-quence of Theorem 3.135. We will only show that the value function is a viscositysupersolution. The proof that v is a viscosity subsolution is easier and uses thesame techniques. To this end, let ψ (t, x) = ϕ (t, x) + δ(t)(1 + |x|21)m be a testfunction and let v + ψ have a global minimum at (t0, x0) ∈ (0, T )× V .

Step 1. We need to show that x0 ∈ V . By (3.467), for every > 0 there existsa (·) ∈ Ut0 such that, writing X (s) for X (s; t0, x0, a(·)), we have

v (t0, x0) + 2 > E t0+

t0

l (X (s) , a(s)) ds+ v (t0 + , X (t0 + ))

.

We can assume that a is F t0,0s -predictable and thus, by Corollary 2.21 and Propo-

sition 3.128-(iii), we can assume that all a(·) are defined on the same reference


probability space ν, i.e. a (·) ∈ Uνt0 . Since for every (t, x) ∈ (0, T )× V

v (t, x)− v (t0, x0) ≥ −ϕ (t, x) + ϕ (t0, x0)− δ(t)(1 + |x|21)m + δ(t0)(1 + |x0|21)m,

we have

2 − E t0+

t0

l (X (s) , a(s)) ds ≥ E [v (t0 + , X (t0 + ))− v (t0, x0)]

≥ E− ϕ (t0 + , X (t0 + )) + ϕ (t0, x0)

−δ(t0 + )1 + |X (t0 + ) |21

m+ δ(t0)

1 + |x0|21

m .

Set λ = inft∈[t0,t0+0] δ (t) for some fixed 0 > 0, and take < 0. Applying Itô’sformula to ϕ (s,X(s)) and δ(s)

1 + |X(s)|21

m, together with identity (3.458), inthe inequality above, and then dividing both sides by , we obtain

− 1

E t0+

t0

l (X (s) , a (s)) ds

≥ −1

E t0+

t0

ϕt (s,X (s)) + AX (s)−B (X (s) , X (s)) , Dϕ (s,X (s))

+ f (s, a (s)) , Dϕ (s,X (s))+1

2Tr

QD2ϕ (s,X (s))

ds

−1

E t0+

t0

δ (s)

1 + |X (s) |21

m+mTr(Q1)

1 + |X (s) |21

m−1

−2mδ(s)|AX (s) |2 + f (s, a (s)) , AX (s)

1 + |X (s) |21

m−1

+2m(m− 1)|Q 12AX (s) |2

1 + |X (s) |21

m−2ds

. (3.492)

By the definition of λ it then follows that

2mλ

E t0+

t0

|X (s) |221 + |X (s) |21

m−1ds

≤ +1

E t0+

t0

− l (X (s) , a (s)) + ϕt (s,X (s))

+ AX (s)−B (X (s) , X (s)) , Dϕ (s,X (s))

+ f (s, a (s)) , Dϕ (s,X (s))+1

2Tr

QD2ϕ (s,X (s))

ds

+1

E t0+

t0

δ (s)

1 + |X (s) |21

m+mTr(Q1)

1 + |X (s) |21

m−1

−2mδ(s) f (s, a (s)) , AX (s)1 + |X (s) |21

m−1

+2m(m− 1)|Q 12AX (s) |2

1 + |X (s) |21

m−2ds

. (3.493)

We now have|l (X (s) , a (s))| ≤ C

1 + |X (s) |k1

,

|ϕt (s,X (s))| ≤ C(1 + |X (s) |),

|AX (s) , Dϕ (s,X (s))| ≤λ

2|X (s) |22 + C

1 + |X (s) |2

,

|B (X (s) , X (s)) , Dϕ (s,X (s))| = |b (X (s) , X (s) , Dϕ (s,X (s)))|

≤ C|X (s) |1|X (s) |2 (1 + |X (s) |) ≤λ

2|X (s) |22 + C

1 + |X (s) |41

,


|TrQD2ϕ (s,X (s))

|, | f (s, a (s)) , Dϕ (s,X (s)) | ≤ C(1 + |X (s) |),

| f (s, a (s)) , AX (s) |1 + |X (s) |21

m−1 ≤ C1 + |X (s) |21

m,

and|Q 1

2AX (s) |21 + |X (s) |21

m−2 ≤ C1 + |X (s) |21

m−1.

Employing the above estimates in (3.493) and then using (3.454) yieldsλ

t0+

t0

E|X (s) |221 + |X (s) |21

m−1ds ≤ C (3.494)

for some constant C independent of . Therefore there exist sequences n → 0 andtn ∈ (t0, t0 + n) such that

E|Xn (tn) |22 ≤ C,

and thus there exist subsequences, still denoted by n, tn, such thatXn (tn) x weakly in L2(Ων ;V2)

for some x ∈ L2(Ων ;V2) (and thus also weakly in L2(Ων ;H)). However, by (3.460),Xn (tn) → x0 strongly in L2(Ων ;H). Therefore, by the uniqueness of the weaklimit in L2(Ων ;H), it follows that x0 = x ∈ V2.

Step 2. We now prove the supersolution inequality. We need to “pass to thelimit" as → 0 in (3.492), at least along a subsequence. This operation is ratherstandard for most of the terms, more precisely for those that only use convergencein the norms of H and V . To explain how we deal with the easy terms, let usconsider the cost term.

Let r ≥ |x0|1. Then, using (3.454), (3.461), (3.463) and (3.464), we have1

E t0+

t0

[l (X (s) , a (s)) ds− l (x0, a (s))] ds

≤ 1

E t0+

t0

σr (|X (s)− x0|1) ds

+1

E t0+

t0

C1 + |X (s) |k1 + |x0|k1

1|X(s)|1>rds

≤ σr

1

E t0+

t0

|X (s)− x0|1ds+ C

1 + |x0|k1

1 + |x0|1r

≤ σr

1

E t0+

t0

ωx0()ds

+ C

1 + |x0|k1

1 + |x0|1r

. (3.495)

The above implies that1

E t0+

t0

[l (X (s) , a (s)) ds− l (x0, a (s))] ds

≤ γ(), (3.496)

where lim→0 γ() = 0. Arguing like in (3.495), and using that (−A)12 f is bounded

in H, (−A)12 f(·, a) is uniformly continuous with values in H, uniformly for a ∈ Λ,

and Q12 (−A)

12 extends to a bounded operator in H, we can deal with all the terms

in (3.492), except the terms

−1

E t0+

t0

AX (s) , Dϕ (s,X (s)) , (3.497)

1

E t0+

t0

B (X (s) , X (s)) , Dϕ (s,X (s)) , (3.498)

1

E t0+

t0

2mδ (s) |AX(s)|21 + |X (s) |21

m−1 (3.499)

which require special consideration.


We first notice that

E1

t0+

t0

δ(s)AX (s)

1 + |X (s) |21

m−12 ds

2

≤ E1

t0+

t0

δ(s)|X (s) |221 + |X (s) |21

m−1ds ≤ C (3.500)

by (3.494). Therefore, there exists a sequence n → 0 and Y ∈ L2(Ων , H) such that

Yn :=1

n

t0+n

t0

δ(s)AXn (s)

1 + |Xn (s) |21

m−12 ds Y in L2(Ων , H)

as n → ∞. However, using arguments similar to these in (3.495), it is easy to seethat

A−1Yn =1

n

t0+n

t0

δ(s)Xn(s)

1 + |Xn(s)|21

m−12 ds →

δ(t0)x0

1 + |x0|21

m−12

strongly in L2(Ων , H). Therefore it follows that

Y =δ(t0)Ax0

1 + |x0|21

m−12 .

Then, using the first inequality of (3.500), we get

lim infn→∞

E 1

n

t0+n

t0

δ(s)|AXn (s) |21 + |Xn (s) |21

m−1ds

≥ δ(t0)|Ax0|21 + |x0|21

m−1.

(3.501)

This takes care of the term (3.499). The same argument also shows that we canassume that

1

n

t0+n

t0

AXn (s) ds Ax0 in L2(Ων , H) as n → ∞. (3.502)

As regards (3.497), denoting by ωϕ a modulus of continuity of Dϕ, we have by(3.494), (3.460), and (3.502)

1

nEt0

t0+n

t0

AXn (s) , Dϕ (s,Xn (s)) ds− Ax0, Dϕ (t0, x0)

≤ 1

n

t0+n

t0

E|AXn (s) |2

12

E (ωϕ (n + |Xn (s)− x0|))2

12ds (3.503)

+

E

1

n

t0+n

t0

AXn (s) ds−Ax0, Dϕ (t0, x0)

→ 0 as n → ∞.


Finally for (3.498), using (3.449), (3.494), (3.454), (3.460), (3.461), and (3.502),1

nE t0+n

t0

b (Xn (s) , Xn (s) , Dϕ (s,Xn (s))) ds− b (x0, x0, Dϕ (t0, x0))

≤ 1

nE t0+n

t0

|Xn (s) |1|Xn (s) |2ωϕ (n + |Xn (s)− x0|) ds

+1

nE t0+n

t0

|Xn (s)− x0|1|Xn (s) |2|Dϕ (t0, x0) |ds

+

1

nE t0+n

t0

b (x0, Xn (s)− x0, Dϕ (t0, x0)) ds

≤ 1

n

t0+n

t0

E|Xn (s) |22

12E|Xn (s) |41

14

E (ωϕ (n + |Xn (s)− x0|))4

14ds

+C1

n

t0+n

t0

E|Xn (s) |22

12E|Xn (s)− x0|21

12 ds (3.504)

+

Ebx0,

1

n

t0+n

t0

Xn (s) ds− x0, Dϕ (t0, Xn (s))

→ 0 as n → ∞.

In particular the last term goes to zero since, by (3.502),

1

n

t0+n

t0

Xn (s) ds x0 in L2(Ων , V2) as n → ∞

andZ → b (x0, Z,Dϕ (t0, x0))

is a bounded linear functional on L2(Ων , V2).Therefore, using (3.496) (and similar estimates for other standard terms),

(3.501), (3.503), and (3.504) in (3.492), we obtain for small n that

−ψt(t0, x0)−1

2Tr

QD2ψ(t0, x0)

− Ax0 −B (x0, x0) , Dψ (t, x0)

+1

nE t0+n

t0

[f (t0, a (s)) ,−Dψ (t0, x0)+ l (x0, a (s))] ds ≤ ω1 (n)

for some modulus ω1. It now remains to take the infimum over a ∈ Λ inside theintegral and then send n → ∞.

Example 3.137 The following example satisfies the assumptions of Theorem3.136. Let

l(x, a) = |curl x|2 + 1

2|a|2,

g(x) = |x|2,f(t, a) = Ka,

where K ∈ L(H;V ) and Λ = BH(0, R) ⊂ H. Such a control and the singularkernel of K can be approximately realized by a suitable Lorentz force distribution inelectrically conducting fluids such as liquid metals and salt water. The Hamiltonianfunction is then

F (x, p) = |curl x|2 + h(K∗p),

where h(·) : H → R is given by

h(z) := infa∈Λ

a, zH +

1

2|a|2


and K∗ is the adjoint of K considered as an operator from H to H. We can in factexplicitly obtain h as

h(z) =

−1

2|z|2 for |z| ≤ R

−R|z|+ 12R

2 for |z| > R.

We also remark that the optimal feedback control here is given formally as

a(t) = Υ(K∗Du(t, x(t))),

where

Υ(z) := Dh(z) =

−z for |z| ≤ R,

−z R|z| for |z| > R.

Under additional conditions on Q, optimal feedback control for this example arediscussed in Section 4.9.1.2 using mild solutions.


The material of Section 3.1.1 about B-continuity is based on [103, 104] and[396]. The formulation and the proof of the exponential moment estimates ofProposition 3.18 is taken from [425] while the rest of Section 3.1.2 mostly follows[104, 286]. More general formulations of Theorem 3.25 are in [420, 312]. Corollary3.26 was first introduced in [104]. Other smooth or partially smooth perturbedoptimization principles can be found in [48, 133, 293].

A first version of a maximum principle for semicontinuous functions in Hilbertspaces appeared in [321]. It was an infinite dimensional version of a maximumprinciple in domains of Rn (see e.g. [101]) and was applicable to a class of boundedsecond order equations (3.68). By a reduction to a finite dimensional case andthe use of the finite dimensional maximum principle it provided test functionswhose second order derivatives satisfied proper inequalities on finite dimensionalsubspaces, and with the remaining parts of second order derivatives becoming neg-ligible for the class of equations considered as the dimension of the finite dimen-sional subspaces increased to +∞. A corrected and simplified proof of this result,which also included its time dependent version, based on the use of so called partialsup-convolutions appeared in [102]. These maximum principles have been adaptedto unbounded equations first in [272, 421, 422] and later in various settings in[242, 244, 245, 286, 288]. The version stated in Theorem 3.27 is general and new aswe tried to formulate it in a way that would be more directly applicable to variousclasses of equations. Its proof draws on the collective body of work from the abovecited papers. A scaling reduction to obtain Corollary 3.29 was introduced in [272].A different type of time dependent maximum principle, similar in the spirit to itsfinite dimensional version in [101], is in [102], see Remark 3.30 for more on this.

The definition of viscosity solution similar to the one presented in Section 3.3was introduced in [421, 422]. It was based on the notion of B-continuous viscositysolution developed by Crandall and P. L. Lions in [103, 104]. An earlier paper[320] dealing with a specific second order HJB equation for an optimal control ofa Zakai equation also used some ideas of the B-continuous viscosity solution. Thematerial of Section 3.3 is mostly based on [421, 422, 286] however the definitions ofviscosity solutions are more general. Lemma 3.37 is taken from [288]. A differentdefinition of viscosity solution for second order equations in Hilbert spaces wasproposed by Ishii in [272]. It was related to the definition for first order equationsin [271]. Ishii’s viscosity sub/super-solutions are allowed to be discontinuous andthe definition uses a special (convex) function to deal with the unboundedness


in the equation. The function is related to the equation and can be thought ofas some energy function for the controlled deterministic/stochastic PDE relatedto the HJB equation. The advantage of this definition is that viscosity solutionscan be relatively easily obtained by Perron’s method and the unbounded operatorA (together with other terms) can be non-linear. However the definition seemsto be difficult to apply to control problems and no attempts have been made inthis direction. The idea of using special functions as part of test functions inthe definition of viscosity solution to exploit the coercivity of the operator A alsoappeared in [71, 106] (see also [183, 243]) and later for second order equations in[242, 244, 245, 288], see Sections 3.9, 3.11, 3.12, 3.13.

We only briefly mentioned bounded equations in Section 3.3.1. The definitionof viscosity solution in Section 3.3.1 is taken from [321] where the theory of suchequations was developed for equations satisfying Hypothesis 3.47. In this paperalso an equivalent definition using second-order jets is discussed. For equationsthat do not satisfy Hypothesis 3.47 a stronger definition of viscosity solution wasintroduced in [319]. It allowed for more general test functions which are not neces-sarily twice Fréchet differentiable. Both papers contain comparison and existenceresults. Uniqueness of solutions is obtained in [319] by a combination of stochasticand analytic techniques. Perron’s method is discussed in [321] while connectionswith stochastic optimal control are discussed in [319]. In particular [319] containsproofs of sub- and super-optimality inequalities of dynamic programming. Regular-ity results for bounded equations and their obstacle problems have been obtainedin [319, 426]. Existence and uniqueness results for bounded equations can alsobe found in [290], in particular one can find there proofs of comparison principlesusing parabolic maximum principle from [102] which allows to relax the way viscos-ity subsolutions and supersolutions attain the initial/terminal values. A Dirichletboundary value problem for a linear equation was investigated in [286] and a risk-sensitive control problem in [424]. An obstacle problem related to optimal stoppingand pricing of American options was studied in [219]. Classical results for boundedlinear equations can be found in [129].

First comparison theorems for B-continuous viscosity sub/super-solutions ofequations discussed in Section 3.5 were proved in [421, 422]. These works dealt withthe case of compact operator B and the proofs of comparison in most part relied onthe combination of techniques developed for first order equations in [103, 104] andthe maximum principle arguments of [102, 321]. Slightly different techniques butalso based on the maximum principle of [321] were used to prove comparison prin-ciple with Ishii’s definition of solution in [272]. Some of Ishii’s methods were laterused in other comparison proofs. Comparison results for equations of Section 3.5without the compactness assumption on B are in [286] and the proofs of comparisonin various special cases are contained in [242, 244, 245, 288, 292]. Comparison the-orem for equations with quadratic gradient terms is in [425]. The papers [272, 288]show comparison for discontinuous viscosity sub- and super-solutions. The materialof Section 3.5 is to some extent new and incorporates formulations and techniques of[272, 286, 421, 422]. The statements of Theorems 3.50, 3.54, 3.56, 3.58 are new andinclude general growth conditions for viscosity sub- and supersolutions. The proofsof the above comparison theorems are also to some extent new. Comparison theo-rems with general growth conditions for B-continuous viscosity sub/super-solutionsof first order equations can be found in [312].

Direct proofs that value functions of stochastic optimal control problems inHilbert spaces are viscosity solutions of their HJB equations were done in variouscases in [245, 319, 320]. An early attempt in this direction was also made in[358]. The general finite time horizon optimal control problem (3.125)-(3.126) and


its connection to B-continuous viscosity solutions of equation (3.128)-(3.129) wasstudied in [286]. Our presentation expands and generalizes [286]. Some resultswhich were part of the folklore of the theory are stated in Section 3.6 for the firsttime. We presented continuity properties of the value functions in both finite andinfinite horizon cases and under both weak and strong B-conditions. Only the weakB-condition case was discussed in [286]. The use of dynamic programming principleis also fully explained and the proofs that value functions are viscosity solutionsof the associated HJB equations are done in all cases. We tried to include all thedetails. The proof of a stronger version of the dynamic programming principle inthe stopping time formulation in Section 3.6.2 uses some arguments from [349].

The material of Section 3.7 about finite dimensional approximations is basedon the results of [421, 422], however it contains some improvements of the resultsand their proofs. The method of finite dimensional approximations provides a wayto construct B-continuous viscosity solutions for equations which may not be HJBequations related to optimal control problems, for instance for Isaacs equations.It requires however that the operator B be compact. The method, together withits basic techniques, was introduced in [103] for first order equations, and waslater generalized to second order equations in [421, 422]. A version of this methodwas also used in [242]. Lemma 3.83-(ii) was proved in [103] by viscosity solutionarguments. Our proof uses direct functional analytic arguments. Other proofs ofexistence employ Perron’s method (see comments in this section in the paragraphabout Perron’s method). For Isaacs equations, probabilistic representation formulascan be obtained [192, 361, 363, 423].

Section 3.8 on singular perturbations is based on [425]. Singular perturbationproblems in finite dimensional spaces have been studied extensively by viscositysolution methods and the reader can consult [30, 195] for results and references.The problems have not been widely investigated yet in Hilbert or other infinitedimensional spaces. Nisio studied such problem in [363] in connection with a risksensitive control problem. Also a singular limit problem related to a risk-sensitivecontrol problem with bounded evolution in a Hilbert space was studied in [424]. In[425] convergence of viscosity solutions of singularly perturbed HJB equations wasused to investigate large deviation problems for stochastic PDE perturbed by smallnoise. The case of integro-PDE was studied in [428]. Both papers [425, 428] usea general PDE approach to large deviations developed in [184] (see also [180, 181,182]).

Perron’s method for viscosity solutions of PDE in finite dimensional spaces wasintroduced by Ishii in [269] (see also [101]). It was extended to bounded equations inHilbert spaces in [321]. Perron’s method provides another way to obtain existenceof viscosity solutions for equations that may not necessarily be of the HJB type, forinstance for Isaacs equations. For unbounded first and second order equations itwas shown to work with Ishii’s definitions of viscosity solution [271, 272] and withthe Tataru-Crandall-Lions definition of viscosity solution [105]. For B-continuousviscosity solutions Perron’s method was introduced in [288] under an assumptionthat the unbounded operator A has some coercivity properties. Perron’s methodrequires the notion of a discontinuous viscosity solution so in [288] a more generaldefinition of a discontinuous viscosity solution using B-semicontinuous envelopeswas introduced. This definition borrowed an idea from the definitions in [271, 272]of combining the upper and lower-semicontinuous envelopes with the radial testfunctions. The method of half-relaxed limits of Barles-Perthame requires compact-ness. The fact that it may not work in infinite dimensional spaces was noticed in[9, 424] (see Example 3.43 in this book). A version of half-relaxed limits presented


in Section 3.9 was developed in [288] where more results on Perron’s method andhalf-relaxed limits can be found.

The material of Section 3.10 is based on [287]. Infinite dimensional Black-Scholes equation was analyzed in [227], where existence of smooth solutions wasproved for smooth data, and an obstacle problem for the Black-Scholes equationwas studied in [454] from the point of view of Bellman’s inclusions. A similarobstacle problem for a related model was studied in [219]. A non-local Black-Scholes-Barenblatt equation associated with the HJMM model driven by a Lévytype noise was investigated in [427].

Section 3.11 follows [245]. First result about viscosity solutions of the HJBequation for control of the DMZ equation appeared in [320], where the equationwas studied in a standard L2 space and the operators Sk

a were bounded multipli-cation operators. The paper [320] used a combination of probabilistic and analytictechniques to deal with the uniqueness of viscosity solution of the HJB equation.In [259] it was shown that the value function is a viscosity solution in a very weaksense when the HJB equation was considered in the space of measures. (A regularityresult for a related equation in the space of measures was obtained in [261].) Otherpaper on the subject is [15]. The approach of [245] used the theory of B-continuousviscosity solutions of [421, 422] together with an idea that originated in [71, 106](also [271, 272] had related ideas) to use a special radial function and the coercivityof operators in the equation to “improve" the points where the maxima/minima oc-cur in the definition of viscosity solution. This idea of using special energy functionrelated to the underlying controlled state equation as a part of test functions wasalso used in may cases for first and second order equations [242, 243, 244, 288, 183].The viscosity solution approach of [245] is also presented in [364], where a differentproof of the dynamic programming principle is given. The book [364] discussesin details, in Chapters 5 and 6, a partially observed optimal control problem, theseparated problem, the optimal control of the DMZ equation, and other related ma-terial. It complements the material in Sections 2.6.6 and 3.11 and gives a slightlydifferent perspective. Our short introduction to variational solutions in Section3.11.1 is based on [93, 220, 296, 298, 385]. Semi-linear stochastic parabolic equa-tions and the DMZ equation are also discussed in [364], where Itô’s formulas areproved for the original test functions used in [245], which are similar but differentfrom the test functions in Section 3.11.5. Our presentation of the various energyand continuous dependence estimates for the DMZ equation follows, with smallchanges, [245]. The reader can also find similar results in [364]. The material onviscosity solutions and the value function of Sections 3.11.5 and 3.11.6 has somedifferences from [245] as we merged it into the presentation of the book and madesome improvements and corrections.

Section 3.12 is based on [242] and fills some missing details there. The HJBequation in this section has second-order coefficients which are not trace class sothe equation is also unbounded in the second-order terms. Since the equation isfully nonlinear it cannot be dealt with by the techniques of mild solutions. Achange of variables is done to convert the equation to a one with bounded second-order terms. This is a rather ad hoc technique. Viscosity solutions of HJB equa-tions with unbounded second-order terms coming from control problems with stateequations driven by cylindrical Wiener processes, have not yet been studied system-atically. The definition of viscosity solution for the converted equation is similar tothis in [71] and uses a special radial function to guarantee that the points wherethe maxima/minima occur belong to a better space (see the previous paragraphfor the discussion about the origins of such definition). A Cauchy problem forequations similar to the “converted" equation was also studied in [439] using the


techniques of [242]. Apart from [242], boundary control problems and their associ-ated HJB equations have been studied by viscosity solution techniques in [451] forthe stochastic case (with noise at the boundary) and in [68, 70, 71, 163, 164] forthe deterministic case. Second-order HJB equations and stochastic boundary con-trol/noise problems have been investigated by mild solutions and Backward SDEin [131, 136, 165, 235, 338, 448, 464, 465], some of them also in connection withstochastic delay equations.

The material of Section 3.13 follows [244]. The definition of viscosity solutionis similar to these of [242, 245] (see previous comments about origins of these defini-tions) however it uses different energy function (a radial function of the | · |1 norm)which reduces the equation to a subspace of the Hilbert space H. In this respect thedefinition is similar to the definitions in [271, 272]. A stationary equation similar to(3.443) was also investigated in [438]. Viscosity solution approaches to first-orderHJB equations associated to optimal control of deterministic Navier-Stokes equa-tions are in [243, 412] (see also [419] for earlier attempts). A PDE-viscosity solutionapproach to large deviations of stochastic two-dimensional Navier-Stokes equationswith small noise intensities is considered in [425] where convergence of viscosity so-lutions of singularly perturbed HJB equations is studied. For results on Kolmogorovand HJB equations by other approaches [26, 25, 115, 117, 188, 328, 401, 443] werefer to Section 4.9.1, and the short discussion at the beginning of Section 3.13.

We did not discuss explicitly in the book Isaacs equations in Hilbert spaceswhich are associated to zero-sum two-player stochastic differential games. For Isaacsequations, one can prove existence of viscosity solutions by showing directly thatthe associated upper/lower value function of the game is a viscosity solution of theupper/lower Isaacs equation. Such results can be found in [192, 361, 363, 423].This however is not easy since the proof of the dynamic programming principle isvery complicated and thus only limited results are available. Related results onrisk-sensitive stochastic control and differential games can be found in [359, 360,362, 363, 424].

