Hiroshi Kunita Stochastic Flows and...

Probability Theory and Stochastic Modelling 92

Hiroshi Kunita

Stochastic Flows and Jump-Diffusions

Probability Theory and Stochastic Modelling

Volume 92

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Hiroshi Kunita

Stochastic Flowsand Jump-Diffusions

123

Hiroshi KunitaKyushu University (emeritus)Fukuoka, Japan

ISSN 2199-3130 ISSN 2199-3149 (electronic)Probability Theory and Stochastic ModellingISBN 978-981-13-3800-7 ISBN 978-981-13-3801-4 (eBook)https://doi.org/10.1007/978-981-13-3801-4

Library of Congress Control Number: 2019930037

Mathematics Subject Classification: 60H05, 60H07, 60H30, 35K08, 35K10, 58J05

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https://doi.org/10.1007/978-981-13-3801-4

This book is dedicated to my family

Preface

The Wiener process and the Poisson random measure are fundamental to the studyof stochastic processes; the former describes a continuous random evolution, andthe latter describes a random phenomenon that occurs at a random time. It wasshown in the 1940s that any Lévy process (process with independent increments) isrepresented by a Wiener process and a Poisson random measure, called the Lévy–Itô representation. Further, Itô defined a stochastic differential equation (SDE) basedon a Wiener process. He defined also an equation based on a Wiener process and aPoisson random measure. In this monograph, we wish to present a modern treatmentof SDE and diffusion and jump-diffusion processes. In the first part, we will showthat solutions of SDE will define stochastic flows of diffeomorphisms. Then, wediscuss the relation between stochastic flows and heat equations. Finally, we willinvestigate fundamental solutions of these heat equations (heat kernels), through thestudy of the Malliavin calculus.

It seems to be traditional that diffusion processes and jump processes arediscussed separately. For the study of the diffusion process, theory of partialdifferential equations is often used, and this fact has attracted a lot of attention. Onthe other hand, the study of jump processes has not developed rapidly. One reasonmight be that for the study of jump processes, we could not find effective tools inanalysis such as the partial differential equation in diffusion processes. However,recently, the Malliavin calculus for Poisson random measure has been developed,and we can apply it to some interesting problems of jump processes.

A purpose of this monograph is that we present these two theories simultane-ously. In each chapter, we start from continuous processes and then proceed toprocesses with jumps. In the first half of the monograph, we present the theoryof diffusion processes and jump-diffusion processes on Euclidean spaces basedon SDEs. The basic tools are Itô’s stochastic calculus. In Chap. 3, we show thatsolutions of these SDEs define stochastic flows of diffeomorphisms. Then inChap. 4, relations between a diffusion (or jump-diffusion) and a heat equation (or aheat equation associated with integro-differential equation) will be studied throughproperties of stochastic flow.

vii

viii Preface

In the latter half of the monograph, we will study the Malliavin calculus onthe Wiener space and that on the space of Poisson random measure. These twotypes of calculus are quite different in detail, but they have some interesting thingsin common. These will be discussed in Chap. 5. Then in Chap. 6, we will applythe Malliavin calculus to diffusions and jump-diffusions determined by SDEs. Wewill obtain smooth densities for transition functions of various types of diffusionsand jump-diffusions. Further, we show that these density functions are fundamentalsolutions for various types of heat equations and backward heat equations; thus, weconstruct fundamental solutions for heat equations and backward heat equations,independent of the theory of partial differential equations. Finally, SDEs onsubdomains of a Euclidean space and those on manifolds will be discussed at theend of Chaps. 6 and 7.

Acknowledgements Most part of this book was written when the author was working onMalliavin calculus for jump processes jointly with Masaaki Tsuchiya and Yasushi Ishikawa in2014–2017. Discussions with them helped me greatly to make and rectify some complicated normestimations, which cannot be avoided for getting the smooth density. The contents of Sects. 5.5,5.6, and 5.7 overlap with the joint work with them [46]. Further, Ishikawa read Chap. 7 carefullyand gave me useful advice. I wish to express my thanks to them both. I would also like to thankthe anonymous referees and a reviewer, who gave me valuable advices for improving the draftmanuscript. Finally, it is my pleasure to thank Masayuki Nakamura, editor at Springer, who helpedme greatly toward the publication of this book.