Other types of equations can be studied using the theory of viscosity solutionspresented in this chapter, for instance obstacle problems for HJB equations relatedto optimal stopping problems, HJB equations for ergodic control problems. Com-parison proofs easily extend to the case of obstacle problems. Explicit literaturehowever is limited. Obstacle problems for bounded equations have been studied in[219, 319, 426]. HJB equations for ergodic control have not been investigated byviscosity solutions. In [226] they have been studied by the perturbation approach tomild solutions and in [206] by BSDE. Likewise HJB equations for singular controlproblems and for state constraints problems have not been studied in the infinitedimensional stochastic case in the viscosity solution framework. A singular sto-chastic control problem with delay is studied by other methods in [5] and [177].Concerning infinite dimensional state constraints control problems and viscositysolutions of the associated HJB equations, the readers may check [172, 173] for thestochastic case and [67, 291, 175, 176, 178] for the deterministic case. A stochasticviability problem for a subset of a Hilbert space was studied by viscosity solutionsin [60].

Also viscosity solution approach to HJB equations for optimal control of sto-chastic delay equations has not been fully explored. Some results on the subject arein [172, 173, 462, 463], see also [460, 461, 175, 176, 178] for the deterministic caseand first order HJB equations. For other methods applicable to HJB equations forcontrol of stochastic delay equations we refer the reader to Chapter 5 (in particularSections 5.5 and 5.6) and Chapter 6 (in particular Section 6.5).


Another unexplored area is viscosity solutions of HJB equations with un-bounded second-order terms which come from control problems with state equa-tions driven by Q-Wiener processes with Tr(Q) = +∞, i.e. such that we may haveTr[(σ(t, x, a)Q

12 )(σ(t, x, a)Q

12 )∗] = +∞. So far [242] has been the only paper on

the subject in a specific case. Up to now viscosity solution theory handles well fullynonlinear but “degenerate" equations while the theory of mild solutions handleswell semilinear but “nondegenerate" equations. One would expect that the theoryof viscosity solutions can be extended to the fully nonlinear “nondegenerate" HJBequations (see also the comments about [98] in the paragraph below discussing pathdependent PDE).

There are also very few explicit results on viscosity solutions of boundary valueproblems in Hilbert spaces. Only some Dirichlet boundary value problems werestudied. There exist comparison theorems (see Section 3.5), however quations stud-ied by viscosity solutions are “degenerate" and hence construction of barriers at theboundary is not easy. Thus value functions may not be continuous up to the bound-ary unless some conditions are imposed on the drift. A Dirichlet boundary valueproblem for a bounded linear equation was investigated in [286] and a boundaryvalue problem for a bounded HJB equation related to a risk-sensitive control prob-lem was studied in [424]. Some results about value functions in bounded sets aresketched in [319]. A related paper for first order HJB equations is [67]. Resultsusing approaches of mild and L2 solutions are limited to linear equations. We referthe reader to [28, 29, 120, 121, 122, 123, 129, 390, 391, 430] and the references therefor more. In [28, 29, 123] Neumann boundary value problems are considered.

An interesting direction in the evolution of the notion of viscosity solution ininfinite dimensional spaces may come from the concept of path dependent PDE.Path dependent PDE come from the study of problems driven by path dependentSDE. In the finite dimensional spaces the notion of path dependent viscosity solutionwas introduced in [148] and this notion was extended to infinite dimensional spacesin [98]. In the Markovian case this approach gives an alternative way to treatHJB equations studied in this book. Its advantage is that it avoids the use of themaximum principle and thus it can be applied to “nondegenerate" equations, thecontinuity assumptions in the | · |−1 norm can be dropped, and the operator A doesnot need to be maximal dissipative. It is not clear if this method can be applied tofully nonlinear equations.

An emerging area of development for second-order equations in infinite dimen-sional spaces seems to be related to PDE in spaces of probability measures, inparticular in the Wasserstein space. A second-order HJB equation in the spaceof probability measures was studied in [261] in connection with partially observedcontrol and regularity of solutions was proved. Similar results were obtained forfirst order equations in [260]. Following the program described in [184], first-orderHJB equations can be used to study large deviations for empirical measures of sto-chastic particle systems. Results in this direction are in [183, 184, 185, 186] (see alsothe references therein). It would be natural to investigate second-order equationsrelated to the Laplace limit for such problems. Equations in the space probabilitymeasures also appear in the form of the so-called Master Equations of Mean FieldGames [37, 36, 59, 72, 73, 74, 86, 314, 216]. These are non-local equations, whichin the case of second-order Mean Field Games are of second-order. So far onlylimited results about existence and in some cases uniqueness of classical and strongsolutions of first and second-order Master Equations was shown in [59, 74, 86, 216].Also an approach proposed by P. L. Lions [72, 314] allows to convert an equationin the Wasserstein space to an equation in the Hilbert space L2, where measureswith finite second moments become random variables in L2 with given laws.


Another emerging direction is HJB integro-PDE in Hilbert spaces which are as-sociated to optimal control problems with state equations driven by Lévy processesor random measures. Viscosity solutions have been introduced for such non-localequations in [428, 429, 427]. Comparison theorems have been proved in [429] andexistence of viscosity solutions and optimal control problems have been studied in[427]. Some linear non-local PDE and properties of transition semigroups for pro-cesses with jumps have been studied by other methods in [8, 311, 378, 392, 393, 394].

APPENDIX A

Notations and Function Spaces

In this appendix, we list the main notation and the definitions of the basicfunctions spaces used throughout the book. Other notation and definitions of func-tions spaces that are used only in specific chapters are not recalled here and areintroduced directly in their chapters.

A.1. Basic Notation

If X is a Banach space we denote its norm by | · |X . If this space is also Hilbert,we denote its inner product by ·, ·X . Given R > 0, BX(x, R) denotes the closedball in X centered at x of radius R. We will omit the subscript X if the context isclear. The dual space of X, i.e. the space of all continuous linear functionals, willbe denoted by X∗. The (operator) norm in X∗ will be denoted by | · |X∗ , and theduality will be denoted by ·, ·X∗,X.

If a sequence (xn)n∈N ⊂ X converges to x ∈ X in the norm topology we writexn → x. If it converges weakly we write xn x.

If X is a Hilbert space and ekk∈N is an orthonormal basis of X we use, forx ∈ X, the notation xk := x, ek. Unless stated explicitly, we will always identifyits dual X∗ with X through the standard Riesz identification.

Given a second Banach space Y with norm | · |Y (and inner product ·, ·Y ifit is also Hilbert) we denote by L(X;Y ) the set of all bounded (continuous) linearoperators T : X → Y with norm TL(X;Y ) := supx∈X,x =0

|Tx|Y|x|X (or simply T),

using for simplicity the notation L(X) when X = Y . L(X) is a Banach algebrawith identity element IX (simply I if unambiguous).

Given a linear (possibly unbounded) operator T : D(T ) ⊂ X → Y such thatD(T ) is dense in X we will denote its adjoint operator by T ∗ : D(T ∗) ⊂ Y ∗ → X∗.If X is a Hilbert space we will denote by S(X) ⊂ L(X) the space of all boundedself-adjoint operators on X.

For k = 1, 2, ... we denote by Xk the product space X × X × · · · × X (ktimes) endowed with the norm |(x1, ..., xk)|Xk :=

|x1|2 + ...+ |xk|2

1/2 and byLk(X;Y ) the set of all bounded multilinear operators T : Xk → Y with normTLk(X;Y ) := supx∈Xk,x =0

|T (x1,...,xn)|Y|(x1,...,xn)|Xk

using for simplicity the notation Lk(X)

when X = Y . It is known (see e.g. [198] p. 318, Theorem A.2.6) that Lk(X;Y ) isisometrically isomorphic to the space L(X,L(X, . . . ,L(X;Y ))).

Given a complex number λ ∈ C we denote by Reλ and Imλ respectively its realand imaginary parts.

A.2. Function spaces

Let Y and Z be two Banach spaces and let X ⊂ Z be endowed with the inducedtopology. We denote by B(X;Y ), Bb(X;Y ), C(X;Y ), UC(X;Y ), Cb(X;Y ) andUCb(X;Y ) the sets of all functions ϕ : X → Y which are, respectively, Borel mea-surable, Borel measurable and bounded, continuous, uniformly continuous, contin-uous and bounded, uniformly continuous and bounded on X. The spaces Bb(X;Y ),

531

532 A. NOTATIONS AND FUNCTION SPACES

Cb(X;Y ) and UCb(X;Y ) are Banach spaces with the usual norm

ϕ0 = supx∈X

|ϕ(x)|Y .

We denote by USC(X;Y ) (respectively, LSC(X;Y )) the space of all upper semi-continuous (respectively, lower semicontinuous) functions f : X → Y .

If Y = R, we will simply write Bb(X), C(X), UC(X), Cb(X), UCb(X),USC(X) and LSC(X) for Bb(X;R), C(X;R), UC(X;R), Cb(X;R), UCb(X;R),USC(X;R) and LSC(X;R).

For a given m > 0 we define Bm(X,Y ) (respectively, Cm(X,Y ) andUCm(X,Y )) to be the set of all functions φ ∈ B(X,Y ) such that the function

ψ(x) :=φ(x)

(1 + |x|2)m/2(A.1)

belongs to Bb(X,Y ) (respectively Cb(X,Y ) and UCb(X,Y )). These spaces of func-tions that have at most polynomial growth of order m are Banach spaces when theyare endowed with the norm

N(φ) := supx∈X

|φ(x)|(1 + |x|2)m/2

.

We will write φBm(X,Y ), φCm(X,Y ), φUCm(X,Y ) to denote these norms, or sim-ply φBm

, φCm, φUCm

when the spaces are clear from the context. The abovedefinition is meaningful also when m = 0 and in such case the spaces Bm(X,Y ),Cm(X,Y ) and UCm(X,Y ) reduce to Bb(X,Y ), Cb(X,Y ) and UCb(X,Y ). We donot use the notation B0(X,Y ), C0(X,Y ) and UC0(X,Y ) to avoid confusion withrespect to the standard notation in the literature. However we often consider thespaces Bm(X,Y ), Cm(X,Y ) and UCm(X,Y ) for m ≥ 0 meaning that also the caseof the spaces Bb(X,Y ), Cb(X,Y ) and UCb(X,Y ) is included.

Let Y be a Banach space and X be an open subset of a Banach space Z. Fork ∈ N, we denote by Bk(X;Y ) (respectively Bk

b (X;Y )) the set of all functions ϕ :X → Y which are Borel measurable (respectively Borel measurable and bounded)on X, together with all their Fréchet derivatives (see Section D.2) up to the orderk. If ϕ ∈ Bk

b (X;Y ) the l-th Fréchet derivative of ϕ is denoted by Dlϕ (or simplyDϕ when l = 1). We set

ϕk = ϕ0 +k

l=1

supx∈X

Dlϕ(x)Ll(Z;Y )

.

Similarly we define the space Ck(X;Y ) (respectively, Ckb (X;Y ), UCk(X;Y ),

UCkb (X;Y )) to be the set of all functions ϕ : X → Y which are continuous (respec-

tively continuous and bounded, uniformly continuous, uniformly continuous andbounded) on X together with all their Fréchet derivatives up to the order k. z

For k ∈ N and m > 0 we denote by Bkm(X;Y ) (respectively, Ck

m(X,Y )and UCk

m(X,Y )) the set of all functions φ ∈ B(X,Y ) such that the correspond-ing function ψ defined in (A.1) belongs to Bk

b (X,Y ) (respectively Ckb (X,Y ) and

UCkb (X,Y )). We set

φBkm(X;Y ) = ψBk

b(X;Y )

where ψ is the function defined in (A.1) corresponding to φ. With this normBk

b (X,Y ), Ckb (X,Y ), UCk

b (X,Y ) are all Banach spaces. For the last two we willdenote the norms by φCk

m(X,Y ), φUCk

m(X,Y ).

If Y = R, we write Bkb (X), Bk

m(X), Ck(X), Ckb (X), Ck

m(X), UCk(X),UCk

b (X), UCkm(X) instead of Bk

b (X;R), Bkm(X;R), Ck(X;R), Ck

b (X;R),Ck

m(X;R), UCk(X;R), UCkb (X;R), UCk

m(X;R), respectively.

A.2. FUNCTION SPACES 533

For α ∈ (0, 1] we denote by C0,α(X;Y ) the space of all Hölder continuousfunctions from X to Y endowed with the semi-norm

[ϕ]0,α = sup

|ϕ(x)− ϕ(y)|Y

|x− y|αZ; x, y ∈ X; x = y

.

The space C0,αb (X;Y ) := Cb(X;Y ) ∩ C0,α(X;Y ) is a Banach space with the norm

ϕ0,α = ϕ0 + [ϕ]0,α

and is contained in UCb(X;Y ). If α = 1 the space C0,1(X;Y ) is the space ofall Lipschitz continuous functions from X to Y . If X is open, convex and ϕ ∈C0,1(X;Y ) is Fréchet differentiable in X then the derivative Dϕ is bounded and

[ϕ]0,1 = supx∈X

Dϕ(x)L(Z;Y ).

We set, for α ∈ (0, 1],

C1,α(X;Y ) :=ϕ ∈ C1(X;Y ) : [Dϕ]0,α < ∞

. (A.2)

The space C1,αb (X;Y ) := C1

b (X;Y )∩C1,α(X;Y ) is a Banach space with the norm

ϕ1,α = ϕ1 + [Dϕ]0,α.

Similarly,C2,α(X;Y ) :=

ϕ ∈ C2(X;Y ) : [D2ϕ]0,α < ∞

. (A.3)

The space C2,αb (X;Y ) := C2

b (X;Y ) ∩ C2,α(X;Y ) is also a Banach space with thenorm

ϕ2,α = ϕ2 + [D2ϕ]0,α.

If Y = R the above spaces are denoted by C0,α(X), C0,αb (X), C1,α(X),

C1,αb (X), C2,α(X), C2,α

b (X).If α ∈ (0, 1], we say that ϕ : X → Y is Hölder (Lipschitz when α = 1)

continuous on bounded subsets of X, if ϕ ∈ C0,α(B(0, R) ∩X;Y ) for every R > 0i.e. if the semi-norm

[ϕ]0,α,R := sup

|ϕ(x)− ϕ(y)|Y

|x− y|αZ; x, y ∈ B(0, R) ∩X; x = y

is finite for every R > 0. The space of such functions is denoted by C0,αloc (X;Y ).

Similarly we define the spaces C1,αloc (X;Y ) and C2,α

loc (X;Y ).If Y = R we write Ck,α(X) instead of Ck,α(X;R) and Ck,α

loc (X) instead ofCk,α

loc (X;R), for k = 0, 1, 2, α ∈ (0, 1].

Let now X be a real separable Hilbert space and Y be a real Banach space. Asusual we set, for an open subset O of Rn, and k ∈ N ∪ ∞,

Ck0 (O;Y ) :=

f ∈ Ck(O;Y ) : f has compact support in O

.

Following [91] Section 0 and [225] Section 2 we define various spaces of cylindricalfunctions. We set, for k ∈ N ∪ ∞,

FCk0 (X;Y ) :=

ϕ : X → Y : ∃n ∈ N, x1, ..., xn ∈ X, f ∈ Ck

0 (Rn;Y )

such that ϕ(x) = f(x, x1, ..., x, xn), ∀x ∈ X . (A.4)

We denote FCk0 (X;R) simply by FCk

0 (X).Given k ∈ N∪∞ and a linear closed operator with dense domain B : D(B) ⊆

X → X, we define

FCk,B0 (X) =

ϕ : X → R : ∃n ∈ N, x1, ..., xn ∈ D(B), f ∈ Ck

0 (Rn;R)such that ϕ(x) = f(x, x1, ..., x, xn), ∀x ∈ X (A.5)


and

FCk,Bb (X) :=

ϕ : X → R : ∃n ∈ N, x1, ..., xn ∈ D(B), f ∈ Ck

b (Rn;R)such that ϕ(x) = f(x, x1, ..., x, xn), ∀x ∈ H . (A.6)

Moreover if E = ekk∈N is an orthonormal basis of X, we introduce, for k ∈N ∪ ∞,

ECk0 (X;Y ) :=

ϕ : X → Y : ∃n ∈ N, f ∈ Ck

0 (Rn;Y ) such thatϕ(x) = f(x, e1, ..., x, en), ∀x ∈ X (A.7)

and, if B : D(B) ⊆ X → X is a linear closed operator with dense domain andE ⊆ D(B),

ECk,B0 (X) :=

ϕ : X → R : ∃n ∈ N, f ∈ Ck

0 (Rn) such thatϕ(x) = f(x, e1, ..., x, en), ∀x ∈ X . (A.8)

Let O be an open subset of Rn and p ∈ [1,+∞). We denote by Lp(O), the set ofall real valued measurable functions1 f : O → R with

O |f(ξ)|pdξ < +∞ (classical

Lebesgue integral); Lp(O) is a Banach space with the usual norm |f |Lp(O) :=

O |f(ξ)|pdξ1/p. We denote by Lp

loc(O) the set of all measurable functions f :O → R such that

K |f(ξ)|pdξ < +∞ for every compact subset K of O. The space

L∞(O) is the quotient space of Bb(O) with respect to the relation of being equala.e. and is a Banach space with the usual ess sup norm. (Obviously the abovespaces can be defined for more general sets O.)

We denote by W k,p(O) (k ∈ N, p ∈ [1,+∞]) the usual Sobolev space of realvalued functions whose distributional derivatives, up to the order k, are p-th powerintegrable (or essentially bounded if p = +∞). Moreover W k,p

0 (O) is the closure ofC∞

0 (O) in W k,p(O). Following the standard convention, sometimes we will writeHk(O) for W k,2(O) and Hk

0 (O) for W k,20 (O). Similarly, for α ≥ 0 and p ∈ [1,+∞],

we denote the fractional Sobolev spaces by Wα,p(O), Wα,p0 (O) (or Hα(O), Hα

0 (O)when p = 2) defined in the usual way (see e.g. [1] Chapter VII). By duality thenone defines, for every α > 0, the negative order spaces H−α(O) setting (Hα

0 (O))∗ =H−α(O) (see e.g. [313] Section 1.12).

If the boundary ∂O is a C∞ manifold of dimension n − 1 in Rn then also thespaces Lp(∂O), Hα(∂O), Wα,p(∂O) can be defined. We refer to [313], Section 1.7or [1], Chapter VII, p. 215.

If Y is a real, separable Banach space and a < b, we define the spaceW 1,p(a, b;Y ) (p ∈ [1,∞]) to be the set of all functions f ∈ Lp(a, b;Y ) whoseweak derivative f (see [435] Chapter III, §1) exists and belongs to Lp(a, b;Y ). It isa Banach space equipped with the norm |f |W 1,p(a,b;Y ) := |f |Lp(a,b;Y ) + |f |Lp(a,b;Y ).

Let X be a subset of a Banach space Z, Y be a Banach space and I be aninterval in R. We define the space

UCxb (I ×X;Y ) :=

ϕ ∈ Cb(I ×X;Y ) :

ϕ(t, x) is uniformly continuous in x, uniformly with respect to t ∈ I. (A.9)

If Y = R, we write UCxb (I ×X) instead of UCx

b (I ×X;R).

1That is, the set of all equivalence classes of such functions with respect to the relation ofa.e. equality.

A.2. FUNCTION SPACES 535

Remark A.1 We recall some useful properties of UCxb (I × X;Y ) (see [82] for

more).(1) If I ⊆ R is compact and u ∈ UCx

b (I×X;Y ), then for every compact setK ⊆ X, the restriction u|I×K of u to I ×K belongs to UCb(I ×K;Y ).Thus, for every compact set K ⊆ X,

u|I×K ∈ Cb(I;Cb(K;Y )). (A.10)In particular u(·, x) is uniformly continuous on I, uniformly with respectto x ∈ K, namely, for every t, s ∈ I, we have

supx∈K

|u(t, x)− u(s, x)|Y ≤ ρK(|t− s|), (A.11)

where ρK is a modulus of continuity (see Appendix D.1) depending onthe compact set K.

(2) UCb(I ×X;Y ) ⊆ UCxb (I ×X;Y ) ⊆ Cb(I ×X;Y ). In particular, as the

uniform continuity is stable with respect to the convergence in the norm ·0, the space UCx

b (I×X;Y ) is a closed subspace of Cb(I×X;Y ). Onthe other hand, if X is the whole space, ϕ ∈ UCb(X) and etA, t ≥ 0is a strongly continuous (not uniformly continuous) semigroup on X,then

w(t, x) = ϕ(etAx)

is a natural example of a function belonging to UCxb (I ×X;Y ) but not

to UCb(I ×X;Y ).

For a given m > 0 we define Bm(I × X,Y ) (respectively, Cm(I × X,Y ) andUCm(I×X,Y )) to be the set of all functions φ ∈ B(I×X,Y ) such that the function

ψ(t, x) :=φ(t, x)

(1 + |x|2)m/2(A.12)

belongs to Bb(I × X,Y ) (respectively Cb(I × X,Y ) and UCb(I × X,Y )). Thesespaces of functions that have at most polynomial growth of order m in the variablex are Banach spaces when they are endowed with the norm

N(φ) := sup(t,x)∈I×X

|φ(t, x)|(1 + |x|2)m/2

.

We will write φBm(I×X,Y ), φCm(I×X,Y ), φUCm(I×X,Y ) to denote these norms.Following [76], we introduce the space

UCxm(I ×X;Y ) :=

ϕ ∈ C(I ×X;Y ) :

ψ(t, x) :=φ(t, x)

(1 + |x|2)m/2∈ UCx

b (I ×X;Y ). (A.13)

It has properties similar to these described in Remark A.1 for u ∈ UCxb (I ×X;Y ).

Let X be an open subset of a Banach space Z, and Y be a Banach space.Let I ⊂ R be open. Given a function u ∈ C(I × X;Y ) which is l times Fréchetdifferentiable in t and k times Fréchet differentiable in x, we denote its l-th partialderivative in t by Dl

tu, and its k-th partial derivative in x by Dkxu. For the low

order derivatives we use the symbols ut, Du, D2u, and so on.We denote by Cl,k(I×X;Y ) (respectively, by Cl,k

b (I×X;Y ), UCl,k(I×X;Y ),UCl,k

b (I×X;Y )) the space of all functions ϕ : I×X → Y that are Fréchet differen-tiable l times in t and k times in x and which are continuous (respectively continuousand bounded, uniformly continuous, uniformly continuous and bounded), together


with their Fréchet derivatives up to these orders. If I is an interval which is notopen then Cl,k(I × X;Y ) is the space of functions from I × X → Y whose re-strictions to Io ×X (where Io is the interior of I) belong to Cl,k(Io ×X;Y ) andsuch that the functions and all their derivatives extend continuously to I×X. Thespaces Cl,k

b (I ×X;Y ), UCl,k(I ×X;Y ), UCl,k(I ×X;Y ) and UCl,kb (I ×X;Y ) are

defined similarly.Cl,k

b (I×X;Y ) and UCl,kb (I×X;Y ) are Banach spaces endowed with the norm

|ϕ|l,k = supx∈X

ϕ(·, x)l + supt∈I

ϕ(t, ·)k

We define Cl,km (I × X;Y ) (respectively UCl,k

m (I × X;Y )) to be the set of allfunctions φ : I×X → Y such that the function ψ in (A.12) belongs to Cl,k

b (I×X;Y )

(respectively to UCl,kb (I × X;Y )). Such spaces are Banach spaces endowed with

the normφCl,k

m (I×X;Y ) := ψCl,k

b(I×X;Y )

where ψ is the function given in (A.12) corresponding to φ.If Y = R we will use the notation Cl,k(I ×X), Cl,k

b (I ×X), UCl,k(I ×X) andUCl,k

b (I ×X) for the above spaces.Let I be an interval in R, X be a real separable Hilbert space and Y be a real

Banach space. Similarly to the time independent case we set, for k ∈ N ∪ ∞,Ck

0 (I × Rn;Y ) :=f ∈ Ck(I × Rn;Y ) : f has compact support in I × Rn.

If k ∈ N ∪ ∞, we denote by FCk0 (I ×X;Y ) the space

FCk0 (I ×X;Y ) :=

ϕ : I ×X → Y : ∃n ∈ N, x1, ..xn ∈ X, f ∈ Ck

0 (I × Rn;Y ),

such that ϕ(x) = f(t, x, x1 , ..., x, xn), ∀(t, x) ∈ I ×X (A.14)

Similarly, for k ∈ N∪∞ and a linear, densely defined closed operator B : D(B) ⊆X → X, we define

FCk,B0 (I ×X) =

ϕ : I ×X → R : ∃n ∈ N, x1, ..xn ∈ D(B), f ∈ Ck

0 (I × Rn;R)such that ϕ(x) = f(x, x1, ..., x, xn), ∀x ∈ X (A.15)

Finally, given an orthonormal basis E = ekn∈N of X, we define, for k ∈ N∪ ∞,

ECk0 (I ×X;Y ) :=

ϕ : I ×X → Y : ∃n ∈ N, f ∈ Ck

0 (I × Rn;Y ), such thatϕ(x) = f(t, x, e1 , ..., x, en), ∀(t, x) ∈ I ×X . (A.16)

APPENDIX B

Linear operators and C0-semigroups

All spaces considered in the book are real. However the spectral theory hasto be done in complex spaces and thus some results presented here require the useof complex spaces. To accommodate real Hilbert and Banach spaces we thus usecomplexification of spaces and operators, which for Hilbert spaces can be done ina natural way. If H is a real Hilbert space, its complexification Hc is defined by

Hc := x = x+ iy : x, y ∈ Hwith standard operations(x+iy)+(z+iw) := (x+z)+i(y+w), (a+ib)(x+iy) = (ax−by)+i(bx+ay), a, b ∈ R,and with the inner product

(x+ iy), (z + iw)c := x, zH + y, wH + i(y, zH − x,wH).

Thus |x + iy|Hc= |(x, y)|H×H . A real Banach space E is complexified in the

same way, however, except for special cases, the product norm is no longer a normbecause the homogeneity condition fails. To define a norm in Ec we first compute

eit(x+ iy) = (x cos t− y sin t) + i(x sin t+ y cos t)

and then define|x+ iy|Ec

:= sup0≤t≤2π

|(x cos t− y sin t, x sin t+ y cos t)|E×E .

We refer the reader to [351, 413] for more on complexification.A linear operator T : D(T ) ⊂ E → E is complexified by setting

D(Tc) := x+ iy : x, y ∈ D(T ), Tc(x+ iy) := Tx+ iTy.

It is easy to see that T ∈ L(E) if and only if Tc ∈ L(Ec), and moreover T = Tc.Also T is invertible if and only if Tc is invertible. It is a standard convention whichwill not be repeated, that the spectrum and the resolvent set of T are understoodto be the spectrum and the resolvent set of Tc. This is how the statements hereshould be understood in the context of real Hilbert and Banach spaces.

Throughout Appendix B, E will be a Banach space endowed with the norm| · |E and H will be a Hilbert space endowed with the inner product ·, ·H and thenorm | · |H .

B.1. Linear operators

For an operator T we denote by D(T ) its domain, by R(T ) its range and bykerT its kernel (or null space).

Definition B.1 (Pseudoinverse) If E is a uniformly convex Banach space, Zis a Banach space, and T ∈ L(E;Z), the pseudoinverse T−1 of T is the linearoperator defined on T (E) ⊆ Z that associates to every element z in T (E) theelement in T−1(z) with minimum norm (for existence of such an element see [144]II.4.29 page 74). Notice that If E is a Hilbert space then we have

R(T−1) = (kerT )⊥.

537

538 B. LINEAR OPERATORS AND C0-SEMIGROUPS

The following result is taken from [130], Proposition B.1, p. 429, where thereader can find its proof.

Proposition B.2 Let E,E1, E2 be three Hilbert spaces, let A1 : E1 → E, A2 :E2 → E be linear bounded operators, let A∗

1 : E → E1 and A∗2 : E → E2 be their

adjoints and finally let A−11 : R(A1) ⊆ E → E1, A−1

2 : R(A2) ⊆ E → E2 be therespective pseudoinverses. Then we have:

(i) R(A1) ⊆ R(A2) if and only if there exists a constant k > 0 such that

|A1x|E1 ≤ k|A

2x|E2 ∀x ∈ E.

(ii) If|A

1x|E1 = |A2x|E2 ∀x ∈ E,

then R(A1)=R(A2) and

|A−11 x|E1 = |A−2

2 x|E2 ∀x ∈ R(A1).

Definition B.3 (Closed operator, Graph norm) Let E and Z be two Banachspaces. A linear operator A : D(A) ⊂ E → Z, is said to be closed if its graph

(x, y) ∈ D(A)× Z : y = Ax

is closed in E×Z. Given a closed operator A : D(A) ⊂ E → Z the graph norm onD(A) is defined as follows:

|x|D(A) :=|x|2E + |Ax|2Z

12 for all x ∈ D(A).

Sometimes an equivalent norm |x|D(A) := |x|E + |Ax|Z is also used.

Proposition B.4 Let E and Z be two Banach spaces. If A : D(A) ⊂ E → Zis a linear, closed operator, then D(A) with the graph norm is a Banach space.Moreover, if D(A) is endowed with the graph norm, A : D(A) → Z is continuous.

Definition B.5 (Resolvent) Consider a linear, closed operator A : D(A) ⊂E → E and define, for λ ∈ C, the operator (λI − A) : D(A) → E. The resolventset of A is defined as follows:

(A) :=λ ∈ C : (λI −A) is invertible and (λI −A)−1 ∈ L(E)

.

σ(A) := C \ (A)

is called the spectrum of A.

Lemma B.6 Let A : D(A) ⊂ E → E be a linear, closed operator. Then theresolvent set (A) is open.

Proof. See [450], Theorem 1 page 201.

Definition B.7 (Closable operator, closure of an operator) Let E and Z betwo Banach spaces. A linear operator A : D(A) ⊂ E → Z is said to be closable ifthe closure of its graph in E × Z is the graph of some (closed) operator. Such anoperator is called the closure of A and it is denoted by A.

Remark B.8 It is easy to see that A : D(A) ⊂ E → Z is closable if and only if,for any sequence xn in D(A) and any y ∈ Z such that

xnn→∞−−−−→

E0 and T (xn)

n→∞−−−−→Z

y,

we have y = 0.

B.2. DISSIPATIVE OPERATORS 539

Definition B.9 (Core of a closed operator) Let E and Z be two Banachspaces. Consider a closed linear operator A : D(A) ⊂ E → Z. A linear subspaceY of D(A) is said to be a core for A if it is dense in D(A) (endowed with itsgraph norm). In other words Y is a core for A if and only if Y ⊆ D(A) and, forany x ∈ D(A) there exists a sequence xn of elements of Y such that xn → x andAxn → Ax.

B.2. Dissipative operators

Definition B.10 (Duality mapping) Let E be a Banach space and E∗ be itsdual. The function J : E → 2E

∗

, defined by

J (x) :=x∗ ∈ E : x∗, xE∗,E = |x|2E = |x∗|2E∗

, for x ∈ E,

is called the duality mapping.

The duality mapping is in general multivalued and, for all x ∈ E, J(x) = ∅.If the dual E∗ is strictly convex, and in particular if E is a Hilbert space, J issingle-valued (see Section 1.1 of [20] and in particular Theorem 1.2).

Lemma B.11 (Kato’s Lemma) Let E be a Banach space and x, y ∈ E. Thereexists w ∈ J (x) such that w, yE∗,E ≥ 0 if and only if

|x|E ≤ |x+ λy|Efor any λ > 0.


In the remainder of this section A will denote a possibly non-linear operator.

Definition B.12 (Dissipative operators) Let E be a Banach space. An oper-ator A : D(A) ⊂ E → E is called dissipative if

w,A(x)−A(y)E∗,E ≤ 0, for all x, y ∈ D(A) and some w ∈ J (x− y). (B.1)

A dissipative operator A is said to be m-dissipative if R(I −A) = E. A dissipativeoperator A is said to be maximal dissipative if there does not exist any properdissipative extension of A.

In the specific case of a linear operator A in a Hilbert space H the expression(B.1) can be rephrased as Ax, xH ≤ 0 for all x ∈ D(A). If the Hilbert space iscomplex one considers its real part: Re Ax, xH ≤ 0.

Definition B.13 (Accretive operators) Let E be a Banach space. An operatorA : D(A) ⊂ E → E is called accretive (respectively m-accretive, maximal accretive)if −A is dissipative (respectively m-dissipative, maximal dissipative).

Remark B.14 It easily follows from the definition that any m-dissipative oper-ator is maximal dissipative. In case of (possibly non-linear) operators in Hilbertspaces the two properties are equivalent, see Remark 3.1 page 101 of [20].

Proposition B.15 An operator A : D(A) ⊂ E → E is dissipative if and onlyif, for any λ > 0, for any x, y ∈ D(A),

|x− y|E ≤ |x− y − λ(A(x)−A(y))|E ,

or equivalently, if there exists λ > 0 with such a property.

Proof. See Proposition 3.1, page 98, of [20].


Remark B.16 Proposition B.15 can be rewritten in an obvious way in the fol-lowing form: A : D(A) ⊂ E → E is dissipative if and only if for any λ > 0, for anyx, y ∈ D(A),

|x− y|E ≤ 1

λ|(λx−A(x))− (λy −A(y))|E ,

or equivalently, if there exists λ > 0 with such a property.

Proposition B.17 A dissipative operator A : D(A) ⊂ E → E is m-dissipativeif and only if R(I − λA) = E for all (equivalently, for some) λ > 0.

Proof. See Proposition 3.3 page 99 of [20].

Proposition B.18 Any linear maximal dissipative operator A in a Hilbert spaceH is closed if and only if it has a dense domain.

Proof. See [383], Theorem 1.1.1, p. 200-201 and Lemma 1.1.3, p. 201.

Proposition B.19 Let E be a Banach space. Let λ0 > 0 and

F : [λ0,+∞) → C0,1(E;E)

be a function such that, for any λ ≥ λ0, F (λ) is injective and, for any λ, µ ≥ λ0,

F (λ) = F (µ) (IE + (µ− λ)F (λ)) .

Then there exists a unique operator A : D(A) ⊆ E → E such that, for any λ ≥ 0,

F (λ) = (λI −A)−1.

If moreover

|F (λ)x− F (λ)y| ≤ 1

λ|x− y|, for all λ ≥ λ0, x, y ∈ E,

then A is m-dissipative.

Proof. Proposition I.3.3, page 13 of [108] ensures the existence of the op-erator A. The second part about the m-dissipativity of A follows from PropositionII.9.6 of [108].

We refer to Chapter 3 of [20], Appendix D of [130], Chapter 5 of [127], Chapter3 of [132], [160], [375], [458] and [459] for more about dissipative and accretiveoperators.

B.3. Trace class and Hilbert-Schmidt operators

Throughout this section H,U, V will denote real, separable Hilbert spaces,·, ·H , ·, ·U , ·, ·V , | · |H , | · |U , | · |V will be respectively the inner products inH,U and V and the related norms.

Definition B.20 A linear operator T ∈ L(U,H) is called nuclear or trace classif T can be represented in the form

T (z) =+∞

k=1

bk z, akU for any z ∈ U ,

where ak and bk are two sequences of elements respectively in U and H such that+∞k=1 |ak|U |bk|H < +∞. We denote the set of the nuclear operators from U to H

by L1(U,H). We write L1(H) instead of L1(H,H).

B.3. TRACE CLASS AND HILBERT-SCHMIDT OPERATORS 541

Proposition B.21 L1(U,H) is a separable Banach space with respect to thenorm

TL1(U,H) := inf

+∞

k=1

|ak|U |bk|H : ak ⊆ U, bk ⊆ H,

and T (z) =+∞

k=1

bk z, akU , ∀z ∈ U

. (B.2)

Proof. See [409] Proposition 2.8, page 21 and the subsequent observations.

Proposition B.22 Given T ∈ L1(H) and an orthonormal basis ek of H, theseries

k∈NTek, ekH

converges absolutely and its sum does not depend on the choice of the basis ek.Proof. See [380], pages 357-358 after the proof of the Proposition A.4.

Definition B.23 Given T ∈ L1(H) and any orthonormal basis ek of H,

Tr(T ) :=

k∈NTek, ekH

is called the trace of T .

We have |Tr(T )| ≤ TL1(H) (see e.g. [380], page 357).

Definition B.24 Let ekk∈N an orthonormal basis of U . The space of Hilbert-Schmidt operators L2(U,H) from U to H is defined by

L2(U,H) :=

T ∈ L(U,H) :

k∈N|Tek|2H < +∞

. (B.3)

We write L2(H) for L2(H,H).

Proposition B.25 The space of Hilbert-Schmidt operators L2(U,H) does notdepend on the choice of orthonormal basis ekk∈N. It is a separable Hilbert spaceif endowed with the inner product

S, T 2 :=∞

k∈NSek, T ekH , S, T ∈ L2(U,H).

The inner product is independent of the choice of basis.

Proof. See Appendix C of [130] after Proposition C.3.

The following proposition follows easily from the definition of the space ofHilbert-Schmidt operators and elementary calculations.

Proposition B.26(i) T ∈ L2(U,H) if and only if T ∗ ∈ L2(H,U). Moreover TL2(U,H) =

T ∗L2(H,U).(ii) If T ∈ L2(U,H) and S ∈ L(H,V ) then ST ∈ L2(U, V ) and

STL2(U,V ) ≤ SL(H,V )TL2(U,H).

If T ∈ L(U,H) and S ∈ L2(H,V ) then ST ∈ L2(U, V ) and

STL2(U,V ) ≤ SL2(H,V )TL(U,H).


Definition B.27 Let U ⊆ H. The embedding U ⊆ H is said to be Hilbert-Schmidt if for some orthonormal basis ekk∈N of U , we have

k∈N|ek|2H < +∞.

Thanks to Proposition B.25 this definition does not depend on the choice ofbasis ek.Proposition B.28 The following properties hold:

(i) If T ∈ L1(U,H) and S ∈ L(H,V ) then ST is in L1(U, V ) andSTL1(U,V ) ≤ SL(H,V )TL1(U,H).

If T ∈ L(U,H) and S ∈ L1(H,V ) then ST is in L1(U, V ) andSTL1(U,V ) ≤ SL1(H,V )TL(U,H).

(ii) If T ∈ L1(U,H) and S ∈ L(H,U) (respectively, T ∈ L(U,H) andS ∈ L1(H,U)) then ST is in L1(U), TS is in L1(H) and

Tr(ST ) = Tr(TS).

(iii) If T ∈ L2(U,H), S ∈ L2(H,V ) then ST ∈ L1(U, V ) andSTL1(U,V ) ≤ SL2(H,V )TL2(U,H).

Moreover, if U = V , then Tr(ST ) = Tr(TS).(iv) L1(U,H) ⊆ L2(U,H).(v) If T ∈ L2(U,H) then T is compact.

Proof. Most claims are proved in Appendix A.2 of [380]. More precisely:(i) is proved in Propositions 4, page 356, (ii) is proved in Proposition A.5-(i), page358, while (iv) and (v) are proved in Proposition A.6, page 359. The proof of (iii)also repeats the proof of Proposition A.5-(ii), page 358, however we include it herefor completeness. Consider an orthonormal basis fkk∈N of H. For every z ∈ Uwe have

Tz =

k∈NTz, fkH fk =

k∈Nz, T ∗fkU fk

soSTz =

k∈Nz, T ∗fkU Sfk (B.4)

and then, using the definition of L1-norm given in (B.2) and the Cauchy-Schwarzinequality, we get

STL1(U,V ) ≤

k∈N|T ∗fk|U |Sfk|V ≤

k∈N|T ∗fk|2U

1/2

k∈N|Sfk|2V

1/2

= T ∗L2(H,U)SL2(H,V ) = TL2(U,H)SL2(H,V ), (B.5)

where in the last step we used Proposition B.26-(i). It also follows from (B.4) andthe Parseval identity that

Tr(ST ) =

k∈NSfk, T ∗fkU =

k∈NTSfk, fkH = Tr(TS).

Observe that in general, if T, S ∈ L(H) and TS ∈ L1(H), this does not nec-

essarily imply ST ∈ L1(H). Indeed consider two operators defined on H × Hby

T =

I I−I −I

, S =

−I II −I

,

B.4. C0-SEMIGROUPS AND RELATED RESULTS 543

where I stands for the identity operator. Then

TS = 0, ST = 2

−I −II I

,

however ST is not trace class if H is infinite dimensional. Another example is givenin [129], page 6.

Notation B.29 We set

L+(H) := T ∈ S(H) : Tx, xH ≥ 0 ∀x ∈ Hand

L+1 (H) := L1(H) ∩ L+(H).

The operators in L+(H) are called positive operators on H. A positive operator Ton H is called strictly positive if it satisfies Tx, xH > 0 for all x ∈ H.

Proposition B.30 An operator T ∈ L+(H) is nuclear if and only if

k∈NTek, ekH < +∞.

for an orthonormal basis en on H. Moreover in this case Tr(T ) = TL1(H).

Proof. See [130], Proposition C.3.

B.4. C0-semigroups and related results

B.4.1. Basic definitions.

Definition B.31 (C0-semigroup) A map S : [0,+∞) → L(E) is called aC0-semigroup (or a strongly continuous semigroup) on E if the following threeconditions are satisfied:

(i) S(0) = I.(ii) For all s, t ∈ [0,+∞), S(t)S(s) = S(t+ s).(iii) For all x ∈ E, the map t → S(t)x is continuous from [0,+∞) to E1.

For C0-semigroups we will use the notation S(t), t ≥ 0 or simply S(t).

Definition B.32 (Generator of a C0-semigroup) Let S(t) be a C0-semigroupon E. The linear operator A : D(A) ⊂ E → E defined as

D(A) :=

x ∈ E : S(t)x−x

t has a limit in E when t → 0+

Ax := limt→0+S(t)x−x

t

is called the infinitesimal generator of S(t).

Proposition B.33 Let S(t) be a C0-semigroup on E. Then there exist M ≥ 1and ω ∈ R such that

S(t) ≤ Meωt, for t ≥ 0. (B.6)

Proof. See for instance [375], Theorem 2.2, Chapter 1, page 4. The infimum of all ω such that (B.6) is satisfied for some Mω is called the type

of the C0-semigroup S(t) and is denoted by ω0, see [35], Part II, Section 2.2. Wehave ω0 ∈ [−∞,+∞). If ω0 < 0 we say that the C0-semigroup S(t) is of negativetype and if ω0 > 0 we say that the C0-semigroup S(t) is of positive type.

1Equivalently one can ask here that limt0 S(t)x = x for all x ∈ E, see e.g. [375] Corollary2.3, p.4.


Definition B.34 (Contraction semigroup) A C0-semigroup S(t) on E iscalled a C0-semigroup of contractions, if (B.6) holds with M = 1,ω = 0.

Definition B.35 (Pseudo-contraction semigroup) A C0-semigroup S(t) onE is called a C0-semigroup of pseudo-contractions if (B.6) holds with M = 1 forsome ω ∈ R.

Definition B.36 (Uniformly bounded semigroup) A C0-semigroup S(t) onE is called uniformly bounded if (B.6) holds with ω = 0 for some M ≥ 1.

We remark that the complexification Sc(t) of a C0-semigroup on E is a C0-semigroup on Ec whose generator is the complexification Ac of the generator A ofS(t).

B.4.2. Hille-Yosida Theorem and Yosida approximation.

Theorem B.37 (Hille-Yosida) A linear operator A : D(A) ⊂ E → E is theinfinitesimal generator of a C0-semigroup S(t) on E satisfying (B.6) if and only if

(1) A is a closed and D(A) is dense in E,(2) (λI −A) is invertible and (λI −A)−1 ∈ L(E) for every λ > ω, and

((λI −A)−1)k ≤ M (λ− ω)−k for all k ∈ N and λ > ω.

Proof. See [222], Theorem 2.13, p. 20 or [375], Theorem 5.3. In fact, see [375], Remark 5.4, we have the following.

Remark B.38 For complex spaces condition (2) in Theorem B.37 can be replacedby λ ∈ C : Reλ > ω ⊂ (A) and, for all k ∈ N and λ ∈ C,Reλ > ω,

((λI −A)−1)k ≤ M (Reλ− ω)−k .

Remark B.39 Considering, if needed, S(t) := e−(ω+)tS(t), > 0, we can al-ways restrict to uniformly bounded C0-semigroups having invertible generators. Inparticular, condition (2) of Theorem B.37 can be assumed to hold with ω = 0.

Definition B.40 (Yosida approximations) Let A : D(A) ⊂ E → E be theinfinitesimal generator of a C0-semigroup S(t) on E satisfying (B.6). For n ∈ Ngreater than ω, define

Jn = n(nI −A)−1. (B.7)The Yosida approximation of A is defined as follows:

An := n2(nI −A)−1 − nI = AJn ∈ L(E). (B.8)

Lemma B.41 Consider A and Jn be as in Definition B.40. Then

Jn ≤ Mn

n− ωfor all n > ω. (B.9)

Proof. It follows directly from condition (2) (k = 1) of Theorem B.37. Proposition B.42 Let A : D(A) ⊂ E → E be the infinitesimal generator of aC0-semigroup on E. Let Jn and An be as in Definition B.40. Then

Jnxn→∞−−−−→

Ex for all x ∈ E (B.10)

andAnx

n→∞−−−−→E

Ax for all x ∈ D(A).

Proof. See [130], Proposition A.4, page 409.

B.4. C0-SEMIGROUPS AND RELATED RESULTS 545

Proposition B.43 Let A : D(A) ⊂ E → E be the infinitesimal generator of aC0-semigroup S(t) on E satisfying (B.6). Let An be the Yosida approximation ofA. Define, for x ∈ E and t ≥ 0,

etAnx :=

j∈N

tjAjnx

j!.

Then,etAn ≤ Met

nω

n−ω . (B.11)

andS(t)x = lim

n→∞etAnx for every x ∈ E (B.12)

uniformly on bounded subsets of [0,+∞).

Proof. See [375], Theorem 5.5, p. 21 and [35], Step 2 of the proof ofTheorem 2.5, p. 102-103.

Expression (B.12) shows how to construct explicitly the semigroup generatedby a linear operator A.

Notation B.44 The semigroup generated by A will be denoted by etA.

Theorem B.45 (Lumer-Phillips) Let H be a separable Hilbert space. Given alinear operator A : D(A) ⊂ H → H, the following facts are equivalent:

(1) A is the generator of a C0-semigroup of contractions on H.(2) D(A) = H and A is maximal dissipative.(3) D(A) = H and A∗ is maximal dissipative.(4) D(A) = H,A is dissipative and R(λ0I −A) = H for some λ0 > 0.

Proof. See [383], Theorem 1.1.3, page 203, Theorem 1.4.2, page 214, and[375], Theorem 4.3, page 14.

The following results is a corollary of the Trotter-Kato Theorem.

Proposition B.46 (Trotter-Kato) Let S(t), Sn(t), n ∈ N, be strongly con-tinuous semigroups on E with generators A and An, respectively. Assume thatD(A) ⊂ D(An) for every n ∈ N and that

S(t), Sn(t) ≤ Meωt, ∀t ≥ 0, n ∈ N

and some constants M ≥ 1 and ω ∈ R. If Anx → Ax for every x ∈ D(A) then

Sn(t)x → S(t)x, ∀t ≥ 0, x ∈ X,

and the limit is uniform in t for t in bounded intervals.

Proof. See [375], Theorem 4.5, p. 88, or [160], Theorem 4.8, p. 209.

Proposition B.47 Let S(t) be a strongly a continuous semigroup on E with thegenerator A. Let Y ⊆ D(A) be a subspace of D(A). Assume that

(i) Y is dense on E,(ii) S(t)(Y ) ⊆ Y for all t ≥ 0.

Then Y is a core for A.

Proof. See [110], Proposition A.19, page 204.


B.4.3. Analytic semigroups and fractional powers of generators.Throughout this section H is a separable Hilbert space. The material about analyticsemigroups requires that H be complex. The statements for real H should be under-stood with the convention that H,A, S(t) are their complexifications Hc, Ac, Sc(t).

Definition B.48 (Differentiable semigroup) A C0-semigroup S(t) on H iscalled differentiable, if for every x ∈ H, t → S(t)x is differentiable for t > 0.

Definition B.49 (Analytic semigroup) A C0-semigroup S(t) on H is calledanalytic if it has an extension G(z) to a sector of the form ∆ := z ∈ C : a <arg(z) < b for some a < 0 < b with the following properties:

(i) z → G(z) is analytic on ∆.(ii) G(0) = I and limz∈∆

z→0G(z)x = x for every x ∈ H.

(iii) G(z1 + z2) = G(z1)G(z2) for z1, z2 ∈ ∆.

Theorem B.50 Let S(t) be a uniformly bounded C0-semigroup on H and let Abe the generator of S(t). Assume that 0 ∈ (A). The following are equivalent:

(1) S(t) can be extended to an analytic semigroup in a sector ∆δ = z ∈C : | arg(z)| < δ, and S(z) is uniformly bounded in every subsector∆δ = z ∈ C : | arg(z)| ≤ δ, δ < δ.

(2) There exists δ ∈0, π

2

and B > 0 such that

Σ :=λ : | arg(λ)| < π

2+ δ

∪ 0 ⊂ (A)

and, for every λ ∈ Σ \ 0,

(λI −A)−1 ≤ B

|λ| .

(3) S(t) is differentiable and there exists a constant C > 0 such that

AS(t) ≤ C

tfor t > 0.

Proof. See [375] Theorem 5.2, page 61.

Theorem B.51 Consider a linear operator A : D(A) ⊂ H → H that generatesa uniformly bounded analytic C0-semigroup S(t), and 0 ∈ (A). Define, for α < 0,

(−A)α :=1

Γ(α)

∞

0t−α−1S(t)dt, (B.13)

where Γ(·) is the Gamma function, and set (−A)0 := I. Then:(i) The integral in (B.13) converges in norm and (−A)α is a well defined

operator in L(H).(ii) (−A)α(−A)β = (−A)α+β for α,β ≤ 0.(iii) (−A)α is injective.

Proof. See [375] pages 70-72.

The operator (−A)α can also be defined by

(−A)α = − 1

2πi

Cλα(λI +A)−1dλ, (B.14)

where the path C is in (−A) and goes from ∞e−iθ to ∞eiθ for θ ∈π2 ,π

, avoiding

the non-positive real axis. Using (B.14) one can define the fractional powers formore general operators, as it is done in classical references like [431] or [375]. Weare only interested in the analytic case in this book.