Fukuoka, Japan Hiroshi Kunita

Introduction

Stochastic differential equations (SDEs) based on Wiener processes have beenstudied extensively, after Itô’s work in the 1940s. One purpose was to constructa diffusion process satisfying the Kolmogorov equation. Results may be found inmonographs of Stroock–Varadhan [109], Ikeda–Watanabe [41], Oksendal [90], andKaratzas–Shreve [55]. Later, the geometric property of solutions was studied. It wasshown that solutions of an SDE based on a Wiener process define a stochastic flowof diffeomorphisms [59].

In 1978, Malliavin [77] introduced an infinite-dimensional differential calculuson a Wiener space. The theory had an interesting application to solutions ofSDEs based on the Wiener process. He applied the theory to the regularity of theheat kernel for hypo-elliptic differential operators. Then, the Malliavin calculusdeveloped rapidly. Contributions were made by Bismut [9], Kusuoka–Stroock[69, 71], Ikeda–Watanabe [40, 41], Watanabe [116, 117], and many others.

At the same period, the Malliavin calculus for jump processes was studied inparallel (see Bismut [10], Bichteler–Gravereau–Jacod [7], and Leandre [74]). Later,Picard [92] proposed another approach to the Malliavin calculus for jump processes.Instead of the Wiener space, he developed the theory on the space of Poisson randommeasure and got a smooth density for the law of a “nondegenerate” jump Markovprocess. Then, Ishikawa–Kunita [45] combined these two theories and got a smoothdensity for the law of a nondegenerate jump-diffusion. Thus, the Malliavin calculuscan be applied to a large class of SDEs.

In this monograph, we will study two types of SDEs defined on Euclidean spaceand manifolds. One is a continuous SDE based on a Wiener process and smoothcoefficients. We will define the SDE by means of Fisk–Stratonovich symmetricintegrals, since its solution has nice geometric properties. The other is an SDEwith jumps based on the Wiener process and the Poisson random measure, wherecoefficients for the continuous part are smooth vector fields and coefficients for jumpparts are diffeomorphic maps. These two SDEs are our basic objects of study. Wewant to show that both of these SDEs generate stochastic flows of diffeomorphismsand these stochastic flows define diffusion processes and jump-diffusion processes.In the course of the argument, we will often consider backward processes, i.e.,

ix

x Introduction

stochastic processes describing the backward time evolution. It will be shownthat inverse maps of a stochastic flow (called the inverse flow) define a backwardMarkov process. Further, we show that the dual process is a backward Markovprocess and it can be defined directly by the inverse flow through an exponentialtransformation. In each chapter, we will start with topics related to Wiener processesand then proceed to those related to Poisson random measures. Investigating thesetwo subjects together, we can understand both the Wiener processes and Poissonrandom measures more strongly.

Chapters 1 and 2 are preliminaries. In Chap. 1, we propose a method of studyingthe smooth density of a given distribution, through its characteristic function(Fourier transform); it will be applied to the density problem of infinitely divisibledistributions. Then, we introduce some basic stochastic processes and backwardstochastic processes. These include Wiener processes, Poisson random measures,martingales, and Markov processes. In Chap. 2, we discuss stochastic integrals. Itôintegrals and Fisk–Stratonovich symmetric integrals based on continuous martin-gales and Wiener processes are defined, and Itô’s formulas are presented. Then,we define stochastic integrals based on (compensated) Poisson random measures.Further, we will give Lp-estimates of these stochastic integrals; these estimateswill be used in Chaps. 3 and 6 for checking that stochastic flows have some niceproperties. The backward stochastic integrals will also be discussed.

In Chap. 3, we will revisit SDEs and stochastic flows, which were discussed bythe author [59, 60]. A continuous SDE on d-dimensional Euclidean space Rd basedon a d ′-dimensional Wiener process (W 1

t , . . . ,Wd ′t ) is given by

dXt =d ′∑

k=1

Vk(Xt , t) ◦ dWkt + V0(Xt , t) dt, (1)

where ◦dWkt denotes the symmetric integral based on the Wiener process Wk

t . Ifcoefficients Vk(x, t), k = 0, . . . , d are of C∞,1

b -class, it is known that the family ofsolutions {Xx,s

t , s < t} of the SDE, starting from x at time s, have a modification{Φs,t (x), s < t}, which is continuous in s, t, x and satisfies (a) Φs,t : Rd → R

d areC∞-diffeomorphisms, (b) Φs,u = Φt,u ◦ Φs,t for any s < t < u almost surely, and(c) Φs,t and Φt,u are independent. {Φs,t } is called a continuous stochastic flow ofdiffeomorphisms defined by the SDE.

A similar problem was studied for SDE based on the Wiener process and thePoisson random measure. Let N(dt dz) be a Poisson random measure on the spaceU = [0, T ] × (Rd ′ \ {0}) with the intensity measure n(dt dz) = dtν(dz), where ν

is a Lévy measure having a “weak drift.” We consider an SDE driven by a Wienerprocess and Poisson random measure:

dXt =d ′∑

k=1

Vk(Xt , t) ◦ dWkt + V0(Xt , t) dt +

∫

|z|>0+(φt,z(Xt−)−Xt−)N(dt dz),

(2)

Introduction xi

where {φt,z} is a family of diffeomorphic maps on Rd with some regularity

conditions (precise conditions will be stated in Sect. 3.2). It was shown that solutionsdefine a stochastic flow of diffeomorphisms stated above (Fujiwara–Kunita [30]).In this monograph, we will prove these facts through discussions of the “masterequation” and “backward SDE.” By using them, some complicated arguments inprevious works [30, 59] are simplified. Further, we will define a backward SDEbased on a Wiener process and another based on Wiener process and Poissonrandom measure. These backward SDEs define backward stochastic flows ofdiffeomorphisms.

The solution of an SDE (or a backward SDE) based on a Wiener process definesa diffusion process (continuous strong Markov process) (or backward diffusionprocess). Further, the solution of an SDE (or backward SDE) based on a Wienerprocesses and a Poisson random measure defines a jump-diffusion (or a backwardjump-diffusion). We will study these diffusion and jump-diffusion processes. Let{Ps,t } be the semigroup defined by Ps,tf (x) = E[f (Φs,t (x))]. In the case ofa diffusion process on R

d , its generator is given by a second-order differentialoperator

A(t)f (x) = 1

2

d ′∑

k=1

Vk(t)2f (x)+ V0(t)f (x), (3)

where Vk(t), k = 0, 1, . . . , m are first-order partial differential operators defined byVk(t)f (x) = ∑

i Vik (x, t)

∂∂xi

f (x). In the case of a jump-diffusion process on Rd ,

the generator is given by an integro-differential operator of the form

AJ (t)f = 1

2

d ′∑

k=1

Vk(t)2f (x)+ V0(t)f (x)+

∫

|z|>0+{f (φt,z(x))− f (x)}ν(dz),

(4)where the last integral is an improper integral.

In Chap. 4, we study the relation between stochastic flows and time-dependentheat equations and backward heat equations associated with the differential operatorA(t) of (3) and integro-differential operator AJ (t) of (4), respectively. For a giventime t1 and a bounded smooth function f1(x), the function v(x, s) := Ps,t1f1(x) =E[f1(Φs,t1(x))] is a smooth function of x. Further, in the case of diffusions, v(x, s)is the unique solution of the final value problem of the time-dependent backwardheat equation:

∂

∂sv(x, s) = −A(s)v(x, s) for s < t1, v(t1, x) = f1(x). (5)

This fact will be extended to a more general class of the operator A(t). Consider

Ac(t)f = 1

2

d ′∑

k=1

(Vk(t)+ ck(t))2f + (V0 + c0)f, (6)

xii Introduction

where c = (ck(x, t); k = 0, 1, . . . , m) are bounded smooth functions. We show thatthe semigroup with the generator Ac(t) is obtained by an exponential transformationbased on ck(x, t); it is given by P c

s,t f (x) = E[f (Φs,t (x))Gs,t (x)], where

Gs,t (x) = exp{∑

k≥1

∫ t

s

ck(Φs,r (x), r) ◦ dWkr +

∫ t

s

c0(Φs,r (x) dr}.