B.5. π-CONVERGENCE, K-CONVERGENCE, π- AND K-CONTINUOUS SEMIGROUPS 547

Definition B.52 Let A be as in Theorem B.51. The fractional powers of −Aare defined as follows:

(i) If α < 0, D((−A)α) = H and (−A)α is defined by (B.13).(ii) If α = 0, D((−A)0) = H and (−A)0 := I.(iii) If α > 0, D((−A)α) = R((−A)−α) and (−A)α := ((−A)−α)

−1.

Theorem B.53 Let A be as in Theorem B.51. The fractional powers of −Asatisfy the following properties:

(i) For all positive α, D((−A)α) is dense in H and (−A)α is a closedoperator.

(ii) If α ≤ β then D((−A)β) ⊂ D((−A)α).(iii) For every real numbers α,β, (−A)α(−A)β = (−A)α+β on

D((−A)β∨(α+β)).


Theorem B.54 Let A be as in Theorem B.51. The following hold:(i) etA(H) ⊂ D((−A)α) for every t > 0 and α ≥ 0.(ii) etA(−A)αx = (−A)αetAx or every x ∈ D((−A)α).(iii) There exists an Mα > 0 such that for every t > 0, the operator

(−A)αetA : H → H is bounded and

(−A)αetA ≤ Mαt−αe−at. (B.15)

(iv) If α ∈ (0, 1] then for every x ∈ D((−A)α)

|etAx− x|H ≤ Cαtα|(−A)αx|H

for some constant Cα independent of x.


B.5. π-convergence, K-convergence, π- and K-continuous semigroups

In this section, where H is always a real separable Hilbert space, we introducethe notions of π-convergence, K-convergence, π-continuous and K-continuous semi-groups and we recall their basic properties as well as other related notions. Forfurther results and details we refer the reader to [386, 387, 389] and [129], Section6.3, for π-convergence and π-continuous semigroups; to [75, 76, 82] and the appen-dix of [79], for K-convergence and K-continuous (also called weakly continuous, seee.g. [75]) semigroups. We also recall the paper [124] that deals with semigroupswhich are not strongly continuous.

Most of the literature on the present subject deals with the spaces Cb(H) andUCb(H), except for [76, 225] which deal with UCm(H) and [225] which deals withCm(H) in the case of K-continuous semigroups, and the final part of [386] (p.293-294, see also [387], Section 6.5) which shows how to extend the results to thespace Bb(H), in the case of π-continuous semigroups. Here we present the resultsfor π-continuous and K-continuous semigroups mainly in the spaces Cm(H) andUCm(H) because these are the spaces most commonly used in this book. In somecases we will also deal with Bm(H).

B.5.1. π-convergence and K-convergence. The definition of π-convergence can be found e.g. in [161], page 111, where it is called bp-convergence(bounded-pointwise), and in [386]; the former in spaces of continuous and boundedfunctions, the latter in spaces of uniformly continuous and bounded functions. For


K-convergence in UCb(H) space the reader is referred to [75], [82] and, for theCm(H) (respectively, UCm(H)) framework, to [225] (respectively, [76]2).

We state all the definitions in this section considering Bm(H) (m ≥ 0) as theenvironment space. The same definitions hold if the basic space Bm(H) is replacedby Cm(H) or UCm(H) (m ≥ 0).

Definition B.55 [π-convergence] Let m ≥ 0. A sequence (fn) ⊆ Bm(H) is saidto be π-convergent to f ∈ Bm(H) and we will write

fnπ−→ f or f = π- lim

n→∞fn

if the following conditions hold:(i) supn∈N fnBm

< +∞.(ii) limn→∞ fn(x) = f(x) for any x ∈ H.

Moreover, given I ⊆ R, t0 ∈ I, a family (ft)t∈I\t0 ⊂ Bm(H) and f ∈ Bm(H) wewrite

ftπ−−−→

t→t0f or f = π-lim

t→t0fn

if, for any sequence tn of elements of I \ t0 converging to t0, we have ftnπ−→ f .

Similarly, given I ⊆ R, a sequence (fn) ⊆ Bm(I×H) is said to be π-convergentto f ∈ Bm(I ×H) and we will write

fnπ−−−−→

n→∞f or f = π- lim

n→∞fn

if the following conditions hold:(i) supn∈N fnBm

< +∞.(ii) limn→∞ fn(t, x) = f(t, x) for any (t, x) ∈ I ×H.

Definition B.56 [K-convergence] A sequence (fn) ⊂ Bm(H) is said to be K-convergent to f ∈ Bm(H) if

supn∈N

fnBm< +∞,

limn→+∞

supx∈K

|fn(x)− f(x)| = 0(B.16)

for every compact set K ⊂ H. In this case we will write

fnK−−−−→

n→∞f or f = K- lim

n→+∞fn.

Moreover, given I ⊆ R, t0 ∈ I and a family (ft)t∈I\t0 ⊂ Bm(H), where I ⊆ Rand f ∈ Bm(H), we write

ftK−−−→

t→t0f or f = K-lim

t→t0ft

if, for any sequence tn of elements of I \ t0 converging to t0, we have ftnK−→ f .

In a similar way, given I ⊆ R, a sequence (fn) ⊂ Bm(I × H) is said to beK-convergent to f ∈ Bm(I ×H) if

supn∈N

fnBm< +∞,

limn→+∞

sup(t,x)∈I0×K

|fn(t, x)− f(t, x)| = 0(B.17)

2In this paper the K-convergence in UCm(H) is called Km-convergence.


for all compact sets I0 ⊆ I and K ⊂ H3. In this case we will write as before

fnK−−−−→

n→∞f or f = K- lim

n→+∞fn.

The two convergencies that we have just introduced can be immediately gen-eralized to the case of sequences in Bm(H,Y ) or in Bm(I × H,Y ) where Y is agiven Banach space. Moreover they induce a series of related concepts like thoseof closedness and density. In the two following definitions we recall some of themtaken from [386, 387] and [82].

Definition B.57 Let m ≥ 0. A subset Y of Bm(H) (respectively Cm(H),UCm(H)) is said to be π-closed if, for any sequence (fn) ⊂ Y and f ∈ Bm(H)(respectively Cm(H), UCm(H)) such that fn

π−→ f , we have f ∈ Y .A subset Y of Bm(H) (respectively Cm(H), UCm(H)) is said to be π-dense

in Bm(H) (respectively Cm(H), UCm(H)) if, for any f ∈ Bm(H) (respectivelyCm(H), UCm(H)), there exists (fn) ⊆ Y such that fn

π−→ f .A linear operator A : D(A) ⊂ Bm(H) → Bm(H) is said to be π-closed if, given

a sequence (fn) ⊆ D(A), the following condition holds:fn

π−→ f and Afnπ−→ g

⇒

f ∈ D(A) and Af = g

.

The same if A : D(A) ⊂ Cm(H) → Cm(H) or A : D(A) ⊂ UCm(H) → UCm(H).

Definition B.58 Let m ≥ 0. A subset Y of Bm(H) (respectively Cm(H),UCm(H)) is said to be K-closed if for any sequence (fn) ⊂ Y and f ∈ Bm(H)(respectively Cm(H), UCm(H)) such that

K- limn→+∞

fn = f,

we have f ∈ Y .A subset Y of Bm(H) (respectively Cm(H), UCm(H)) is said to be K-dense if

for any f ∈ Bm(H) (respectively Cm(H), UCm(H)) there exists a sequence (fn) ⊆Y such that

f = K- limn→+∞

fn.

A linear operator A : D(A) ⊂ Bm(H) → Bm(H) is said to be K-closed if, givena sequence (fn) ⊂ D(A) such that

K- limn→+∞

fn = f and K- limn→+∞

Afn = g,

we havef ∈ D(A) and Af = g.

The same if A : D(A) ⊂ Cm(H) → Cm(H) or A : D(A) ⊂ UCm(H) → UCm(H).Let A : D(A) ⊂ Bm(H) → Bm(H) and B : D(B) ⊂ Bm(H) → Bm(H) be two

linear operators and assume that A ⊂ B and that B is K-closed. We say that Bis the K-closure of A, and we write B = AK, if for every f ∈ D(B) there exists asequence (fn) ⊂ D(A) such that

K- limn→+∞

fn = f

K- limn→+∞

Afn = Bf.(B.18)

3In the literature (see e.g. [82, 233, 234]) this definition is given taking the supremum overI×K even when I is not compact. We prefer to use the above definition as it is more coherent withthe concept of K-convergence and it does not change anything in the results and in the proofs.


The same if A : D(A) ⊂ Cm(H) → Cm(H), B : D(B) ⊂ Cm(H) → Cm(H) or ifA : D(A) ⊂ UCm(H) → UCm(H), B : D(B) ⊂ UCm(H) → UCm(H).

Motivated by Theorem 4.5 of [225] we introduce the notions of K-core andπ-core, .

Definition B.59 (K-core and π-core) Let m ≥ 0. Let the operator A : D(A) ⊆Bm(H) → Bm(H) be K-closed (respectively π-closed). A linear subspace Y of D(A)is a K-core (respectively, a π-core) for A if for any f ∈ D(A) there exists a sequencefn of elements of Y such that

K- limn→+∞

fn = f and K- limn→+∞

Afn = Af

(respectively,

π- limn→+∞

fn = f and π- limn→+∞

Afn = Af).

The same holds if A : D(A) ⊂ Cm(H) → Cm(H) or A : D(A) ⊂ UCm(H) →UCm(H).

Remark B.60 Let (fn) ⊆ Bm(H) be a sequence which π-converges to a functionf : H → R. Since measurability preserves over pointwise limits (see e.g. Lemma1.8) then it must be f ∈ Bm(H).

Similarly, since converging sequences in H are compact, one can prove that,if (fn) ⊆ Cm(H) is a sequence which K-converges to a function f : H → R, thenf ∈ Cm(H) (i.e. Cm(H) is K-closed in Bm(H)).

On the other hand, if fnK−−−−→

n→∞f and (fn) ⊆ UCm(H) it is not true, in general,

that f ∈ UCm(H). Indeed, using Lemma B.77 below, one easily sees that UCm(H)is K-dense in Cm(H). Similarly, if fn

π−−−−→n→∞

f and (fn) ⊆ Cm(H) it is not true, ingeneral, that f ∈ Cm(H).

Remark B.61 Concerning π-convergence, in [386], Theorem 2.2 (see also [387],Theorem 6.2.3), the author introduces a “natural” Hausdorff locally convex topol-ogy τ0, not metrizable and not sequentially complete in UCb(H), whose convergentsequences are exactly π-convergent sequences. As stated in [386], end of Section 5,the same holds in Cb(H). The lack of completeness of τ0 relies on the fact that con-tinuity (and, a fortiori, uniform continuity) is not preserved under π-convergence,as observed in the previous remark.

Similarly, concerning K-convergence, in [225] it is shown (see in particularProposition 2.3) that in the so-called mixed topology τM, which is a locally con-vex and complete one, introduced in [447], convergent sequences are precisely K-convergent sequences. The result is obtained in Cm(H) for m ≥ 0. Completeness ofτM relies on the fact that continuity is preserved under K-convergence, as observedin the previous remark. However, completeness of such topology is not guaranteedif we consider it on UCm(H), as observed in the introduction of [225].

In [161] p. 495-496 a topology in the space Bb(H) is given whose convergentsequences coincide with π-convergent sequences.

It is clear that alls the concepts introduced in Definitions B.57, B.58 and B.59can also be seen as topological concepts in the topologies just described.

Remark B.62 In [327] (Definition 2.1 an Remark 2.6, see also [392][Section 5.2])the author also considers π-convergence for multisequences and defines the conceptsof π-closedness, π-density (Definition 2.5 there) and π-core (Definition 2.10) withmultisequences. It seems that using multisequences the theory works as well sincethe dominated convergence is still preserved under π-convergent multisequences.


Since in this book we do not need such more general setting we keep the definitionswith single sequences which, apriori, are not equivalent to the analogous definitionsin [327].

B.5.2. π- and K-continuous semigroups and their generators. We usethe definitions of the previous section to introduce π-continuous and K-continuoussemigroups. The theory of such semigroups has been developed in the literaturemainly using UCb(H) as the environment space but the definitions and the resultscan be easily adapted to the Cb(H) (or also to the Cm(H) and UCm(H)) framework.We will present the UCb(H) setting, making few remarks on how to generalize theresults to Cb(H) (or also to Cm(H) and UCm(H)).

B.5.2.1. The definitions.

Definition B.63 (π-continuous semigroup) Let S(t) be a semigroup ofbounded linear operators on UCb(H), namely, for any f ∈ UCb(H) and s, t ∈ R+,S(t + s)f = S(t)S(s)f and S(0)f = f . We say that S(t) is a π-continuous semi-group on UCb(H) of class Gπ(M,ω) if the following conditions hold:

(i) There exist M ≥ 1 and ω ∈ R s.t.

S(t)L(UCb(H)) ≤ Meωt, t ≥ 0.

(ii) For any (fn) ⊆ UCb(H), f ∈ UCb(H) s.t. fnπ−→ f we have

S(t)fnπ−−−−→

n→∞S(t)f for all t ≥ 0.

(iii) For any x ∈ H and f ∈ UCb(H), the map [0,+∞) → R, t → (S(t)f)(x)is continuous.

Definition B.64 (K-continuous semigroup) Let S(t) be a semigroup ofbounded linear operators on UCb(H), namely, for any f ∈ UCb(H) and s, t ∈ R+,S(t + s)f = S(t)S(s)f and S(0)f = f . We say that S(t) is a K-continuous semi-group (or a weakly continuous semigroup) of class GK(M,ω) on UCb(H) if thefollowing conditions hold:

(i) There exist M ≥ 1 and ω ∈ R s.t.

S(t)L(UCb(H)) ≤ Meωt, t ≥ 0.

(ii) For any (fn) ⊆ UCb(H), f ∈ UCb(H) s.t. fnK−→ f we have

S(t)fnK−−−−→

n→∞S(t)f for all t ≥ 0. The limit is uniform in t ∈ [0, T ]

for any T > 0.(iii) For every f ∈ UCb(H) and t0 ≥ 0 we have

K- limt→t0

S(t)f = S(t0)f.

(iv) For any T > 0 and f ∈ UCb(H), the family of functions

S(t)f : t ∈ [0, T ] is equi-uniformly continuous in UCb(H).

We now give some observations concerning:• The relationship between the above two definitions and the definition

of a strongly continuous semigroup.• The extension of the above two definitions to more general spaces.• The relationship of the notions of K and π-continuous semigroups with

strongly continuous semigroups in coarser topologies.• The main reason why they are introduced: to deal with transition semi-

groups.


Remark B.65 The above definitions were introduced first in [386, 387] (for π-semigroups) and in [75] (for K-semigroups, there called weakly continuous). Weobserve the following.

(1) Differently from the case of strongly continuous semigroups (see Propo-sition B.33), condition (i) has to be included in both definitions abovebecause it is not known if it follows from other assumptions.

(2) Conditions (ii) in both definitions gives a kind of continuity with respectto the π or K-convergence. In condition (ii) for a K-continuous semi-group (following [75]) the limit is required to be uniform in t ∈ [0, T ].Such a requirement is avoided in the definition given in [76]: we keepit here since it is verified in all cases we consider. Also this is not thecase for π-continuous semigroups. There are examples of π-continuoussemigroups where the limit is not uniform (see Remark 6.2.2 in [387]).

(3) Conditions (iii) in both definitions are analogues of condition (iii) inDefinition B.31. However here the equivalence mentioned there in thefootnote is not obvious. Indeed condition (iii) for a π-continuous semi-group clearly implies that

π- limt→0+

S(t)f = f, (B.19)

while the opposite is not obvious: the above implies right continuity ofthe map t → (S(t)f)(x) but it is not known if also left continuity holds(see Remark 6.2.5 (a) in [387]). In [386] Remark 2.5 (see also Remark6.2.5 (b) in [387]), the author proves that the equivalence holds for π-semigroups defined on Cb(M) for a compact metric space M . Similarlyfor a K-continuous semigroup is not obvious if condition (iii) for t0 = 0implies that the same holds true for all t0 ≥ 0.

The reason why in both cases the stronger conditions in the defini-tions are used is that the continuity of the map t → S(t)f(x) is useful tosimplify various steps in the proofs. Moreover the stronger conditionsare verified in the cases we consider in this book. We finally observe thatin the original definition of K-continuous semigroup (see [75], Definition2.1) the weaker condition (B.19) is used but it seems that the authorstill uses continuity of the map t → S(t)f(x) (proof of Proposition 3.1,[75]).

(4) Condition (iv) for a K-continuous semigroup extends the regularity intime of the function (t, x) → (S(t)f)(x) requiring uniformity in time ofthe modulus of continuity in x. We point out that in the original defi-nition of K-continuous semigroups in [75] the uniformity in conditions(ii) and (iv) are required on [0,+∞) since in this paper the author dealswith semigroups of negative type which we do not want to assume here.

Remark B.66 The definition of a π-continuous semigroup can also be made inCb(H) or in Bb(H) by simply substituting UCb(H) with Cb(H) or Bb(H) (see [386]pages 293-294). Similarly we can define them in UCm(H), Cm(H), Bm(H).

Concerning K-continuous semigroups, they can be easily defined in UCm(H)by simply substituting UCb(H) with UCm(H) as it is done in Section 2.2 of [76].On the other hand, to define K-continuous semigroups in Cb(H) and in Cm(H)one should also substantially modify (or erase) condition (iv) of Definition B.64.This is done in Theorem 4.1 if [225] where (iv) is implicitly substituted with thelocal equicontinuity of the family of operators S(t), t ≥ 0. The extension of the


definition of K-continuous semigroups to spaces Bb(H) or Bm(H) is not consideredin the literature.

Remark B.67 Recalling Remark B.61, consider a π-continuous (respectivelyK-continuous) semigroup S(t) acting on the space UCb(H) endowed with thetopology τ0 generated by the π-convergence (respectively τM generated by the K-convergence). By Definition B.63-(ii) (respectively B.64-(ii)), for every t ≥ 0, S(t)is sequentially continuous but it is not known if it is also continuous. In [225] theauthors show that the transition semigroups (introduced in the following section)are strongly continuous with respect to the mixed topology τM so, in particularthey are continuous in such topology.

Remark B.68 The reason why K-continuous semigroups, and later, π-continuous semigroups, were introduced (in [75] and in [387], respectively) is theneed for studying4 Markov transition semigroups associated with finite and infinitedimensional SDE since such semigroups are naturally not C0-semigroups: as shownfor instance in Example 6.1 of [75], already in spatial dimension 1, the Ornstein-Uhlenbeck semigroup is not strongly continuous. Nevertheless by a simple applica-tion of the dominated convergence theorem, one can see that all Markov transitionsemigroups defined in (1.95) are π-continuous semigroups (see also Definition 3.5 of[386] and the subsequent comments). Moreover, with a slightly more complicatedproof it can also be proved that such semigroups are K-continuous.

On the other hand, it is not true, as one may expect, that all strongly con-tinuous semigroups are also π-continuous or K-continuous. Indeed it is shown in[387], Proposition 6.2.4 and the subsequent observation, that the class of uniformlycontinuous semigroups on UCb(R) is not contained in the class of π-continuoussemigroups nor in that of K-continuous semigroups. Moreover, even though theclass of π-continuous semigroups has been introduced to study a wider set of prob-lems (see Remark 6.2.2 of [387]), it is not clear if all K-continuous semigroup arealso π-continuous semigroups.

B.5.2.2. The generators. Similarly to what we have for C0-semigroups, we candefine the generators of π-continuous and K-continuous semigroups. Given a semi-group of bounded operators S(t), we will write

∆h :=S(h)− I

h.

Definition B.69 Let S(t) be a π-continuous semigroup on UCb(H). We definethe infinitesimal generator A of S(t) as follows:

D(A) := f ∈ UCb(H) : there exists g ∈ UCb(H) s.t. π- limt→0+ ∆hf = g(Af)(x) := limt→0 ∆tf(x), for x ∈ H.

Definition B.70 Let S(t) be a K-continuous semigroup on UCb(H). We definethe infinitesimal generator A of S(t) as follows:

D(A) := f ∈ UCb(H) : there exists g ∈ UCb(H) s.t. K- limt→0+ ∆hf = g(Af)(x) := limt→0 ∆tf(x), for x ∈ H.

Remark B.71 In [75, 79, 80, 82] the generator A of a K-continuous semigroupS(t) is defined in a different way, by using the resolvent. Indeed A is the unique

4Other approaches are possible to deal with such semigroups (see e.g. the theory of semi-groups on general locally convex spaces [277, 278]) but, as remarked in the introduction of [386],the use of such a theory for the above goal would be much more complicated.


closed linear operator such that, for all λ > ω, f ∈ UCb(H), x ∈ H, we have

(λI −A)−1f(x) =

+∞

0e−λtS(t)f(x)dt.

In fact the two definitions are equivalent. The equivalence is implicitly provedfor π-continuous semigroups in [387][Proposition 6.2.11] (see also [386][Proposition2.5]) with a proof that easily adapts to K-continuous semigroups too. Moreoverthe equivalence of the two definitions is also proved in [225][Remark 4.3], for K-continuous transition semigroups (which is the case of interest in this book).

In [387], Theorem 6.2.13, the author shows that, if a π-continuous semigroupis also K-continuous, the two generators coincide.

We finally observe that the generators can be introduced, exactly in the sameway, if we consider, as said in Remark B.66, π- continuous semigroups on UCm(H),Cm(H), Bm(H) and K-continuous semigroups in UCm(H), Cm(H), for some m ≥0.

B.5.2.3. Cauchy problems for π-continuous and K-continuous semigroups.The following propositions establish the relationship between π-continuous/K-continuous semigroups, their generators and homogeneous Cauchy problems.

Proposition B.72 Let A be the generator of a π-continuous semigroup S(t) onUCb(H). Then:

(i) D(A) is π-dense in UCb(H).(ii) A is a closed and π-closed operator on UCb(H).(iii) For any f ∈ D(A):

(a) S(t)f ∈ D(A) and AS(t)f = S(t)Af , for any t ≥ 0.(b) For any x ∈ H the mapping

(0,+∞) → R, t → (S(t)f)(x)

is continuously differentiable andd

dtS(t)f(x) = A(S(t)f)(x)

for all t > 0.

Proof. See [387] Proposition 6.2.9 and 6.2.7. Closedness of A follows fromRemark B.71.

Proposition B.73 Let A be the generator of a K-continuous semigroup S(t)on UCb(H). Then:

(i) D(A) is K-dense in UCb(H).(ii) A is a closed and K-closed operator on UCb(H).(iii) For every f ∈ D(A):

(a) S(t)f ∈ D(A) and AS(t)f = S(t)Af , for any t ≥ 0.(b) For any x ∈ H the mapping:

(0,+∞) → R, t → (S(t)f)(x)

is continuously differentiable andd

dtS(t)f(x) = A(S(t)f)(x)

for all t > 0.

Proof. See [82] Proposition 2.9 and Remark 2.10. Closedness of A followsfrom Remark B.71.


Remark B.74 In the case of a K-continuous semigroup with generator A, if weconsider the function u : [0, T ]×H → R defined as

(t, x) → (S(t)ϕ) (x), (B.20)

which should be the natural solution of the homogeneous Cauchy problem corre-sponding to the operator A, then u belongs to UCx

b ([0, T ]×H)5 and not, in general,to C([0, T ];UCb(H)) = UCb([0, T ] ×H). Indeed, if for every datum ϕ ∈ UCb(H),u ∈ C([0, T ];UCb(H)), then A generates a C0-semigroup. Similarly, in the case ofπ-continuous semigroups the function in (B.20) does not belong to UCb([0, T ]×H)in general, but it belongs in a natural way to a space called Cπ([0, T ];UCb(H))(requiring global boundedness and separate continuity), introduced in Definition7.2.1 on page 161 of [387], which is strictly bigger than UCx

b ([0, T ]×H).Moreover, in general the UCb(H)-valued function t → etAϕ is not even measur-

able. In fact measurability of this map implies strong continuity of the semigroup(see [450], pages 233-234).

The above propositions can be extended, as said in Remark B.66, to the caseof π- continuous semigroups in UCm(H), Cm(H), Bm(H), and K-continuous semi-groups in UCm(H), Cm(H), for some m ≥ 0.

We now consider non-homogeneous Cauchy problems. Similarly to the case ofC0-semigroups (see e.g. [35] Chapter II.1.3), given a π-continuous or a K-continuoussemigroup S(t) and the related generator A, one can consider the following nonhomogeneous Cauchy problem on UCb(H)6:

ddtv(t) = Av(t) + f(t), t ∈ (0, T ],

v(0) = ϕ ∈ UCb(H),(B.21)

where f : (0, T ] → UCb(H).In analogy with what is usually done for C0-semigroups, the mild solution of

problem (B.21) is, by definition, the function u : [0, T ]×H → R given by

u(t, x) = [S(t)ϕ](x) +

t

0[S(t− s)f(s)](x)ds. (B.22)

Remark B.75 This definition assumes that, for every t > 0 and x ∈ H, the map

[0, t] → R, s → [S(t− s)f(s)](x)

is measurable. This is true under mild assumptions on f but one has to be carefulsince, even when f above is constant, as recalled in Remark B.74, for a fixedt ∈ [0, T ], the function

[0, t] → UCb(H), s → S(t− s)f

may not be measurable if the semigroup S(t) is not strongly continuous.