Further, v(x, s) := P cs,t1

f1(x) is the unique solution of the final value problem (5)associated with the operator Ac(t). For jump-diffusion processes, we will alsoextend the integro-differential operator AJ (t) to another one, which will be denotedby A

c,dJ (t). Then, we will study the final value problem of the backward heat

equation associated with the operator Ac,dJ (t) (see Sect. 4.5).

We are also interested in the initial value problem of the time-dependent heatequation associated with the operator Ac(t) given by (6):

∂

∂tu(t, x) = Ac(t)u(t, x) for t > t0, u(t0, x) = f0(x). (7)

For this problem, we solve SDE (1) to the backward direction and obtain a backwardstochastic flow {Φs,t }; then, we define a backward semigroup by P c

s,t f (x) :=E[f (Φs,t (x))Gs,t (x)

], where Gs,t (x) is the exponential functional associated with

the backward flow {Φs,t }. Then, if f0(x) is a bounded smooth function, the solutionof the forward equation (7) exists uniquely, and it is represented by u(x, t) =P ct0,t

f0(x).We stress that the final function f1 (or initial function f0) is smooth in these

studies. Indeed, the smoothness of functions v(x, s) = P cs,t f1(x), etc. with respect

to x follows from the smoothness of f1(x) and the stochastic flow Φs,t (x) withrespect to x, a.s. If the function f1 is not smooth, we need additional arguments forthe solution of equations (5) and (7), which will be discussed in Chap. 6 using theMalliavin calculus.

Another subject of Chap. 4 is the investigation of the dual of a given diffusionand a jump-diffusion with respect to the Lebesgue measure. It will be shown that thedual of these processes can be constructed through the change-of-variable formulaconcerning stochastic flows {Φs,t }; the stochastic process defined by the inversemaps Xs = Ψs,t (x) = Φ−1

s,t (x) should be a dual process of Xt = Φs,t (x), and itis a backward diffusion or a backward jump-diffusion with respect to s, where t isthe initial time of the process. The dual semigroup of the semigroup {Ps,t } is thendefined by using the inverse flow {Ψs,t } as

P ∗s,t g(x) = E[g(Ψs,t (x)) det∇Ψs,t (x)],

where ∇Ψs,t is the Jacobian matrix of the diffeomorphism Ψs,t ;Rd → Rd .

Introduction xiii

In the latter half of this monograph, we will study the Malliavin calculus on theWiener space and the space of Poisson random measure, called Poisson space; wewill apply it for proving the existence of fundamental solutions for heat equationsdiscussed in Chap. 4.

In Chap. 5, we will discuss the Malliavin calculus on the Wiener space and thaton the Poisson space (space of Poisson random measure) separately. Then, we willcombine these two. For the Malliavin calculus on the Wiener space, we will restrictour attention to the problem of finding the smooth density for laws of Wienerfunctionals. Our discussion is motivated by Malliavin and Watanabe (Watanabe[116, 117]), but we will take a simple and direct approach; we will not considerthe Ornstein–Uhlenbeck semigroup on the Wiener space. Instead, we study thederivative operator Dt and its adjoint δ (Skorohod integral) directly. Then, we givean estimate of Skorohod integrals using Lp-Sobolev norms; a new proof is given forTheorem 5.2.1. A criterion for the smooth density of laws of Wiener functional willbe given in terms of the celebrated “Malliavin covariance” in Sect. 5.3.

Another reason why we do not follow Ornstein–Uhlenbeck semigroup argumentis that a similar fact is not known for Poisson space; in fact, we want to bringtogether the Malliavin calculus on the Wiener space and that on the Poisson space ina unified method. In Sects. 5.4, 5.5, 5.6, 5.7, and 5.8, we will discuss the Malliavincalculus on the Poisson space, which is characterized by a Lévy measure ν. A basicassumption for the Poisson random measure is that the characterizing Lévy measureis nondegenerate and satisfies the order condition at the center. Here, the origin 0 isregarded as the center of the Lévy measure defined on R

d ′ \ {0}. We will see that thedifference operator Du and its adjoint operator δ defined by Picard [92] work wellas Dt and δ do on the Wiener space.