Results on uniqueness and regularity of mild solutions and their relationshipwith other concepts of solutions (in particular approximations by classical solutions)are contained in [82, 76] for the case of K-continuous semigroups and in Chapter 7of [387, 389] for π-continuous semigroups. They are applied mainly to obtain suit-able approximations of mild solutions of Kolmogorov equations which are used inChapter 4. Such results are presented in Section B.7 for the Kolmogorov equationswe are interested in.

5Indeed as remarked in [82], Remark 2.6, this is equivalent to requiring that points (iii) and(iv) of Definition B.64 be satisfied.

6Clearly, extending the concepts of π-semigroups and K-semigroups it is possible to considerthe same problem in the spaces Cb(H), Cm(H), UCm(H), Bm(H).


Several other results on π-continuous and K-continuous semigroups are ob-tained in [386, 387, 389] and [75, 76, 82]. We recall in particular the possibility ofproving an analogue of Hille-Yosida theorem in both cases.

B.6. Approximation of continuous functions through K-convergence

Recall that H is a real separable Hilbert space. When dimH = +∞, thespace UC2

b (H) is not dense in UCb(H) (see [354]). We refer to Appendix D.3 formore on approximations in Hilbert spaces. However, in many cases (in particularto prove verification theorems for optimal control problems), we need to be ableto approximate functions in Cm(H), m ≥ 0, (or UCm(H)) by smooth (at leastC2) functions with special properties. We can substitute uniform convergence byπ-convergence or K-convergence which are good enough to apply the dominatedconvergence theorem. We start with the following lemma which is a small variationof Lemma 5.2 of [82].

Lemma B.76 Let A be the generator of the strongly continuous semigroup S(t) =etA in H. Let Jn as in (B.7) and let Pnn∈N be a sequence of finite dimensionalorthogonal projections on H strongly convergent to the identity operator. Then, forall compact subsets I0 ⊂ R and K ⊂ H, the sets

R1(K) =: Pnx : x ∈ K, n ∈ N ⊂ H,

R1(I0 ×K) =: (t, Pnx) : t ∈ I0, x ∈ K, n ∈ N ⊂ R×H,

R2(K) =: PnJnx : x ∈ K, n ∈ N ⊂ H

are relatively compact.

Proof. We show the claim for R2(K). The argument for R1(K) andR1(I0 ×K) is the same and easier. Let

Pnj

Jnjxj

j∈IN

be a sequence in R2(K).From compactness of K it follows that there exists an increasing subsequence jkand x ∈ K such that

xjk → x, as k → +∞.

If there exists C > 0 such that njk ≤ C, for all jk, then we can suppose njk = n,for all jk and then

limk→+∞

PnjkJnj

kxjk = PnJnx.

Otherwise, let us suppose limk→+∞ njk = +∞. Then, using (B.9) and (B.10),

|PnjkJnj

kxjk − x| ≤ |Pnj

kJnj

k(xjk − x)|+ |Pnj

k(Jnj

kx− x)|+ |Pnj

kx− x|

≤ Mnjk

njk + ω|xjk − x|+ |Jnj

kx− x|+ |Pnj

kx− x| → 0 as k → +∞.

The following lemma generalizes Lemma 2.6, page 25 in [225].

Lemma B.77 Let m ≥ 0 and let I ⊆ R be an interval. Then FC∞0 (H) is K-dense

in Cm(H) and FC∞0 (I ×H) is K-dense in Cm(I ×H). Moreover, for every closed

operator B with dense domain we also have that FC∞,B0 (H) is K-dense in Cm(H)

and FC∞,B0 (I ×H) is K-dense in Cm(I ×H).

Proof. Take an orthonormal basis E = (en)n∈N of H and, for x ∈ H, letx =

∞i=1 xiei. For every n ∈ N let Pn be the orthogonal projection onto the n

dimensional subspace of H spanned by e1, . . . en. Define

Πn : H → Rn, Πx = (x1, . . . , xn),

Qn : Rn → H, Qn(x1, . . . , xn) = x1e1 + · · ·+ xnen,

B.6. APPROXIMATION OF CONTINUOUS FUNCTIONS THROUGH K-CONVERGENCE 557

and recall that Pn = Qn Πn. Let ϕ ∈ Cm(H). Given a family of C∞ mollifiersηk : Rn → R with support in B(0, 1/k), we define the regularizing convolutions (ase.g. [379])

ψnk (x) =

Rn

ϕ(Qny)ηk(Πnx− y)dy =

Rn

ϕ(Pnx−Qny)ηk(y)dy.

Observe that, by the definition, we have for x ∈ H,

|ψkn(x)| ≤ sup

|x1|≤|Pnx|+1/k|ϕ(x1)| ≤ sup

|x1|≤|x|+1/k|ϕ(x1)|,

so|ψn

k (x)|(1 + |x|2)m/2

≤ sup|x1|≤|x|+1/k

|ϕ(x1)|

(1 + |x1|2)m/2

(1 + |x1|2)m/2

(1 + |x|2)m/2

≤ ϕCm(H) sup|x1|≤|x|+1/k

(1 + |x1|2)m/2

(1 + |x|2)m/2.

Now it is easy to see that

sup|x1|≤|x|+1/k

1 + |x1|21 + |x|2 = 1 + ρ(1/k)

for some modulus ρ. Hence we get

ψnk Cm(H) ≤ ϕCm(H)(1 + ρ(1/k))m/2. (B.23)

Now, from the properties of finite dimensional convolutions, we easily observethat, setting ξn(x) := ϕ(Pnx), the sequence ψn

k k∈N converges to ξn uniformlyon bounded sets of Rn. For every n ∈ N, let k(n) ∈ N be such that

sup|x|≤n

|ψnk(n)(x)− ξn(x)| ≤

1

n. (B.24)

Therefore, if we set ψn = ψnk(n), we have, for any compact set K ⊂ H and n ≥

supx∈K |x|,supx∈K

|ψn(x)− ϕ(x)| ≤ supx∈K

|ψn(x)− ξn(x)|+ supx∈K

|ξn(x)− ϕ(x)|

≤ 1

n+ sup

x∈K|ξn(x)− ϕ(x)|. (B.25)

By Lemma B.76 and the continuity of ϕ it follows that, for all x ∈ K and n ∈ N,

|ξn(x)− ϕ(x)| ≤ ρR1(K)|Pnx− x|, (B.26)

where ρR1(K) is the modulus of continuity of ϕ over the compact set R1(K). Sincelimn→+∞ supx∈K |Pnx− x| = 0, it then follows from (B.23) and (B.25), that

K- limn→+∞

ψn = ϕ. (B.27)

To end the proof of the first statement in the time independent case it is enoughto define ϕn(x) := ψn(x)θ(|Pnx|2/n2), where θ ∈ C∞(R) is a cut-off function suchthat 0 ≤ θ ≤ 1, θ(r) = 1 for |r| ≤ 1 and θ(r) = 0 for |r| ≥ 2.

Consider now the time dependent case. Let f ∈ Cm(I ×H). First let I = R,and define fn following the same procedure used for ϕn. Given a family of C∞

mollifiers ηk : R × Rn → R with support in B(0, 1/k) ⊂ R × Rn, we define theregularizing convolutions

gnk (t, x) =

Rn+1

f(s,Qny)ηk(t− s,Πnx− y)dsdy

=

Rn+1

f(t− s, Pnx−Qny)ηk(s, y)dsdy.


Similarly to the time independent case we prove that

gnk Cm(I×H) ≤ fCm(I×H)(1 + ρ(1/k))m/2. (B.28)

Furthermore we define hn(t, x) := f(t, Pnx) and gn(t, x) = gnk(n)(t, x) as we did inthe time independent case for ξn and ψn, respectively. Hence we obtain

K- limn→+∞

gn = f. (B.29)

Then, defining fn(t, x) := gn(t, x)θ(|Pnx|2/n2) with θ as in the time independentcase, we get the claim.

If I = [a, b] for a < b ∈ R we first define f(t, x) = f(a, x) for t < a andf(t, x) = f(b, x) for t > b. We then take the approximations fn defined above andfinally define fn as the restrictions of fn to I × H. In other cases, we first takea sequence of suitable approximations. For example, if I = (a, b) is open we firstdefine fh on R×H for h > 4/(b−a) by fh(t, x) = χh(t)f(t, x), where χ is continuous,0 ≤ χh ≤ 1,χh(t) = 1 for t ∈ [a+2/h, b− 2/h], χh(t) = 0 for t ∈ [a+1/h, b− 1/h].Then for every h we approximate the functions fh by a sequence fh,nn chosen asabove. The diagonal sequence fnn := fn,nn K-converges to f . Indeed, givenI0 and K compact subsets of (a, b) and H, respectively, we have

supI0×K

|fn,n(t, x)− f(t, x)| ≤ supI0×K

|fn,n(x)− fn(x)|+ supI0×K

|fn(x)− f(x)|.

The second term of the right hand side converges to 0 since, by construction, fnK-converges to f ; for the first term we observe that, replicating (B.25) in thiscase, it remains to estimate supI0×K |fn(t, Pnx) − fn(t, x)|. By construction, if nis sufficiently large, then fn = f on I0 ×K. Hence, arguing as in (B.26) and usingthe compactness of R1(I0 ×K) from Lemma B.76, we get the claim.

Regarding the second statement, it is enough to take an orthonormal basis Ewhich is contained in D(B), it always exists since D(B) is dense.

B.7. Approximation of solutions of Kolmogorov equations throughK-convergence

Let H be a real separable Hilbert space and T > 0. We consider the followinginitial value problem for a linear Kolmogorov equation7

ut =1

2Tr [QD2u] + x,A∗Du+ Cu+ f(t, x) = 0, (t, x) ∈ (0, T ]×H,

u(0, x) = ϕ(x), x ∈ H,(B.30)

where C ∈ R, ϕ ∈ C(H), f ∈ C((0, T ]×H) and A, Q satisfy the following.

Hypothesis B.78(i) A is the infinitesimal generator of a strongly continuous semigroup

etA , t ≥ 0 on H with etA ≤ Meωt for all t ≥ 0 for given M ≥ 1,ω ∈ R.

(ii) Q ∈ L+(H) is such that, setting, for all x ∈ H, Qtx = t0 esAQesA

∗

x ds,the operator Qt is nuclear for all t ≥ 0.

7Often (see e.g. [129] Chapter 6) the term x,A∗Dv is replaced by the term Ax,Dv that is

well defined if x ∈ D(A). Here we use the former expression because we will look for more regularsolutions, having the derivative in D(A∗) (see the notion of classical solution used in Section 6.2of [129]).

B.7. APPROXIMATION OF SOLUTIONS OF KOLMOGOROV EQUATIONS THROUGH K-CONVERGENCE559

If we call, formally, A to be the operator associated to the second and thirdterms of (B.30) and, still formally, Rt to be the corresponding semigroup, then wecan rewrite (B.30) in the following mild form

u(t, x) = eCtRt[ϕ](x) +

t

0eC(t−s)Rt−s[f(s, ·)](x)ds (B.31)

and call the function on the right hand side the mild solution of (B.30). In whatfollows we provide some useful approximation results for solutions of such equations.Similar results are available in the literature (see e.g. [82, 233, 387]) but only whenthe data ϕ and f are bounded and in a slightly different setting. Here we allow thedata to have polynomial growth and take a setting which is more suitable for thepurpose of this book (see Remark B.95 for more on this).

In the next three subsections we first present (Subsection B.7.1) suitable defi-nitions of classical and strong solutions of (B.30); then (Subsection B.7.2) we defineprecisely Rt, the mild solutions, and connect the generator A of Rt with (B.30)through the operators A0 and A0; finally, (Subsection B.7.3) we show a usefulapproximation result.

B.7.1. Classical and strong solutions of (B.30). Let us introduce theoperator A0 on C(H) as follows (compare it with the operators A1 in Section 4.5and A0 in Section 4.6):

D(A0) =φ ∈ UC2

b (H) : A∗Dφ ∈ UCb(H,H), D2φ ∈ UCb(H,L1(H))

A0φ = 12Tr [QD2φ] + x,A∗Dφ .

(B.32)We also consider the restriction A0 ⊆ A0 defined as in [82][Section 5] (see also[389][Section 5] or [387][Section 7.4.3]),

D(A0) =φ ∈ UC2

b (H) : A∗Dφ ∈ UCb(H,H),

x,A∗Dφ ∈ UCb(H) and D2φ ∈ UCb(H,L1(H))

A0φ = 12Tr [QD2φ] + x,A∗Dφ .

(B.33)

Remark B.79 The operator A0 can be seen as an unbounded operator on Cm(H)for all m ≥ 0 since, by the definition of D(A0), one easily sees that A0φ ∈ Cb(H)(and so it also belongs to Cm(H)) for all φ ∈ D(A0). On the other hand thisis not the case for A0. Indeed, for a generic φ ∈ D(A0) we can only say thatA0φ ∈ Cm(H) for m ≥ 1 since the term x,A∗Dφ may be unbounded. The sameconsiderations holds if we consider the operators in UCm(H).

Both operators are used to define suitable approximations of the solution of(B.30). In most of the literature only A0 is used. Here we see that, for approxima-tion purposes, also A0 can be used with some advantages.

We endow D(A0) with the norm

φD(A0) := φ0 + Dφ0 + A∗Dφ0 + supx∈H

D2φ(x)L1(H), (B.34)

while in D(A0) we take the norm

φD(A0):= φ0 + Dφ0 + A∗Dφ0 + x,A∗Dφ 0 + sup

x∈HD2φ(x)L1(H).

(B.35)


Arguing as in Theorem 2.7 of [126], it can be proved that both D(A0) and D(A0)with the above norms are Banach spaces8. However the Banach structure is notessential for our purposes even if it simplifies the notation.

We now give two notions of classical solutions of (B.30). The first, a morerestrictive one, uses the operator A0, and is in line with what is done e.g. inDefinition 4.6 of [233] or Definition 4.1 of [234].

Definition B.80 u ∈ Cb([0, T ]×H) is a classical solution of (B.30) in D(A0)if

u(·, x) ∈ C1([0, T ]), ∀x ∈ H

u(t, ·) ∈ D(A0) for any t ∈ [0, T ] and supt∈[0,T ] u(t, ·)D(A0)< +∞

Du, D2u, A∗Du, A0u ∈ Cb([0, T ]×H)

(B.36)

and u satisfies, for every x ∈ H, equation (B.30).

The second definition is similar to Definition 4.87 and uses the operator A0.

Definition B.81 u ∈ Cb([0, T ]×H) is a classical solution of (B.30) in D(A0)if

u(·, x) ∈ C1([0, T ]), ∀x ∈ H

u(t, ·) ∈ D(A0) for any t ∈ [0, T ] and supt∈[0,T ] u(t, ·)D(A0) < +∞Du, D2u, A∗Du, ∈ Cb([0, T ]×H)

(B.37)

and u satisfies, for every x ∈ H, equation (B.30).

Remark B.82 Concerning the above definitions we observe the following.(1) If we compare these definitions with the one given in Section 6.2 of

[129], we see that the classical solutions in the sense of both Defini-tions B.80 and B.81 are more regular. Indeed the goal here is not (asin Section 6.2 of [129]) to prove that mild solutions are classical, butto approximate mild solutions with classical solutions to which we canapply Itô’s/Dynkin formula (see Section 1.7); hence we want the ap-proximating solutions to be as regular as possible.

(2) The definitions above are the same no matter if we work in the spacesCm(H) (as we mainly do) or UCm(H), (m ≥ 0).

(3) Note that, if u is a classical solution from Definition B.80, then A0u,and consequently also ut, belong to Cb([0, T ]×H). On the other hand,if u is a classical solution from Definition B.81, then A0u, and then alsout, only belong to C1([0, T ]×H).

(4) If we compare Definition B.80 with Definition 4.6 of [233] and Def-inition 4.1 of [234] we see that, in the last two, the functionsu, ut, Du,D2u,A∗Du, A0u are required to belong to UCx

b ([0, T ] × H).The reason is that elements of this space possess, roughly speaking,the maximal joint regularity one can expect even when f = 0, C = 0and ϕ ∈ UC∞

b (H). Indeed in such case the mild solution is u(t, x) =RT−tϕ(x) (where Rt is the Ornstein-Uhlenbeck semigroup correspond-ing to A and Q) which is in UCx

b ([0, T ]×H) but not in UCb([0, T ]×H),see Subsection B.7.2. Here we decided to ask for less joint regularity,i.e. u,Du,D2u,A∗Du ∈ Cb([0, T ] × H) since this last space is morecommonly used in the literature on PDEs in infinite dimensions and

8The definition of the domain of the operator studied in this paper is slightly different butthe arguments there can be adapted easily.


since all the results we need, in particular Itô’s/Dynkin formula, stillhold.

(5) The definitions implicitly require regularity of data. Indeed the secondrequirement of (B.36) implies that ϕ ∈ D(A0) (and similarly for (B.37))while, from the last requirement we see that necessarily f ∈ Cb([0, T ]×H).

We now pass to the definition of a strong solution i.e. approximation of classicalsolutions. It is substantially a variation of Definition 4.7 of [233] (see also Definition4.3 of [234]) in the sense that we use, as underlying space, the space Cm instead ofthe space UCb. The definition below is a special case of Definition 4.89 and usessome definitions given in Chapter 4 which, for the reader’s convenience, we repeatin the forms needed here.

We recall first the definition of the class of weights I1 given in (4.17):

I1 :=η(·) : (0,+∞) → (0,+∞) integrable on (0, T ) for all T > 0

. (B.38)

Then we recall (see (4.21)) that, given η ∈ I1, a function f : (0, T ]×H → Z (whereZ is a given real separable Hilbert space) belongs to Cm,η((0, T ]×H;Z) if

f ∈ Cm([τ, T ]×H;Z) ∀τ ∈ (0, T ) and η−1f ∈ Cm((0, T ]×H;Z). (B.39)When Z = R, we omit it as usual, simply writing Cm,η((0, T ] ×H). Furthermorewe give a variant of the definition of K-convergence (see Definition 4.88).

Definition B.83 Let m ≥ 0, η ∈ I1 and let Z be a real Hilbert space. We saythat a sequence fnn∈N ⊆ Cm,η((0, T ] × H;Z) K-converges to f ∈ Cm,η((0, T ] ×H;Z) if

supn∈N

fnCm,η((0,T ]×H;Z) < +∞,

limn→+∞

sup(t,x)∈(0,T ]×K

η(t)−1|fn(t, x)− f(t, x)| = 0, (B.40)

for every compact set K ⊂ H. In such case we write K-limn→+∞ fn = f inCm,η((0, T ]×H;Z).

Below are the definitions of K-strong solutions.

Definition B.84 Let m ≥ 0 and η ∈ I1. Let ϕ ∈ Cm(H) and f ∈ Cm,η((0, T ]×H). We say that a function u ∈ Cm([0, T ] × H) is a K-strong solution in D(A0)of (B.30) if u(t, ·) is Fréchet differentiable for any t ∈ (0, T ] and there exist threesequences un ⊂ Cb([0, T ] × H), ϕn ⊂ D(A0), fn ⊂ Cm,η((0, T ] × H) suchthat:

(i) For every n ∈ N, un is a classical solution in D(A0) (from DefinitionB.80) of

wt + A0w + Cw + fn = 0w(T ) = ϕn.

(B.41)

(ii) The following limits hold

K- limn→+∞

ϕn = ϕ in Cm(H)

K- limn→+∞

un = u in Cm([0, T ]×H)

and, for some η1 ∈ I1 (possibly different from η),

K- limn→+∞

fn = f in Cm,η((0, T ]×H)

K- limn→+∞

Dun = Du in Cm,η1((0, T ]×H;H).


Finally we say that a function u ∈ Cm([0, T ]×H) is a K-strong solution in D(A0)of (B.30) if all the above holds substituting D(A0) with D(A0).

Remark B.85 The spirit of this definition is substantially the one of the so-called strong solutions in Friedrichs sense (terminology dating back to [200]) forabstract Cauchy problems, see e.g. Definition 4.1.1 of [323]. It is useful to connectthe concept of mild solution with the one of classical solution proving that mildsolutions are indeed strong solutions. We remark two key points here: first, theconvergences in (ii) are asked to hold in the K-sense, and second, the convergenceof the derivatives is also required. The reason for the former is that uniform con-vergence in the whole H would be a requirement which is too strong as UC2

b (H)(and so D(A0)) is not dense in UCb(H) when H is infinite dimensional, see [354]and also [387], Remark 2.2.11. On the other hand the use of π-convergence (as in[389][Section 4] or [387][Chapter 7]) is possible but in the cases we treat here itwould give weaker results, as K-convergence can always be proved when data arecontinuous. The latter is motivated by the use of this definition for applicationsto HJB equations (as in Chapter 4) which are written in the mild form of (B.31)with f depending on u and Du (see e.g. (4.1) and (4.5)). Hence convergence of thederivatives allows to pass to the limit in such a mild form.

B.7.2. The Ornstein-Uhlenbeck semigroup associated to (B.30), themild solutions and their properties. Assume that Hypothesis B.78 holds andtake a generalized reference probability space µ = (Ω,F , Fss≥0,P,W ), where Wis a Wiener process on H with covariance operator I (the identity).

Consider the following linear stochastic equation on H

dX(t) = AX(t) dt+√QdW (t)

X(0) = x.(B.42)

Under Hypothesis B.78 the problem (B.42) has an H-valued mild solution fromDefinition 1.113 (see Theorem 1.144). Such solution is mean square continuous (seeProposition 1.138 or also [130][Theorem 5.2-(i)]) and is denoted by X(t, x). Wedenote by Rt, t ≥ 0 or simply by Rt, the corresponding transition semigroup (seeSection 1.6) on Bm(H):

Rtϕ(x) = E [ϕ(X(t, x))] =

Hϕ(y)N (etAx,Qt)(dy) =

Hϕ(y+etAx)N (0, Qt)(dy),

(B.43)where N (z,Q) is the Gaussian measure introduced in Definition 1.57. See Subsec-tions 4.3.1.1 and 4.3.1.2 for more on this.

We immediately see that, when ϕ ∈ Bm(H) and f ∈ Bm([0, T ] × H), thefunction u in (B.31) is well defined. Thus we can give the precise definition of themild solution of equation (B.30).

Definition B.86 Given ϕ ∈ Bm(H) and f ∈ Bm([0, T ] × H), the function ugiven by (B.31) is well defined on [0, T ]×H and is called the mild solution of (B.30).

We now explain the connection between equation (B.30) and the notion of themild solution. We start with the following result.

Proposition B.87 Under Hypothesis B.78, Rt, t ≥ 0, is a K-continuoussemigroup and a π-continuous semigroup in Cm(H) and UCm(H) (m ≥ 0). More-over it is not strongly continuous in these spaces unless A = 0. Finally, as t → 0+,we have Rtφ → φ in UCb(H) if and only if φ(etA·) → φ(·) in UCb(H).

Proof. For the K-continuity, see [75], Proposition 6.2, and [387][Section6.3.3], when m = 0. Moreover, see [76] for the UCm(H) case and Theorem 4.1 of


[225] for the case of Cm(H) (recalling the change in the definition mentioned inRemark B.66).

For the π-continuity, see [387][Section 6.3.3], and [386][p. 293-294]: there theproof is done in UCb(H) but it can be easily extended to the cases of Cm(H) andUCm(H).

Regarding the non-strong continuity, see [75][Example 6.1] and [387][Section6.3.3]. Finally, for the last statement, one can see [129], Proposition 6.3.1.

From now on Am will denote the generator of the transition semigroup Rt as aK-continuous semigroup in Cm(H) (or UCm(H) when specified), as defined in theprevious section. We first give a useful lemma.

Lemma B.88 Let Hypothesis B.78 be satisfied.(i) For all t ≥ 0, the operator Rt maps D(A0) into itself, and for all T > 0

there exists a constant LT > 0 such that

Rt[φ]D(A0) ≤ LT φD(A0), ∀ t ∈ [0, T ], φ ∈ D(A0). (B.44)

The same holds for D(A0).(ii) For all m ≥ 1 we have A0 ⊆ Am and, for any φ ∈ D(A0) we have

d

dt(Rtφ) = A0Rtφ = RtA0φ, t ≥ 0. (B.45)

Similarly, for all m ≥ 0 A0 ⊆ Am and, for any φ ∈ D(A0) we haved

dt(Rtφ) = A0Rtφ = RtA0φ, t ≥ 0. (B.46)

(iii) Let T > 0. For every f ∈ Cb([0, T ] × H) such that, for all t ∈ [0, T ],f(t, ·) ∈ D(A0) and

f(t, ·)D(A0) ≤ g0(t)

for a suitable g0 ∈ L1(0, T ;R+), let us set

g(t, x) =

t

0Rt−s[f(s, ·)](x) ds, ∀x ∈ H. (B.47)

Then g ∈ Cb([0, T ] × H) and for every t ∈ [0, T ], g(t, ·) ∈ D(A0).Moreover

g(t, ·)D(A0) ≤ LT g0L1 , ∀ t ∈ [0, T ]. (B.48)

The same holds if we replace A0 by A0.(iv) Let m ≥ 0 and let λ > m(ω ∨ 0), where ω is given in Hypothesis B.78.

Then (λI − Am)−1 exists and it is a bounded operator. Moreover itmaps D(A0) into itself. The same is true for D(A0).

Proof. Proof of (i).If φ ∈ D(A0) we have, using the last equality of (B.43) and straightforward

computations, for x, h, k ∈ H,

DRt[φ](x), h =

H

Dφ(y + etAx), etAh

N (0, Qt)(dy) = Rt[

Dφ(·), etAh

],

(B.49)

A∗DRt[φ](x), h =

H

A∗Dφ(y + etAx), etAh

N (0, Qt)(dy) = Rt[

A∗Dφ(·), etAh

],

(B.50)D2Rt[φ](x)h, k

=

H

D2φ(y + etAx)etAk, etAh

N (0, Qt)(dy) = Rt[

D2φ(·)etAk, etAh

].

(B.51)


and, for a given orthonormal basis en,

TrD2Rt[φ](x)

=

+∞

n=0

D2Rt[φ](x)en, en

=

H

+∞

n=0

D2φ(y + etAx)etAen, e

tAenN (0, Qt)(dy).

(B.52)Hence by simple computations, we get that Rt[φ] ∈ D(A0) and

Rt[φ]D(A0) ≤ φ0 +Meωt[Dφ0 + A∗Dφ0] +M2e2ωtDφL1(H),

which gives the claim. In the case of D(A0) we also need to estimate the termA∗DRt[φ](x), x. We have, thanks to (B.50),

A∗DRt[φ](x), x =

H

A∗Dφ(y + etAx), y + etAx

N (0, Qt)(dy)

−

H

A∗Dφ(y + etAx), y

N (0, Qt)(dy).

Since, by the Hölder inequality and by Proposition 1.58,

H

A∗Dφ(y + etAx), y

N (0, Qt)(dy)

≤ A∗Dφ0(Tr(Qt))1/2,

we obtain

supx∈H

| < A∗DRt[φ](x), x > | ≤ supx∈H

| < A∗Dφ(x), x > |+ A∗Dφ0(Tr(Qt))1/2.

which gives the claim.Proof of (ii).From Dynkin’s formula of Proposition 1.159 we easily get that (here X(s, x) is

the solution of (B.42))Rh[φ](x)− φ(x)

t=

1

hE (φ(X(h, x))− φ(x)) =

=1

hE h

0

1

2Tr [QD2φ(X(s, x))] + A∗Dφ(X(s, x)), X(s, x)

ds.

It thus follows from the dominated convergence theorem that

limh→0

Rh[φ](x)− φ(x)

t= A0φ(x)

which, by the definition of generator (Definition B.70), implies A0 ⊂ Am and A0 ⊂Am, too. Now equations (B.45) and (B.46) follow immediately from PropositionB.73-(iii) and from point (i) of this proposition.

Proof of (iii).9 Since f ∈ Cb([0, T ]×H), using the first equality in (B.43) andthe mean square continuity of X(t, x), one can prove exactly as in Proposition 4.38-(ii), that also g ∈ Cb([0, T ]×H). Moreover, since f(t, ·) ∈ D(A0), for all t ∈ [0, T ],by part (i) it follows that, for 0 ≤ s ≤ t ≤ T , Rt−s[f(s, ·)] ∈ D(A0) and

Rt−s[f(s, ·)]D(A0) ≤ LT f(s, ·)D(A0) ≤ LT g0(s).

Now, using that the derivative operator is closed, we get that g(t, ·) is Fréchetdifferentiable and, for t ∈ [0, T ], x, h ∈ H,

Dg(t, x), h = t

0DRt−s[f(s, ·)](x), h ds =

t

0Rt−s

Df(s, ·), e(t−s)Ah

(x)ds,

(B.53)where we exploited (B.49) in the last equality. The integrand is Borel measurable ins since it is the limit of the difference quotients; the continuity of Dg(t, x) is proved

9This proof is partially similar to the proof of Lemma 4.5 of [389].


again as in Proposition 4.38-(ii). Similarly, using (B.50) we get for t ∈ [0, T ],x, h ∈ H,

A∗Dg(t, x), h = t

0A∗DRt−s[f(s, ·)](x), h ds =

t

0Rt−s

A∗Df(s, ·), e(t−s)Ah

(x)ds.

(B.54)Moreover, iterating the argument and exploiting (B.49) in the last equality, weprove that g(t, ·) is twice Fréchet differentiable and, for t ∈ [0, T ], x, h, k ∈ H,D2g(t, x)h, k

=

t

0

D2Rt−s[f(s, ·)](x)h, k

ds =

t

0Rt−s

D2f(s, ·)e(t−s)Ah, e(t−s)Ak

(x)ds.

(B.55)The integrand is Borel measurable in s since it is the limit of the difference quotientsand the continuity of D2g(t, x)h for each h is proved again as in Proposition 4.38-(ii). This does not even imply the measurability of Dg(t, x) in general, but it isenough to prove the continuity of the trace. Indeed

Tr(D2g(t, x)) =+∞

n=0

D2g(t, x)en, en

= (B.56)

t

0

H

+∞

n=0

D2f(s, y + etAx)e(t−s)Aen, e

(t−s)AenN (0, Qt−s)(dy)ds.(B.57)

As before the integrand is Borel measurable in (s, y), hence continuity can be provedas in Proposition 4.38-(ii).

In the case of D(A0) we also need to estimate the term A∗Dg(t, x), x. Wehave, thanks to (B.54),

A∗Dg(t, x), x =

t

0A∗DRt−s[f(s, ·)](x), x ds =

t

0Rt−s

A∗Df(s, ·), e(t−s)Ax

(x)ds

=

t

0

H

A∗Df(s, y + e(t−s)Ax), y + e(t−s)Ax

N (0, Qt−s)(dy)ds

− t

0

H

A∗Dφ(y + e(t−s)Ax), y

N (0, Qt−s)(dy)ds.

The conclusion now follows as in part (i) using Hölder’s inequality and Proposition1.58.

Proof of (iv). Recall first that, for any φ ∈ Cm(H), x ∈ H, the functiont → e−λtRt[φ](x) is integrable when λ > m(ω ∨ 0),due to the first estimate ofTheorem 4.32. Thus by Remark B.71, we have that λ is in the resolvent set of Am

and

(λI −Am)−1φ =

+∞

0e−λtRt[φ](x)dt.

Taking φ ∈ D(A0) the required conclusion follows using the same arguments asthese in the proof of part (iii), which here are even easier since we do not have thetime dependency of the integral.

We now pass to the following result.

Proposition B.89 Let Hypothesis B.78 be satisfied and let T > 0.(i) Let ϕ ∈ D(A0) and f ∈ Cb([0, T ] × H) such that, for all t ∈ [0, T ],

f(t, ·) ∈ D(A0) and

f(t, ·)D(A0) ≤ g0(t)

for a suitable g0 ∈ L1(0, T ;R+). Then the function u defined in (B.31)is a classical solution in D(A0) of (B.30).


(ii) If the assumption of point (i) hold with D(A0) in place of D(A0) thenthe function u defined in (B.31) is a classical solution in D(A0) of(B.30).

Proof. We take C = 0 as the case C = 0 can be obtained by a straightfor-ward change of variable. We start proving (i). As a first step we need to prove thatu satisfies the regularity required in (B.37). Indeed, by Lemma B.88, in particular(B.44) and (B.48), it immediately follows that the second line of (B.37) is true. Thethird line of (B.37) is obtained, for the first term in (B.31), from (B.49), (B.50),(B.51), (B.52), and for the convolution term, from (B.53), (B.54), (B.55), (B.56).Hence continuity is immediately deduced. The first line of (B.37) follows from thenext computation which also shows that u satisfies (B.30).

We apply Dynkin’s formula of Proposition 1.159 to the process ϕ(X(s, x)) get-ting, on the interval [0, t],

Eϕ(X(t, x)) = ϕ(x) + E T

tA∗Dϕ(X(s, x)), X(s, x) ds

+1

2Tr

QD2ϕ(X(s, x))

ds. (B.58)

Now we compute the left derivative of u at t = 0: first we observe that, by thedefinition of u,

u(h, x)− u(0, x)

h=

Rh[ϕ](x)− ϕ(x)

h+

1

h

h

0Rh−s[f(s, ·)](x)ds.

By the continuity of the integrands in (B.58) we obtain that, when h 0, the firstterm of the above right hand side converges to

A∗Dϕ(x), x+ 1

2Tr[QD2ϕ(x)]

Similarly by the continuity of the integrand in the second term of the right hand sidewe get that this term, when h 0, converges to f(0, x). We then have, denotingby D+

t the right time derivative,

D+t u(0, x) = lim

h0

u(h, x)− u(0, x)

h=

= A∗Dϕ(x), x+ 1

2Tr[QD2ϕ(x)] + f(0, x)

so the equation is satisfied for t = 0. For t > 0 we observe that, by the semigroupproperty (see Theorem 1.152),

u(t+ h, x) = Rh[u(t, ·)](x) + t+h

tRt+h−s[f(s, ·)](x)ds,

hence we haveu(t+ h, x)− u(t, x)

h=

Rh[u(t, ·)](x)− u(t, x)

h+

1

h

t+h

tRt+h−s[f(s, ·)](x)ds

and, arguing as for the case t = 0 but replacing ϕ by u(t, ·), we get

D+t u(t, x) = x,A∗Du(t, x)+ 1

2Tr

QD2u(t, x)

+ f(t, x).

Now the right hand side of the above identity is a continuous function on [0, T ]×Hand consequently, by Lemma 3.2.4 of [129], un(·, x) is continuously differentiableand satisfies equation (B.30).

The proof of part (ii) is almost exactly the same. The only difference is that,thanks to the regularity of the data, also A0u and ut are bounded.


B.7.3. The approximation results. We start with the following resultabout operators A0, A0 and Am (recall that ω below is the one given by HypothesisB.78).

Proposition B.90 Let Hypothesis B.78 hold. We have the following:

(i) For any m ≥ 1, we have A0 ⊆ Am, and D(A0) is a K-core for Am

in Cm(H). In particular A0 is K-closable in Cm(H) and its K-closureA0

K coincides with Am.Moreover, for all m ≥ 0 and λ > m(ω ∨ 0), the set

(λI −Am)−1FC∞,A∗

0 (H):=

φ ∈ Cb(H) : (λI −Am)φ ∈ FC∞,A∗

0 (H)

is contained in D(A0) and it is always a core for Am.Finally, the set FC∞,A∗

0 (H) ⊆ D(A0) is always a core for Am whenm ≥ 1, but not, in general, when m ∈ [0, 1).

(ii) For any m ≥ 0, we have A0 ⊆ Am, D(A0) is a K-core for Am in

Cm(H) In particular A0 is K-closable in Cm(H) and its K-closure A0

K

coincides with Am.Moreover, for all m ≥ 0 and λ > m(ω ∨ 0), the set (λI −Am)−1

FC∞,A∗

0 (H)

is contained in D(A0) and it is always a corefor Am when m ≥ 0.Finally the set FC∞,A∗

0 (H) is in general not contained in D(A0).

Proof. Proof of part (i).For any m ≥ 1, we have A0 ⊆ Am by Lemma B.88-(ii). To prove that D(A0) is

a K-core for Am in Cm(H) we take any φ ∈ D(Am) and set, for some λ > m(ω∨0)g := (λI − Am)φ ∈ Cm(H). Then we consider the approximating sequence gnin FC∞,A∗

0 (H) ⊆ D(A0) given by Lemma B.77. By construction we have gnK→ g.

Define φn := (λI − Am)−1gn. By Lemma B.88-(iv) we have φn ∈ D(A0) and, bythe dominated convergence theorem, we get φn

K→ φ. Moreover

A0φn = Amφn = λφn − (λI −Am)φn = λφn − gn,

hence also A0φnK→ Amφ.

Clearly FC∞,A∗

0 (H) ⊆ D(A0) and, since (λI−Am)(D(A0)) ⊆ D(A0) (LemmaB.88- (iv)), then it must be


0 (H)⊆ D(A0).

The choice of the sequence above implies that such set is a core for Am.The fact that the set FC∞,A∗

0 (H) is a core for Am when m ≥ 1 follows usingLemma 4.4 in [225]. This lemma generalizes a well known result about cores (seee.g. [160][Proposition 1.7, p.53] implying that a K-dense subspace D of Cm(H)

which is invariant for Rt is always a core. Indeed let φ ∈ FC∞,A∗

0 (H) and letf : Rn → R and x1, . . . xn ∈ D(A∗) be such that

φ(x) = f(x, x1 , . . . , x, xn).


Then

Rt[φ](x) =

Hφ(y + etAx)N (0, Qt)(dy)

=

Hfy + etAx, x1

, . . . ,

y + etAx, xn

N (0, Qt)(dy)

=

Hfy, x1+

x, etA

∗

x1

, . . . , y, xn+

x, etA

∗

xn

N (0, Qt)(dy)

= g

x, etA∗

x1

, . . . ,

x, etA

∗

xn

for a suitable function g ∈ FC∞,A∗

0 (H). Since x1, . . . xn ∈ D(A∗) we haveetA

∗

x1, . . . etA∗

xn ∈ D(A∗) and the claim is proved.However when m ∈ [0, 1) it is not true in general that, for φ ∈ FC∞,A∗

0 (H),one has A0φ ∈ Cm(H) so the core property in general fails contrary to what isstated in[225][Theorem 4.5] (see on this [392][Remark 5.11]).

The proof of part (ii) is exactly the same except for the fact that, since ingeneral FC∞,A∗

0 (H) ⊆ D(A0), we need to prove directly that


0 (H)⊆ D(A0).

In fact we can prove more, as in Proposition 4.6 of [389], that

(λI −Am)−1 (D(A0)) ⊆ D(A0). (B.59)

To do this it is enough to show that, given φ ∈ (λI − Am)−1 (D(A0)), the mapx → A∗Dφ(x), x is bounded, which is equivalent to prove that A0φ is bounded,since the second order term is always bounded when φ ∈ D(A0). Let f ∈ D(A0)and let φ := (λI −Am)−1f . Then f = (λI −Am)φ. Since φ ∈ D(A0) we also havef = (λI −A0)φ which gives A0φ = λφ− f . Since the right hand side is bounded,then also A0φ is bounded.

We are going to use, in addition to Hypothesis B.78, the following assumption(compare it with Hypotheses 4.22 and 4.25).

Hypothesis B.91 The operators A and Q satisfy the following:

(i) For all t > 0 we have etA(H) ⊆ Q1/2t (H).

(ii) Defining Γ(t) := Q−1/2t etA we have that the map t → Γ(t) (which is

always decreasing) belongs to I1.Remark B.92 If Hypothesis B.91 holds, then the semigroup Rt enjoys thesmoothing property stated in Theorem 4.32 with the estimate, for all f ∈ Cm(H),

DRtfCm(H) ≤ C(m)em(ω∨0)tΓ(t) fCm(H). (B.60)

for some constant C(m) ≥ 1. In particular Hypothesis B.91 implies that, for m ≥ 0,D(Am) ⊆ C1

m(H). Indeed let φ ∈ D(Am). Then, taking λ > m(ω ∨ 0) we musthave, for some f ∈ Cm(H),

φ(x) = (λI −Am)−1f(x) =

+∞

0e−λtRtf(x)dt, x ∈ H.

Estimate (B.60) and the closedness of the derivative operator imply then

Dφ(x) =

+∞

0e−λtDRtf(x)dt, x ∈ H.


This fact also implies that the approximating sequence φn of elements in D(A0)in the proof of the previous proposition can be chosen such that

φnK→ φ, A0φn

K→ Aφ, DφnK→ Dφ.

Here is the final result.

Theorem B.93 Let Hypotheses B.78 and B.91 hold. Let m ≥ 0 and η(t) =Γ(t) for t > 0.

(i) Let ϕ ∈ Cm(H) and f ∈ Cm,η((0, T ] × H). Then the mild solutionu (given by (B.31)) of the Cauchy problem (B.30) is also a K-strongsolution of (B.30) both in D(A0) and in D(A0).

(ii) If in addition ϕ ∈ C1m(H), f ∈ Cm([0, T ] × H), f is differentiable in

the x variable and Df ∈ Cm,η((0, T ] × H;H), then the approximatingsequences un,ϕn, fn defining the K-strong solution u in part (i) (bothin D(A0) and in D(A0)) can be chosen such that, for some η1 ∈ I1,

K- limn→+∞

Dϕn = Dϕ, in Cm(H;H),

K- limn→+∞

Dfn = Df, in Cm,η((0, T ]×H;H),

K- limn→+∞

Dun = Du, in Cm((0, T ]×H;H),

K- limn→+∞

D2unh = D2uh in Cm,η1((0, T ]×H;H), ∀h ∈ H.

Proof. A similar result is proved in [233][Proposition 4.10] when m = 0.Here we give a complete proof.

Proof of (i). We first approximate ϕ by a sequence ϕn ⊂ FC∞,A∗

0 (H) givenby Lemma B.77. Moreover setting f = η−1f we approximate f ∈ Cm((0, T ] ×H)

by a sequence fn ⊂ FC∞,A∗

0 ((0, T ]×H) given by Lemma B.77. We then definefn := ηfn and set

un(t, x) := Rt[ϕn](x) +

t

0Rt−s[fn(s, ·)](x)ds (B.61)

By construction we know that ϕn and fn satisfy the assumptions of part (i) ofProposition B.89 and so un is a classical solution of (B.30) in D(A0). Moreover,still by construction, we have

K- limn→+∞

ϕn = ϕ, in Cm(H),

K- limn→+∞

fn = f, in Cm,η((0, T ]×H).

By Proposition 4.38 (i) and (ii) we get u ∈ Cm([0, T ]×H) while using dominatedconvergence we obtain K- limn→+∞ un = u in Cm([0, T ] ×H). Up to now we didnot use Hypothesis B.91. We use it now to show that, as required, u(t; ·) is Fréchetdifferentiable for any t ∈ (0, T ] and that Dun K-converges to Du. Indeed, usingProposition 4.38, Remark 4.39 and the closedness of the derivative operator we getthe required differentiability and that

Du(t, x) = DRt[ϕ](x) +

t

0DRt−s[f(s, ·)](x)ds.


Note that Du ∈ Cm,η1([0, T ]×H;H), where

η1(t) = η(t) ∨ t

0η(s)η(t− s)ds

.

Finally the required convergence of Dun to Du in Cm,η1([0, T ]×H;H) follows usingthe representation formula (4.62) for the derivatives of Rt[ϕ] and Rt−s[f(s, ·)], andthen applying the dominated convergence theorem.

The above approximating sequences un,ϕn, fn are not suitable, in general, foru to be a K-strong solution in D(A0) since the sequences ϕn and fn(t, ·) may notbelong to D(A0), as we discussed in the proof of Proposition B.90. To get anapproximating sequence un for K-strong solution in D(A0) it is enough to define,as in [389][Section 4],

un(t, x) := Jnun(t, x) = Rt[Jnϕn](x) +

t

0Rt−s[Jnfn(s, ·)](x)ds,

where, as in (B.7), we set Jn := n(nI −Am)−1. By Lemma B.88-(iv) we easily seethat un(t, ·) ∈ D(A0) for all t ∈ [0, T ]. All the other required properties can beproved exactly as we did for un.

Proof of (ii). Assume now that, in addition, ϕ ∈ C1m(H), f ∈ Cm([0, T ]×H),

f is differentiable in the x variable and Df ∈ Cm,η((0, T ]×H;H). Take sequencesϕn ⊂ FC∞,A∗

0 (H) and fn ⊂ FC∞,A∗

0 ((0, T ] × H) given by Lemma B.77 andtake un given by (B.61). It is not difficult to see, looking at the proof of LemmaB.77, that for such sequences we also have, under the assumptions on ϕ and f ,

K- limn→+∞

Dϕn = Dϕ, in Cm(H;H),

K- limn→+∞

Dfn = Df, in Cm,η((0, T ]×H;H)

To prove the regularity of u and the remaining convergences we observe first that,using (B.49) and (B.53), we have, for t ∈ (0, T ], x, h ∈ H,

DRt[ϕ](x), h = Rt[Dϕ(·), etAh

], (B.62)

D

t

0Rt−s[f(s, ·)](x)ds, h

=

t

0Rt−s

Df(s, ·), e(t−s)Ah

(x)ds, (B.63)

so the required regularity of u follows again using the smoothing property and theestimates of Theorem 4.32. On the other hand the formulae (B.62) and (B.63) alsohold for ϕn and fn, hence the required convergences simply hold by applying thedominated convergence theorem as in the first part of the proof.

The proof of (ii) for the strong solution in D(A0) follows the same argumentsand replacing ϕn by Jnϕn and fn(s, ·) by Jnfn(s, ·), s ∈ (0, T ]. Remark B.94 In Definition B.84 (of strong solution), as noted in Remark B.85,we also required convergence of the derivatives. If we did not ask such convergence,then part (i) of Theorem B.93 would be true without using Hypothesis B.91, asnoted in the body of the proof. In such a case the first part of Theorem B.93may be seen as a particular case of a more general result for a class of abstractK-continuous semigroups (see [82] Theorem 4.10 for the case m = 0).

Remark B.95 We point out some general observations that can be extractedfrom the results of this section.

First of all, when one needs to approximate continuous functions in infinitedimensions by C2 functions through K-convergence, a straightforward way (not theonly one, see e.g. [389][Section 4] for an alternative approach) is to use cylindrical


functions (as in Lemma B.77). The use of cylindrical functions allows to find ap-proximating sequences required in the definition of strong solutions, as in TheoremB.93, however it is not adequate to find approximations belonging to D(A0) sincethe map

x → A∗Dφ(x), xmay be unbounded for φ ∈ FC∞,A∗

0 (H). This also implies, as observed in[392][Remark 5.11], that FC∞,A∗

0 (H) is in general not a K-core for Am, m ∈ [0, 1).Thus we decided to use separately the two operators A0 and A0. They have

their advantages and drawbacks. Indeed, using A0 allows to use directly the cylin-drical functions to find approximations to the mild solutions of (B.30) but, sincethe K-closure of A0 is Am only for m ≥ 1, it is not suitable to look at the generatorAm for m ∈ [0, 1) (Proposition B.90). On the other hand, using A0 calls for morecomplicated approximations but allows to study the generator Am for all m ≥ 0.For these reasons A0 is used when we want to find strong solutions to HJB equa-tions in Sections 4.5 and 4.7, while A0 is used when we use the generator Am in thespaces Cb(H) to study mild solutions to the infinite horizon problem in Subsection4.6.2.

APPENDIX C

Parabolic equations with non-homogeneousboundary conditions

In this section we show how to rewrite some classes of parabolic equationswith control and noise on the boundary using the infinite dimensional formalism.We focus on two particular cases: Dirichlet and Neumann boundary conditions.Throughout the section O will denote a bounded domain (open and connected) inRd with regular boundary ∂O.

We begin introducing and recalling some properties of the Dirichlet and Neu-mann maps.

C.1. Dirichlet and Neumann maps

Consider the Laplace equation with Dirichlet boundary condition

∆ξy(ξ) = 0, ξ ∈ O

y(ξ) = γ(ξ), ξ ∈ ∂O.(C.1)

Theorem C.1 Given s ≥ 0 and a boundary condition γ ∈ Hs(∂O), there existsa unique solution Dγ ∈ Hs+1/2(O) of the problem (C.1). Moreover, for all s ≥ 0,the operator

D : Hs(∂O) → Hs+1/2(O)

γ → Dγ(C.2)

is continuous.

Proof. See [313] Theorem 5.4, page 165, Theorem 6.6, page 177 and The-orem 7.3, page 187.

Definition C.2 The operator D introduced in (C.2) is called the Dirichlet map.

Proposition C.3 Consider the heat equation with zero Dirichlet boundary con-dition

∂∂sy(s, ξ) = ∆ξy(s, ξ), (s, ξ) ∈ (0, T )×O

y(s, ξ) = 0, (s, ξ) ∈ (0, T )× ∂O

y(0, ξ) = x(ξ), ξ ∈ O,

(C.3)

where the initial datum x belongs to L2(O). For s ≥ 0 denote by SD(s)x the solutionof (C.3) at time s. Then SD(t) is a C0-semigroup on L2(O). Its generator AD isgiven by:

D(AD) := H10 (O) ∩H2(O)

ADφ := ∆φ.(C.4)

AD is maximal dissipative, self-adjoint, invertible, A−1D ∈ L(L2(O)), and SD(t) is

analytic.

573

574 C. PARABOLIC EQUATIONS WITH BOUNDARY TERMS

Proof. This is a standard result. We refer for instance to Theorem 12.40of [397] for the proof that the C0-semigroup SD(t) is analytic and that A−1

D ∈L(L2(O)) (it is included in the definition of an analytic semigroup given there).

Proposition C.4 The Dirichlet map D introduced in (C.2) is continuous as alinear map between the spaces L2(∂O) and D((−AD)1/4−) for all > 0:

D : L2(∂O) → D((−AD)1/4−). (C.5)

Proof. See [308], Section 6.1.

Similarly we consider the following problem with Neumann boundary condition:

∆ξy(ξ) = λy(ξ), ξ ∈ O∂∂ny(ξ) = γ(ξ), ξ ∈ ∂O,

(C.6)

where n is the outward unit normal vector and ∂∂n is the normal derivative.

Theorem C.5 Let λ > 0. Given any s ≥ 0 and a boundary condition γ ∈Hs(∂O), there exists a unique solution Nλγ ∈ Hs+3/2(O) of the problem (C.6).Moreover, for all s ≥ 0, the operator

Nλ : Hs(∂O) → Hs+3/2(O)

γ → Nλγ(C.7)

is continuous.

Proof. See [313] Theorem 5.4, page 165, Theorem 6.6, page 177 and The-orem 7.3, page 187.

Definition C.6 The operator Nλ introduced in (C.7) is called the Neumannmap.

Proposition C.7 Consider the following heat equation with zero Neumannboundary condition

∂∂sy(s, ξ) = ∆ξy(s, ξ), (s, ξ) ∈ (0, T )×O∂∂ny(s, ξ) = 0, (s, ξ) ∈ (0, T )× ∂O

y(0, ξ) = x(ξ), ξ ∈ O

(C.8)

where the initial datum x belongs to L2(O). For s ≥ 0 denote by SN (s)x the solutionof (C.8) at time s. Then SN (t) is a C0-semigroup on L2(O). Its generator AN isgiven by

D(AN ) :=u ∈ H2(O) : ∂u

∂n = 0 on ∂O

ANφ := ∆φ.(C.9)

If λ ≥ 0 then (AN − λI) is maximal dissipative, self-adjoint, and et(AN−λI) isanalytic. If λ > 0 then (AN − λI) is invertible and (AN − λI)−1 ∈ L(L2(O)).

Proof. For the proof of analyticity, see for instance [2].

Proposition C.8 For any > 0, the Neumann map Nλ introduced in (C.7) iscontinuous as a linear map between the spaces L2(∂O) and D((−AN + λI)3/4−):

Nλ : L2(∂O) → D((−AN + λI)3/4−). (C.10)

Proof. See [308], Section 6.1.

C.2. NON-ZERO BOUNDARY CONDITIONS, THE DIRICHLET CASE 575

We remark that, see [308], Section 6.1, in fact we have

D((−AD)α) = H2α(O), 0 < α <1

4,

D((−AD)α) = H2α0 (O), 0 < α <

3

4,α = 1

4, (C.11)

D((−AN + λI)α) = H2α(O), 0 < α <3

4,λ > 0.

Moreover, for α ≥ 0,

|x|H2α(O) ≤ Cα|(−AD)αx|, x ∈ D((−AD)α), (C.12)

|x|H2α(O) ≤ Cα,λ|(−AN + λI)αx|, x ∈ D((−AN + λI)α), λ > 0. (C.13)We also recall the Sobolev embeddings (see e.g. [1], Theorem 7.57).

Theorem C.9 Let O be a domain in Rd with C1 boundary, s > 0, 1 < p < +∞.The following embeddings are continuous:

• If d > sp then W s,p(O) → Lq(O) for p ≤ q ≤ dp/(d− sp).• If d = sp then W s,p(O) → Lq(O) for p ≤ q < +∞.• If d < (s− j)p for some nonnegative integer j then W s,p(O) → Cj

b (O).

For more on Sobolev embeddings we refer to [1], Chapters V and VII.

C.2. Non-zero boundary conditions, the Dirichlet case

In this and in the following subsections, we show how to rewrite some classesof parabolic equations with control and noise on the boundary using the infinitedimensional formalism.

We will always work on the time interval [t, T ] where t and T are such that0 ≤ t < T . We consider the initial datum at time t instead of time 0 to be consistentwith the notation we use in Chapter 2.

We consider the following problem:

∂∂sy(s, ξ) = ∆ξy(s, ξ) + f(s, ξ), (s, ξ) ∈ (t, T )×O

y(s, ξ) = γ(s, ξ), (s, ξ) ∈ (t, T )× ∂O

y(t, ξ) = x(ξ), ξ ∈ O.

(C.14)

Until the end of the section, H and Λ denote respectively the Hilbert spaces L2(O)and L2(∂O), and AD is the operator defined in (C.4). Recall from Theorem C.1that the Dirichlet map D : Λ → H is continuous.

Lemma C.10 Assume that A is the generator of a C0-semigroup on H. Supposethat η ∈ W 1,1(t, T ;H), x ∈ D(A) and

z(s) = e(s−t)Ax+

s

te(s−r)Aη(r)dr, for s ∈ [t, T ].

Then z ∈ C1([t, T ], H) ∩ C([t, T ], D(A)) and

Az(s) = Ae(s−t)Ax+

s

te(s−r)A d

drη(r)dr + e(s−t)Aη(t)− η(s), for s ∈ [t, T ].

Proof. See [127] Lemma 13.2.2, Chapter 13, page 242.

Proposition C.11 Assume that y ∈ C∞([t, T ] × O) is a classical solution of(C.14). Then, denoting X(s) = y(s, ·), the solution can be written as

X(s) = e(s−t)ADx −AD

s

te(s−r)ADDγ(r)dr +

s

te(s−r)ADf(r)dr. (C.15)


Proof. We follow a well known procedure, see for instance [127] Chapter13 or [312] Chapter 9, Section 1.1.

Since y is smooth, by classical theory we have that Dγ(s) is smooth and more-over d

dsDγ(s) = D ddsγ(s) for t < s < T . In particular the function

z(s) := X(s)−Dγ(s) for s ∈ [t, T ] (C.16)is in C2((t, T );H) ∩ C([t, T ];D(AD)) and

d

dsz(s) =

d

dsX(s)− d

dsDγ(s) = ∆ξX(s)−∆ξDγ(s) +∆ξDγ(s)

− d

dsDγ(s) + f(s) = ADz(s)−D

d

dsγ(s) + f(s),

where in the last equality we used that ∆ξDγ(s) = 0. Observe that the expressionADz(s) is well defined because z(s) belongs to D(AD) while neither X(s) nor Dγ(s)are contained in D(AD). In particular, z(·) is a strict solution (see [35], Definition3.1, page 129) of the following evolution equation in H:

ddsz(s) = ADz(s)−D d

dsγ(s) + f(s)z(t) = x−Dγ(t).

(C.17)

The strict solution of (C.17) can be written in the mild form (see e.g. [35]Chapter II.1, Lemma 3.2, page 135)

z(s) = e(s−t)AD [x−Dγ(t)] +

s

te(s−r)AD

−D

d

drγ(r) + f(r)

dr.

Therefore

X(s) = z(s) +Dγ(s)

= e(s−t)AD [x−Dγ(t)] +

s

te(s−r)AD

−D

d

drγ(r) + f(r)

dr +Dγ(s). (C.18)

which, upon using Lemma C.10, yields

X(s) = e(s−t)ADx−AD

s

te(s−r)ADDγ(r)dr +

s

te(s−r)ADf(r)dr.

Observe that equation (C.15) can be seen as a mild form of equation

ddsX(s) = ADX(s)−ADDγ(s) + f(s)

X(t) = x ∈ H.(C.19)

Define for > 0GD := (−AD)1/4−D. (C.20)

By (C.5), GD ∈ L(Λ, H). Thus (if γ(·) ∈ L1(t, T ;Λ))

AD

s

te(s−r)ADDγ(r)dr =

s

t(−AD)3/4+e(r−t)ADGDγ(r)dr. (C.21)

Hence we can rewrite

X(s) = e(s−t)ADx− s

t(−AD)3/4+e(s−r)ADDγ(r)dr+

s

te(s−r)ADf(r)dr. (C.22)

Notation C.12 Expression (C.22) is called the mild form of equation (C.19)(and of (C.14)).

Notation C.13 We used the letter X for the unknown in the equation to beconsistent with the notation we use in Chapter 2 where the variable y is only usedfor the equation in the PDE form.

C.3. NON-ZERO BOUNDARY CONDITIONS, THE NEUMANN CASE 577

Let Q ∈ L+(H)1 and letΩ,F , F t

ss∈[t,T ] ,P,WQ

be a generalized reference

probability space.We consider now the following stochastic parabolic equation:

dy(s, ξ) = [∆ξy(s, ξ) + f(s, y(s, ξ))] ds+ g(s, y(s, ξ))dWQ(s)(ξ), on (t, T )×O

y(s, ξ) = γ(s, ξ), on (t, T )× ∂O

y(t, ξ) = x(ξ), on O(C.23)

where f, g : [t, T ] × R × Ω → R and γ : [t, T ] × ∂O × Ω → R are appropriatelymeasurable functions.

Denote b(s, y)(·) := f(s, y(s, ·)) and [σ(s, y)z](·) := g(s, y(s, ·))z(·) and considerthe following integral equation

X(s) = e(s−t)ADx+

s

te(s−r)ADb(r,X(r))dr+

s

t(−AD)3/4+e(s−r)ADGDγ(r)dr

+

s

te(s−r)ADσ(r,X(r))dWQ(r) P-a.e.. (C.24)

Similarly to the deterministic case it can be viewed as the mild form of theequation

dX(s) = [ADX(s)−ADDγ(s) + b(s,X(s))] ds+ σ(s,X(s))dWQ(s)

X(t) = x ∈ H.(C.25)

Notation C.14 Equation (C.24) is called the mild form of equation (C.23) (andof equation (C.25)). Its solution is given by Definition 1.113, see Remark 1.114.Thanks to (C.20) we can rewrite the term −ADD in (C.25) as (−AD)3/4+GD in(C.24).

Conditions under which some equations of the form (C.23) have unique mildsolutions are given in Theorem 1.135 part (a). If σ ∈ L2(Ξ0, H) then the stochasticterm in (C.24) must be given proper interpretation and the same is also true about(C.33).

C.3. Non-zero boundary conditions, the Neumann case

The Neumann case is similar to the Dirichlet case which was explained inSubsection C.2.


∂∂sy(s, ξ) = ∆ξy(s, ξ) + f(s, ξ), (s, ξ) ∈ (t, T )×O∂∂ny(s, ξ) = γ(s, ξ), (s, ξ) ∈ (t, T )× ∂O

y(t, ξ) = x(ξ).

(C.26)

As before we denote respectively by H and Λ the Hilbert spaces L2(O) andL2(∂O). AN is the generator of the C0-semigroup associated to heat equationswith zero Neumann boundary conditions defined in (C.9), and λ > 0. Using thesame arguments as these in the proof of Theorem C.15 one can show the followingproposition.

1With the notation of Chapter 1, this means that we assume in this case Ξ = H.


Proposition C.15 If y ∈ C∞([t, T ]×O) is a classical solution of (C.26) thenX(s) := y(s, ·) can be written as

X(s) = e(s−t)ANx− (AN − λI)

s

te(s−r)ANNλγ(r)dr

+

s

te(s−r)AN f(r)dr. (C.27)

The previous expression can be seen as the mild form of the equation

ddsX(s) = ANX(s) + (λI −AN )Nλγ(s) + f(s)X(t) = x ∈ H.

(C.28)

If γ ∈ L1(t, T ;Λ), thanks to (C.10) we have

−(AN−λI)

s

te(s−r)ANNλγ(r)dr =

s

t(λI−AN )1/4+e(s−r)ANGNγ(r)dr, (C.29)

whereGN := (λI −AN )3/4−Nλ ∈ L(Λ, H). (C.30)

Therefore we can rewrite (C.27) as

X(s) = e(s−t)ANx+

s

t(λI −AN )1/4+e(s−r)ANGNγ(r)dr

+

s

te(s−r)AN f(r)dr. (C.31)

Notation C.16 Equation (C.31) is called the mild solution of (C.26) and (C.28).

To define mild form of the stochastic parabolic equation

dy(s, ξ) = [∆ξy(s, ξ) + f(s, y(s, ξ))] ds+ g(s, y(s, ξ))dWQ(s)(ξ), on (t, T )×O∂∂ny(s, ξ) = γ(s, ξ), on (t, T )× ∂O

y(t, ξ) = x(ξ), on O(C.32)

we consider the integral equation

X(s) = e(s−t)ANx+

s

te(s−r)AN b(r,X(r))dr+

s

t(λI−AN )1/4+e(s−r)ANGNγ(r)dr

+

s

te(s−r)ANσ(r,X(r))dWQ(r) P-a.e., (C.33)

where b(s, y)(·) := f(s, y(·)) and [σ(s, y)z](·) := g(s, y(·))z(·). Conditions underwhich some equations of the form (C.32) have unique mild solutions are given inTheorem 1.135 part (b).

Equation (C.33) is in fact the mild form of the problem

dX(s) = [ANX(s) + (λI −AN )Nλγ(s) + b(s,X(s))] ds+ σ(s,X(s))dWQ(s)

X(t) = x.(C.34)

Notation C.17 Equation (C.33) is called the mild form of equation (C.32) (andof equation (C.34)). Thanks to (C.30) we can rewrite the term (λI − AN )Nλ in(C.34) as (λI −AN )1/4+GN in (C.33).

C.5. BOUNDARY NOISE, DIRICHLET CASE 579

Conditions under which some equations of the form (C.32) have unique mildsolutions are given in Theorem 1.135 part (b).

C.4. Boundary noise, Neumann case

Let H,Λ, Q, andΩ,F , F t

ss∈[t,T ] ,P,WQ

be defined as in Subsection C.2.


∂∂sy(s, ξ) = ∆ξy(s, ξ) + f(s, y(s, ξ)), (s, ξ) ∈ (t, T )×O∂∂ny(s, ξ) = h(s, y(s, ξ))dWQ(s)

ds + g(s, ξ), (s, ξ) ∈ (t, T )× ∂O

y(t, ξ) = x(ξ)

(C.35)

where f, h : [t, T ] × R × Ω → R and g : [t, T ] × ∂O × Ω → R are appropriatelymeasurable functions.

As in Subsection C.3, GN := (λI − AN )3/4−Nλ ∈ L(Λ, H), λ > 0, AN isdefined by (C.9) and Nλ by (C.7).

To rewrite the equation in an infinite-dimensional setting in H, we followthe approach used in Subsections C.2 and C.3. The idea is to consider formallythe boundary term as a (particularly irregular) boundary condition correspond-ing to γ appearing in (C.26). So, denoting as before b(s, y)(·) := f(s, y(·)) and[σ(s, y)z](·) := h(s, y(·))z(·), we define the mild form of (C.35) as

X(s) = e(s−t)ANx+

s

te(s−r)AN b(r,X(r))dr

+

s

t(λI −AN )1/4+e(s−r)ANGNg(r)dr

+

s

t(λI −AN )1/4+e(s−r)ANGNσ(r,X(r))dWQ(r) P-a.e. (C.36)

The above expression can also be referred to as the mild form of the infinite di-mensional problem

dX(s) = [ANX(s) + (λI −AN )Nλg(s) + b(s,X(s))] ds+ (λI −AN )Nλσ(s,X(s))dWQ(s)

X(t) = x ∈ H,(C.37)

where, thanks to (C.30), the term (λI − AN )Nλg(s) is rewritten as (λI −AN )1/4+GNg(s) and the stochastic term is rewritten similarly.

Theorem 1.135 part (b) provides conditions under which some equations of theform (C.35) have unique continuous mild solutions.

Notation C.18 The solution of (C.36) is called the mild solution of (C.35) (andof (C.37)).

C.5. Boundary noise, Dirichlet case

To apply the approach of Subsection C.4 to the Dirichlet-boundary-noise ver-sion of the problem (C.35) we would need to give a meaning to the term

(−AD)3/4+

s

te(s−r)ADGDσ(r,X(r))dWQ(r),

where AD and GD ∈ L(Λ, H) are introduced respectively in (C.4) and (C.20).However estimate (B.15) does not allow to prove the convergence of the integral

T

t

(−AD)3/4+e(T−r)ADGD

2

L2(Λ,H)dr.


One way to resolve this problem is to look for a solution of the stochastic PDEin the completion of H in the weaker norm |x|∗ := |(−AD)−αx| for some α > 0. Itis, roughly speaking, a space of distributions. This approach was used e.g. in [125],see in particular Proposition 3.1, page 176.

Another way, used e.g. by [6, 46, 165], is to look for a solution in a weighted L2

space. Consider for example a simple case of the following problem on the positivehalf-line

∂∂sy(s, ξ) =

∂2

∂ξ2 y(s, ξ) + f(s, y(s, ξ)), (s, ξ) ∈ (t, T )× R+

y(s, 0) = h(s, y(t, 0))dW (s)ds + g(s), s ∈ (t, T )

y(t, ξ) = x(ξ),

(C.38)

where W a one-dimensional Brownian motion.Define the weight η(ξ) := min1, ξ1+θ for some θ > 0 and the weighted L2

space

L2η :=

p : [0,+∞) → R measurable :

+∞

0p2(ξ)η(ξ)dξ < ∞

.

It is a real separable Hilbert space. The inner product in L2η is given by

p, qη :=

+∞

0p(ξ)q(ξ)η(ξ)dξ

and it induces the usual norm

|p|η =

+∞

0p(ξ)2η(ξ)dξ

1/2

.

Proposition C.19 The heat semigroup with zero Dirichlet-boundary conditionsextends to a C0-semigroup on L2

η. The semigroup is analytic. Denote its generatorby Aη. For λ > 0, (λI −Aη) is invertible and the Dirichlet map

Dλa = φ ⇐⇒

(λI − ∂2x)φ[ξ] = 0 for all ξ > 0

φ(0) = a

is linear and continuous from R to D((λI −Aη)α) for all α ∈ [0, 1/2 + θ/4).

Proof. See Proposition 2.1 and Lemma 2.2 in [165].

Using the above proposition we can proceed in a fashion similar to that ofSubsection C.4. We fix λ > 0 and we choose

αθ :=1

2+

θ

8.

We define Gη := (λI − Aη)αθDλ ∈ L(R;L2η) and rewrite (C.38) as an integral

equation

X(s) = e(s−t)Aηx+

s

te(s−r)Aηb(r,X(r))dr

+

s

t(λI −Aη)

(1−αθ)e(s−r)AηGηg(r)dr

+

s

t(λI −Aη)

(1−αθ)e(s−r)AηGησ(r,X(r))dWQ(r) P-a.e. (C.39)

C.5. BOUNDARY NOISE, DIRICHLET CASE 581

This integral equation is in fact the mild form of the evolution equation inL2(t, T ;R)

dX(s) =AηX(s) + (λI −Aη)(1−αθ)Gηg(s) + b(s,X(s))

ds

+(λI −Aη)(1−αθ)Gησ(s,X(s))dWQ(s)

X(t) = x ∈ L2(t, T ;R),(C.40)

and we remark that (λI −Aη)(1−αθ)Gη = (λI −Aη)Dλ.One can use Theorem 1.135 part (b) to obtain existence and uniqueness of mild

solution X(·) of (C.40).

APPENDIX D

Functions, derivatives and approximations

In this appendix we collect some standard definitions related to functions whichare used in the text and are recalled here for the reader’s convenience. In the secondpart of this appendix we recall the definition and some properties of the sup-infconvolutions.

D.1. Continuity properties, modulus of continuity

Definition D.1 (Modulus) We say that a function ρ : [0,+∞) → [0,+∞)is a modulus (of continuity) if ρ is continuous, nondecreasing, subadditive, andρ(0) = 0.

In the literature the subadditivity property in the definition of a modulus isnot always required and continuity is sometimes required only at 0. The followingtheorem shows that one can use Definition D.1 without loss of generality.

Theorem D.2 Consider η : R+ → R+, non-decreasing, such that lims→0 η(s) =η(0) = 0, and

lims→∞

η(s)

s< +∞.

Then there exists a modulus ρ (in the sense of Definition D.1) such that ρ(s) ≥ η(s)for all s ∈ R+.

Proof. See Theorem 1 page 406 in [11].

The modulus ρ in Theorem D.2 can be also assumed to be concave. Thus wewill always assume that a modulus is concave.

Remark D.3 Thanks to the subadditivity, we have the following property forany modulus ρ(·): given any ε > 0, there exists Cε > 0 such that

ρ(r) ≤ ε+ Cεr for every r ≥ 0. (D.1)

Definition D.4 (Local Modulus) A function ρ : [0,+∞)× [0,+∞) → [0,+∞)is called a local modulus if the following three conditions are satisfied:

(i) ρ is continuous and nondecreasing in both variables.(ii) ρ is subadditive in the first variable.(iii) ρ(0, r) = 0 for every r ≥ 0.

Definition D.5 Consider two Banach spaces E0 and E1. Given ϕ ∈UC(E0;E1), we define its modulus of continuity ρ[ϕ](·) as follows:

ρ[ϕ]() := supx,y∈E0

|ϕ(x)− ϕ(y)|E1 : |x− y|E0 ≤ ε , for ε ≥ 0. (D.2)

583

584 D. REAL VALUED FUNCTIONS

Of course |ϕ(x)− ϕ(y)|E1 ≤ ρ[ϕ](|x− y|E0) for every x, y ∈ E0. Any moduluswith such property is also called a modulus of continuity of ϕ. In particular, forevery ϕ ∈ UC(E0;E1) we can always find two positive constants C0, C1 such that

|ϕ(x)|E1 ≤ C0 + C1|x|E0 , for every x ∈ E0.

Definition D.6 (Local boundedness from above (below)) Let G ⊂ E0. Afunction u : G → R is said to be locally bounded from above (respectively, below)if, for every R > 0, u is bounded from above (below) on G ∩ (BE0(0, R)). u is saidto be locally bounded if it is both locally bounded from above and from below.

Definition D.7 (Local uniform continuity) Let G ⊂ E0. A function u : G →R is said to be locally uniformly continuous if, for every R > 0, its restriction toG ∩ (BE0(0, R)) is uniformly continuous.

Definition D.8 (Local uniform convergence) Given G ⊂ E0, and un, u ∈C(G), we say that un converge locally uniformly to u if, for every R > 0, therestrictions of un to G ∩ (BE0(0, R)) converge uniformly to the restriction of u toG ∩ (BE0(0, R)).

Definition D.9 (Upper/lower semicontinuous envelope) Let G ⊂ E0. Con-sider a function u : G → R. Its upper (respectively, lower) semicontinuous envelopeis defined as follows:

u∗ : G → Ru∗(x) := lim sup

y∈Gy→x

u(y) (respectively, u∗(x) := lim infy∈Gy→x

u(y)).

Definition D.10 (Strict maximum/minimum) Let G ⊂ E0. We say thata function u : G → R ∪ −∞ (respectively, u : G → R ∪ +∞) has a strictmaximum (respectively, strict minimum) at x ∈ G if u has a maximum (respectively,minimum) at x and whenever xn ∈ G are such that u(xn) → u(x) then xn → x.Similarly we define a strict local maximum and a strict local minimum.

D.2. Fréchet and Gâteaux derivatives

Throughout this section Y, Z are Banach spaces, X is an open subset of Z, andH is a real separable Hilbert space with inner product ·, ·.Definition D.11 (Fréchet derivative) A function u : X → Y is said to beFréchet differentiable at a point x ∈ X if there exists a linear functional Du(x) ∈L(Z;Y ) such that

lim|x−x|Z→0

|u(x)− u(x)−Du(x)(x− x)|Y|x− x|Z

= 0.

Notation D.12 We consider a continuous function u : X → Y having Fréchetderivative at all x ∈ X. If the function

X → L(Z, Y )

x → Du(x)

is continuous (L(Z, Y ) is endowed with the operator norm) then u is said to becontinuously (Fréchet) differentiable on X.

The k-th order Fréchet derivative Dku of u is defined inductively

D1u = Du, Dku = D(Dk−1f), k = 1, 2, 3, . . .

D.3. INF-SUP CONVOLUTIONS 585

(see e.g. [198] p.186). If Dku is defined at a point x ∈ X then we say that u is k timesFréchet differentiable at x and we call Dku(x) the k-th Fréchet derivative of u atx. Clearly Dku(x) ∈ L(Z,L(Z, . . . ,L(Z, Y )) . . . ). Since this space is isometricallyisomorphic to Lk(Z;Y ), we will always consider Dku(x) as an element of this lastspace endowed with its natural norm TLk(Z;Y ) := supz∈Zk,z =0

|T (z1,...,zn)|Y|(z1,...,zn)|Zk

.If X is an open subset of H and Y = R then thanks to the Riesz Representation

Theorem, we can identify the linear functional Du(x) with the element y ∈ H suchthat y, x = Du(x)(x) for all x ∈ H. Abusing slightly notation, we will denote yby Du(x). The second order derivative D2u(x) can be identified with a symmetricbilinear form in L2(H;R) (see e.g. [198] 3.5.4 and 3.5.5, p.190). So we can identifyD2u(x) with the unique T ∈ S(H) having the property that

Tx, y = D2u(x)(x)(y) = D2u(x)(y)(x) for all x, y ∈ H.

With an abuse of notation we will again denote T by D2u(x).The following is a special case of the Generalized Taylor’s Theorem.

Theorem D.13 Let the function f : U(x) ⊂ H → R be defined on an openneighborhood U(x), of x, and let f ∈ C2(U(x)). Then, for all h in a neighborhoodof the origin in H, we have

f(x+ h) = f(x) + Df(x), h+ D2f(x)h, h+ o(|h|2)Proof. See [457], page 148. We now introduce now the notion of Gâteaux derivative and we give a special

case of the Generalized Taylor’s Theorem for Gâteaux differentiable functions.

Definition D.14 (Gâteaux derivative) A function u : X → Y is said to beGâteaux differentiable at a point x ∈ X if there exists a linear functional ∇u(x) ∈L(Z;Y ) such that, for any y ∈ Z with |y|Z = 1,

limt→0+

|u(x+ ty)− u(x)− t∇u(x)(y)|Yt

= 0. (D.3)

Theorem D.15 Let the function f : U(x) ⊂ H → R be defined on an open,convex neighborhood U(x) of x. Consider h ∈ Z such that x + h ∈ U(x). Supposethat f is Gâteaux differentiable at any point x ∈ U(x) and that the mapping

y → ∇f(y)(h)

is continuous on U(x). Then

f(x+ h) = f(x) +

1

0∇f(x+ th)(h)dt.

Proof. See Theorem 4.A, page 148 of [457]. Other type of derivatives are also used in the book. The basic material on

directional derivatives can be found in Section 6.1.2. The concepts of G-directionalderivative, G-Gâteaux derivative and G-Fréchet derivative, used in Chapters 4 and5, are introduced in Section 4.2.1.

D.3. Inf-Sup convolutions

We recall here some results from [310] on approximations of bounded uniformlycontinuous functions in Hilbert spaces. Consider a real and separable Hilbert spaceH with inner product ·, ·, and dim(H) = +∞. Since closed balls are not compactin H, functions in Cb(H) cannot be approximated uniformly on bounded sets byfunctions in UCb(H) and consequently by functions in C1

b (H). It was proved in [302]that UCb(H)∩C∞(H) is dense in UCb(H). However, it was observed in [354] that,


differently from what we have in the finite dimensional case, UC2b (H) is not a dense

subset of UCb(H). Nevertheless the inclusion UC1,1b (H) ⊂ UCb(H) remains dense.

Lasry and P. L. Lions introduced in [310] an explicit way to approximate functionsin UCb(H) by elements of UC1,1

b (H), the so-called inf-sup-convolutions and sup-inf-convolutions. These explicit approximations have many other interesting properties,for instance they preserve order and commute with translations. More informationabout them can be found in [310] and [129]. We also remark that P. L. Lions provedin [319] that functions in UC1,1

b (H) can be uniformly approximated by functionsin UC1,1

b (H) with uniformly continuous second order partial derivatives, for whichItô’s formula can be applied (see the proof of Lemma IV.1 and Lemma III.2 in[319]). The technique of [319] is based on limits of mollifications over increasingfinite dimensional subspaces of H. Also in [387, 388] it is proved that the spaceUC2

s (H), the subspace of UC1,1b (H) admitting weakly uniformly continuous second

Hadamard derivative, is dense in UCb(H).

Definition D.16 (Semiconvex and semiconcave functions) A function u :H → R is said to be semiconcave (respectively, semiconvex) if there exists a constantM ≥ 0 such that

x → u(x)−M |x|2 (respectively, x → u(x) +M |x|2)is concave (respectively, convex).

Definition D.17 (Sup-convolution and Inf-convolution) Given u ∈ Cb(H)and > 0, we define the inf-convolution of u as

u(x) := infy∈H

u(y) +

|x− y|22

, x ∈ H

and its sup-convolution as

u(x) := supy∈H

u(y)− |x− y|2

2

, x ∈ H.

We set u0(x) = u0(x) := u(x).

Proposition D.18 For all , δ ≥ 0 and u ∈ UCb(H), u, u ∈ UCb(H), and(u)δ = u+δ, (u)δ = u+δ. Moreover, for all u, v ∈ UCb(H) and > 0, we havethe following properties:

(1) u − v0 ≤ u− v0 and u − v0 ≤ u− v0.(2) ρ[u] ≤ ρ[u] and ρ[u] ≤ ρ[u].(3) lim→0 u = u and lim→0 u = u in UCb(H), more precisely

u − u0 ≤ ρ[u](2u0) and u − u0 ≤ ρ[u](2

u0).

(4) x → u(x)− |x|22 is concave.

(5) x → u(x) + |x|22 is convex.

(6) u and u are Lipschitz continuous, and, for all x, y ∈ H,

|u(x)− u(y)||x− y| ,

|u(x)− u(y)||x− y| ≤ 2

u0√

Proof. Most of the claims are proved in [129], Propositions C.3.2, C.3.3,and C.3.4. We sketch the proof of (6) for u.

Let x, y ∈ H. We define the function v : R → R by

v(t) := u(x+ t(y − x)).

Since u is semiconvex, v is semiconvex and hence locally Lipschitz and differentiablea.e. Let 0 < t < 1 be a point of differentiability of v and let ϕ ∈ C1(R) be such

D.3. INF-SUP CONVOLUTIONS 587

that v − ϕ has a strict maximum at t. By Theorem 3.25, for every n there arean ∈ R, pn ∈ H, |an|+ |pn| < 1/n such that

u(z)− |x+ t(y − x)− z|22

− ϕ(t) + pn, z+ ant (D.4)

has a global maximum over [0, 1]×H at some point (tn, zn). Since the maximum ofv−ϕ at t was strict, it is easy to see that we must have tn → t and |zn| ≤ C, n ∈ Nfor some C. Moreover, setting z = zn and differentiating the function in (D.4) withrespect to t we obtain

Dϕ(tn) =x+ tn(y − x)− zn, x− y

+ an. (D.5)

Taking z = x+ tn(y − x) we get

u(x+ tn(y − x)) + pn, x+ tn(y − x) ≤ u(zn)−|x+ tn(y − x)− zn|2

2+ pn, zn,

which implies|x+ tn(y − x)− zn|2

2≤ u(zn)−u(x+tn(y−x))+pn, zn−x−tn(y−x) ≤ 2u0+

C1

n.

Using this and letting n → +∞ in (D.5) we thus obtain

|Dϕ(t)| ≤ 2u0√

|y − x|. (D.6)

We can thus conclude

|u(y)− u(x)| = |v(1)− v(0)| ≤ 1

0|v(t)|dt ≤ 2

u0√

|y − x|.

We remark that the above proof shows that one can in fact obtain

|u(y)− u(x)| ≤ t√|y − x|,

where t is defined in Proposition D.21.We also remark that it follows from semiconvexity of u that if v(t) exists, we

must havev(t) = p, y − x for every p ∈ D−u(x+ t(y − x)),

where D−u(x+ t(y−x)) is the subdifferential of u at (x+ t(y−x)) (see DefinitionE.1). Moreover, D−u(z) is non-empty for every z. In fact for a semiconvex functionw : H → R,

D−w(z) = convp : Dw(zn) p, zn → z, (D.7)where zn above are points of Fréchet differentiability of w (see [48], page 522).Lipschitz functions on H are Fréchet differentiable on a dense subset of H byPreiss’s theorem.

Finally we remark that if u is differentiable at some point x then Du(x) =(y − x)/, where y is a unique point such that

u(x) = u(y)− |x− y|22

. (D.8)

To show this let ϕ ∈ C1(H) be such that u − ϕ has a global strict maximum atx and ϕ(x) = |x|2 if |x| is large enough. By Theorem 3.25 for every n there arepn, qn ∈ H, |pn|+ |qn| < 1/n such that

u(y)− |x− y|22

− ϕ(x) + pn, x+ qn, y (D.9)


has a global maximum over H ×H at some point (xn, yn). Since the maximum ofu − ϕ at x was strict, it is easy to see that we must have xn → x and

u(yn)−|xn − yn|2

2→ u(x)

as n → +∞. Moreover, setting y = yn and differentiating the function in (D.9)with respect to x we obtain

Dϕ(xn) = −xn − yn

+ pn,

which implies that limn→+∞ yn = y for some point y,

Du(x) = Dϕ(x) =y − x

, (D.10)

and (D.8) is satisfied. Since (D.10) holds for every point y satisfying (D.8), it mustbe unique.

Definition D.19 (Inf-sup and sup-inf-convolutions) Given ε > 0 and afunction u ∈ Cb(H), we define the inf-sup convolution (respectively, sup-inf-convolution) of u as

u := (u)

2 (respectively, u := (u)

2).

More explicitly we have, for x ∈ H,

u(x) = supz∈H

infy∈H

u(y) +

1

2|z − y|2 − 1

|z − x|2

(respectively,

u(x) = infz∈H

supy∈H

u(y)− 1

2|z − y|2 + 1

|z − x|2

.)

Proposition D.20 The inf-sup-convolution and the sup-inf-convolution pre-serve the order. In other words, given u, v ∈ UCb(H) such that

u(x) ≥ v(x), for all x ∈ H,

we have uε(x) ≥ vε(x) and uε(x) ≥ vε(x) for all x ∈ H. Moreover the inf-sup-convolution and the sup-inf-convolution commute with translations, i.e. forevery y ∈ H and translation τy : x → x − y, we have (τyu)ε(x) = (τyuε)(x) and(τyu)ε(x) = (τyuε)(x) for all x ∈ H.

Proof. See [310]. Proposition D.21 Let u ∈ UCb(H). Let t be the maximum positive root ofthe equation

t2 = 2ρ[u](t),

which implies that t−1/2 → 0 as → 0. Then u and u belong to UC1,1b (H) and,

for all x, y ∈ H, the following properties hold:(1) infy∈H u(y) ≤ u(x) ≤ u(x) ≤ u(x) ≤ supy∈H u(y).(2) |u(x)− u(y)| ≤ ρ[u](|x− y|) and |u(x)− u(y)| ≤ ρ[u](|x− y|).(3) u − u0, u − u0 ≤ ρ[u](t).(4) Du0, Du0 ≤ t

.

(5) |Du(x)−Du(y)|, |Du(x)−Du(y)| ≤ 2 |x− y|.

Proof. See [310], pages 260-261.

Remark D.22 In fact x → u(x)−|x|22 is concave and x → u(x)+

|x|2 is convex.

Similarly x → u(x)− |x|2 is concave and x → u(x) +

|x|22 is convex.

D.4. TWO VERSIONS OF GRONWALL’S LEMMA 589

The inf-sup and sup-inf-convolutions can also be used to approximate moregeneral functions.

Proposition D.23 If u ∈ C(H) and there exists C > 0 such that |u(x)| ≤C(1 + |x|2) then, for all small enough, the inf-sup convolution u and the sup-infconvolution u are well defined, they belong to C1,1(H) and they converge pointwiseto u when → 0. Moreover, if u is locally uniformly continuous then u and u

converge to u locally uniformly.

Proof. See [310] page 261 (iii).

We remark that to obtain pointwise convergence in the above proposition onecan replace u ∈ C(H) by u ∈ USC(H) for u, and by u ∈ LSC(H) for u.

D.4. Two versions of Gronwall’s Lemma

We recall two versions of Gronwall’s Lemma. The first is a well known resultwhile the second is more specialized. (See e.g. [161] or [198] pages 95-97 for similarversions of Gronwall’s Lemma). .

Proposition D.24 (Gronwall’s Lemma 1) Let T ∈ [0,+∞) ∪ +∞ andI = [t0, T ). Let a(·) be a nonnegative, measurable, nondecreasing function on I.Let b(·) be a non-negative, locally integrable function on I. Suppose that u(·) is anon-negative function such that b(·)u(·) is locally integrable on I. Assume also that

u(s) ≤ a(s) +

s

t0

b(r)u(r)dr, for a.e. s ∈ I. (D.11)

Then

u(s) ≤ a(s) exp

s

t0

b(r)dr

, for a.e. s ∈ I,

Proof. Define

v(s) = exp

− s

t0

b(r)dr

s

t0

b(r)u(r)dr, s ∈ I.

Then, for a.e. s ∈ I, v(s) exists and

v(s) =

u(s)−

s

t0

b(r)u(r)dr

b(s) exp

− s

t0

b(r)dr

.

So, using (D.11) and integrating we have

v(s) ≤ s

t0

a(r)b(r) exp

− r

t0

b(τ)dτ

dr.

Now, since s

t0

b(r)u(r)dr = v(s) exp

s

t0

b(r)dr

, s ∈ I,

by (D.11) we get

u(s) ≤ a(s) +

s

t0

b(r)u(r)dr = a(s) + exp

s

t0

b(r)dr

v(s)

≤ a(s) + exp

s

t0

b(r)dr

s

t0

a(r)b(r) exp

− r

t0

b(τ)dτ

dr

= a(s) +

s

t0

a(r)b(r) exp

s

rb(τ)dτ

dr.


Since the function a(·) is nondecreasing the above implies

u(s) ≤ a(s) +

s

t0

a(s)b(r) exp

s

rb(τ)dτ

dr

= a(s) +

−a(s) exp

s

rb(τ)dτ

r=s

r=t0

= a(s) exp

s

t0

b(r)dr

which is the claim. Proposition D.25 (Gronwall’s Lemma 2) Let T ∈ [0,+∞) ∪ +∞, b ≥ 0,β > 0. Let a(·) be a non-negative, locally integrable function on [0, T ). Supposethat u(·) is a non-negative, locally integrable function on [0, T ) such that

u(s) ≤ a(s) + b

s

0(s− r)β−1u(r)dr, for a.e. s ∈ [0, T ].

Thenu(s) ≤ a(s) + θ

s

0E

β(θ(s− r))a(r)dr, for a.e. s ∈ [0, T ],

where, for s > 0, Γ(s) := +∞0 rs−1e−rdr, and θ and Eβ(s) are defined as

θ := (bΓ(β))1/β

and

Eβ(s) :=+∞

n=0

snβ

Γ(nβ + 1).

The function Eβ(s) = d

dsEβ(s) has the following properties: (i) Eβ(s) = sβ−1

Γ(β) +

o(sβ−1) as s → 0+, (ii) Eβ(s) =

es

β + o(es) as s → +∞.

As a particular case, if a, b, T ∈ R+ and α,β ∈ [0, 1), there exists M ∈ R(depending on b, T , α and β) such that any integrable function u : [0, T ] → R suchthat

0 ≤ u(s) ≤ as−α + b

s

0(s− r)−βu(r)dr, for a.e. s ∈ [0, T ],

satisfies0 ≤ u(s) ≤ aMs−α, for a.e. s ∈ [0, T ].

Proof. See Lemma 7.1.1, Chapter 7, page 188 of [256]. The second claimis again in [256], Subsection 1.2.1, page 6.

APPENDIX E

Viscosity solutions in RN

We collect some basic definitions and results about viscosity solutions in finitedimensional spaces. We refer the reader to [101] and the books [30, 31, 195] formore information on the subject.

E.1. Second order jets

Let O be an open subset of RN .

Definition E.1 (Sub- and superdifferentials) Let u : O → R be an uppersemicontinuous function. The superdifferential of u at a point x ∈ O is defined as

D+u(x) :=

p ∈ RN : lim sup

y→x,y∈O

u(y)− u(x)− p, y − x|y − x| ≤ 0

.

Similarly, given a lower semicontinuous function u : O → R, the subdifferentialof u at a point x ∈ O is defined as

D−u(x) :=

p ∈ RN : lim inf

y→x,y∈O

u(y)− u(x)− p, y − x|y − x| ≥ 0

.

Lemma E.2 Let u : O → R be an upper semicontinuous function, and x ∈ O.Then p ∈ D+u(x) if and only if there exists a function φ ∈ C1(RN ) such that u−φattains a strict global maximum at x and

(φ(x), Dφ(x)) = (u(x), p).

Similarly, if u : O → R is a lower semicontinuous function, and x ∈ O, thenp ∈ D−u(x) if and only if there exists a function φ ∈ C1(RN ) such that u − φattains a strict global minimum at x and

(φ(x), Dφ(x)) = (u(x), p).

Proof. See [449], Lemma 2.7, page 173 or [162], page 544.

We remark that Definition E.1 is exactly the same and Lemma E.2 is true ifRN is replaced by a real Hilbert space.

Definition E.3 (Second order sub- and superjets) Let u : O → R be anupper semicontinuous function, and x ∈ RN . The set

J2,+u(x) :=

(p,X) ∈ RN × S(RN ) :

lim supy→x,y∈O

u(y)− u(x)− p, y − x − 12 X(y − x), (y − x)

|y − x|2 ≤ 0

591

592 E. VISCOSITY SOLUTIONS IN RN

is called the second order superjet of u at x. Similarly, given a lower semicontin-uous function u : O → R, and x ∈ O, the set

J2,−u(x) :=

(p,X) ∈ RN × S(RN ) :

lim infy→x,y∈O

u(y)− u(x)− p, y − x − 12 X(y − x), (y − x)

|y − x|2 ≥ 0

is called the second order subjet of u at x.

Lemma E.4 Let u : O → R be an upper semicontinuous function, and x ∈ O.Then (p,X) belongs to J2,+u(x) if and only if there exists a function φ ∈ C2(RN )such that u− φ attains a strict global maximum at x and

φ(x), Dφ(x), D2φ(x)

= (u(x), p,X).

Similarly, if u : O → R is a lower semicontinuous function, and x ∈ O, then(p,X) ∈ J2,−u(x) if and only if there exists a function φ ∈ C2(RN ) such that u−φattains a strict global minimum at x and

φ(x), Dφ(x), D2φ(x)

= (u(x), p,X).

Proof. See [449] Lemma 5.4 page 193 or [195], Lemma 4.1, page 211.

Definition E.5 (Closure of second order sub- and superjets) Let u : O → Rbe an upper semicontinuous function, and x ∈ O. We define

J2,+

u(x) :=

(p,X) ∈ RN × S(RN ) : there exist xn ∈ O

and (pn, Xn) ∈ J2,+u(xn) s.t. (xn, u(xn), pn, Xn)n→∞−−−−→ (x, u(x), p,X)

.

Similarly, given a lower semicontinuous function u : O → R and x ∈ O, we define

J2,−

u(x) :=

(p,X) ∈ RN × S(RN ) : there exist xn ∈ O

and (pn, Xn) ∈ J2,−u(xn) s.t. (xn, u(xn), pn, Xn)n→∞−−−−→ (x, u(x), p,X)

.

Remark E.6 Note that the definition is a little different from what one wouldexpect as closures of set-valued mappings. Indeed we also ask u(xn) → u(x). Thisform of the closures of the semijets was first introduced in [273].

Definition E.7 (Parabolic second order sub- and superjets) Let T > 0. Letu : (0, T ) × O → R be an upper semicontinuous function, and (t, x) ∈ (0, T ) × O.The set

P2,+u(t, x) :=

(a, p,X) ∈ R× RN × S(RN ) :

lim sup(s,y)→(t,x)

u(s, y)− u(t, x)− a(s− t)− p, y − x − 12 X(y − x), (y − x)

|s− t|+ |y − x|2 ≤ 0

E.2. DEFINITION OF VISCOSITY SOLUTION 593

is called the parabolic second order superjet of u at (t, x). Similarly, given a lowersemicontinuous function u : (0, T )×O → R, and (t, x) ∈ (0, T )×O, the set

P2,−u(t, x) :=

(a, p,X) ∈ R× RN × S(RN ) :

lim inf(s,y)→(t,x)

u(s, y)− u(t, x)− a(s− t)− p, y − x − 12 X(y − x), (y − x)

|s− t|+ |y − x|2 ≥ 0

is called the parabolic second order subjet of u at (t, x).

The closures of the parabolic second order sub- and superjetsP2,−

u(t, x),P2,+u(t, x) are defined in the same way as J

2,−u(x), J

2,+u(x).

A parabolic analogue of Lemma E.4 is also true, see [195], Lemma 4.1, page 211.

E.2. Definition of viscosity solution

O be an open subset of RN . Consider an equation

F (x, u,Du,D2u) = 0 in O, (E.1)

where F : O × R × RN × S(RN ) → R is continuous, non-decreasing in the secondvariable, and degenerate elliptic, i.e. for every (x, r, p) ∈ O × R× RN , and X,Y ∈S(RN ),

F (x, r, p,X) ≤ F (x, r, p, Y ) if X ≥ Y.

Definition E.8 An upper semi-continuous function u : O → R is a viscositysubsolution of (E.1) if

F (x, u(x), p,X) ≤ 0 if x ∈ O and (p,X) ∈ J2,+u(x).

A lower semi-continuous function u : O → R is a viscosity supersolution of (E.1)if

F (x, u(x), p,X) ≥ 0 if x ∈ O and (p,X) ∈ J2,−u(x).

A viscosity solution of (E.1) is a function which is both a viscosity subsolution anda viscosity supersolution of (E.1).

We remark that since F is continuous, we obtain an equivalent def-inition if J2,+u(x), J2,−u(x) in Definition E.8 are replaced respectively byJ2,+

u(x), J2,−

u(x). Moreover, in light of Lemma E.4, Definition E.8 is equivalentto the following definition using test functions.

Definition E.9 An upper semi-continuous function u : O → R is a viscositysubsolution of (E.1) if whenever u − ϕ has a local maximum at a point x ∈ O fora test function ϕ ∈ C2(O) then

F (x, u(x), Dϕ(x), D2ϕ(x)) ≤ 0.

A lower semi-continuous function u : O → R is a viscosity supersolution of (E.1) ifwhenever u−ϕ has a local minimum at a point x ∈ O for a test function ϕ ∈ C2(O)then

F (x, u(x), Dϕ(x), D2ϕ(x)) ≥ 0.

A viscosity solution of (E.1) is a function which is both a viscosity subsolution anda viscosity supersolution of (E.1).

Viscosity sub/supersolutions and solutions of parabolic initial value problems

ut + F (t, x, u,Du,D2u) = 0 in (0, T )×O,u(0, x) = g(x) on O (E.2)

594 E. VISCOSITY SOLUTIONS IN RN

are defined in the same way if we replace J2,−u(x), J2,+u(x) in Definition E.8 byP2,−u(t, x),P2,+u(t, x) and use test functions ϕ which are once continuously differ-entiable in t and twice continuously differentiable in x on (0, T ) ×O in DefinitionE.9.

It is often useful to use the notion of a discontinuous viscosity solution. Afunction u is a discontinuous viscosity subsolution if u∗ is a viscosity subsolution,and u is a discontinuous viscosity supersolution if u∗ is a viscosity supersolution.

E.3. Finite dimensional maximum principles

The following form of the finite-dimensional maximum principle was introducedin [100] and is sometimes referred to as Crandall-Ishii lemma.

Theorem E.10 (Maximum principle) Let N ∈ N and O be an open subsetof RN . Let ui : O → R, i = 1, 2, be two upper semicontinuous functions, andφ ∈ C2(O ×O). Set, for x = (x1, x2) ∈ O ×O,

w(x) := u1(x1) + u2(x2).

Suppose that w−φ has a local maximum at x = (x1, x2) ∈ O. Then, for each ε > 0,there exist Xi ∈ S(RN ) such that

(Dxiφ(x), Xi) ∈ J

2,+ui(xi) for i = 1, 2

and

−1

ε+D2φ(x)

I ≤

X1 00 X2

≤ D2φ(x) + ε

D2φ(x)

2.

Proof. Theorem E.10 is a particular case of Theorem 3.2 of [101]. Its proofis given in the appendix of [101].

The following is a parabolic version of Theorem E.10 and is taken from [100],see also [101], Theorem 8.2 or [195], Theorem 6.1, page 216.

Theorem E.11 (Parabolic maximum principle) Let T > 0, N ∈ N, and O bean open subset of RN . Let ui : (0, T )×O → R, i = 1, 2 be two upper semicontinuousfunctions, and φ : (0, T ) × O × O → R be once continuously differentiable in tand twice continuously differentiable in x = (x1, x2) ∈ R2N . Set, for (t, x) =(t, x1, x2) ∈ (0, T )×O ×O,

w(t, x) := u1(t, x1) + u2(t, x2).

Suppose that w − φ has a local maximum at (t, x) = (t, x1, x2) ∈ (0, T ) × R2N .Assume moreover that there is r > 0 such that for every M > 0 there is C > 0 suchthat for i = 1, 2

bi ≤ C whenever (bi, pi, Xi) ∈ P2,+ui(t, xi),|xi − xi|+ |t− t| ≤ r and |ui(t, xi)|+ |pi|+ Xi ≤ M.

(E.3)

Then, for each ε > 0, there exist bi ∈ R, Xi ∈ S(RN ) such that

(bi, Dxiφ(t, x), Xi) ∈ P2,+

ui(t, xi) for i = 1, 2, b1 + b2 = ϕt(t, x),

and

−1

ε+D2φ(t, x)

I ≤

X1 00 X2

≤ D2φ(t, x) + ε

D2φ(t, x)

2.

We remark that the somewhat strange looking condition E.3 is satisfied if u1 isa viscosity subsolution and −u2 is a viscosity supersolution of a parabolic equation.

E.4. PERRON’S METHOD 595

E.4. Perron’s method

Perron’s method is an easy and very general procedure to obtain existence ofviscosity solutions. Consider a parabolic initial value problem (E.2), where O = RN

and F : [0, T ] × RN × R × RN × S(RN ) → R is continuous, non-decreasing in thethird variable, and degenerate elliptic. This is the only case that will be used inthis book.

Suppose that we have a viscosity supersolution u of (E.2) and a viscosity sub-solution u of (E.2) such that u ≤ u and u(0, x) = u(0, x) = g(x). Suppose moreoverthat the equation satisfies the following comparison property: If u is a viscosity sub-solution of (E.2) and v is a viscosity supersolution of (E.2) such that u∗ ≤ u, v ≤ u∗,then u ≤ v. We then have the following theorem. Its proof follows standard argu-ments, see for instance [101], Section 4, pages 22-24.

Theorem E.12 (Perron’s method) If the assumptions of this subsection aresatisfied then the function

w(t, x) = supu(t, x) : u ≤ u ≤ u, u is a viscosity subsolution of (E.2)is a viscosity solution of (E.2).

We remark that when applying Perron’s method it is often more convenientto use the notion of discontinuous viscosity solution. The comparison property isthen not needed and one always has that the function w, defined as the supremumof discontinuous viscosity subsolutions u such that u ≤ u ≤ u is a discontinuousviscosity solution.

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