Criteria of the smooth density for Poisson functionals are more complicated. Weneed a family of Lp-Sobolev norms conditioned to a family of neighborhoods of thecenter of the Lévy measure. It will be given in Sects. 5.5, 5.6, 5.7, and 5.8. Further,in Sects. 5.9, 5.10, and 5.11, we will study the Malliavin calculus on the product ofthe Wiener space and the Poisson space. A criterion for the smooth density of thelaw of a Wiener–Poisson functional will be given after introducing the “Malliavincovariance at the center.”

In the application of the Malliavin calculus to solutions of an SDE, properties ofstochastic flows defined by the SDE are needed. In Chap. 6, we study the existenceof smooth densities of laws of a nondegenerate diffusion and a nondegeneratejump-diffusion defined on a Euclidean space. The class of nondegenerate diffusionsincludes elliptic diffusions and hypo-elliptic diffusions. Further, the class of non-degenerate jump-diffusions includes pseudo-elliptic jump-diffusions. Let P c

s,t (or

Pc,ds,t ) be the semigroup associated with the generator Ac(t) (or A

c,dJ (t)). It will be

shown that its transition functions P cs,t (x, ·) (or P c,d

s,t (x, ·)) have densities pcs,t (x, y)

(or pc,ds,t (x, y)), which are smooth with respect to both variables x and y, and further,

the family of the densities is the fundamental solution of the backward heat equationassociated with the operator Ac(t) (or Ac,d

J (t)); the fundamental solution of the heatequation will be obtained as a family of density functions of a backward transition

xiv Introduction

function P cs,t (x, E) associated with the semigroup {P c

s,t }, etc. Thus, initial–finalvalue problems (5) and (7) are solved for any bounded continuous functions f0and f1, respectively.

In Sects. 6.7 and 6.8, for elliptic diffusions and pseudo-elliptic jump-diffusions,we will discuss the short-time asymptotics of the transition density functions as t ↓s, making use of the Malliavin calculus. Our Malliavin calculus cannot be appliedto jump processes or jump-diffusion processes which admit big jumps. Indeed, inorder to apply the Malliavin calculus, solution Xt of the SDE should be at least anelement of L∞− = ⋂

p>1 Lp, and the fact excludes solutions of SDEs with big

jumps. In Sect. 6.9, we consider such processes: we first truncate big jumps andget the smooth density. Then, we add big jumps and show that this action shouldpreserve the smooth density, where the short-time asymptotics of the fundamentalsolution will be utilized.

It is hard to apply the Malliavin calculus directly to (jump) diffusions on abounded domain of a Euclidean space or those defined on a manifold. In Sect. 6.10,we consider a process killed outside of a bounded domain of a Euclidean space. Inorder to get a smooth density for the killed process, we need two facts. One is ashort-time estimate of the density of a non-killed process. The other is a potentialtheoretic argument of a strong Markov process using hitting times. We show that thedensity function qc

s,t (x, y) of the killed transition function is smooth with respectto x and y; further, we show that qc

s,t (x, y) is the fundamental solution for thebackward heat equation (5) on an arbitrary bounded domain of a Euclidean spacewith the Dirichlet boundary condition.

Finally, in Chap. 7, we study SDEs on a manifold. Stochastic flow generated byan SDE on a manifold will be discussed in Sect. 7.1. Diffusions, jump-diffusions,and their duals will be treated in Sects. 7.2 and 7.3. Then, the smooth density for a(jump) diffusion on a manifold will be obtained by piecing together killed densitieson local charts. It will be discussed in Sects. 7.4 and 7.5.

A Guide for Readers Discussions of the Malliavin calculus for Poisson randommeasures contain some complicated and technical arguments. For the beginner orthe reader who is mainly interested in Wiener processes and diffusion processes, wesuggest to skip these arguments at the first reading. After Chap. 4, we could proceedin the following way:

Chapter 5, Sects. 5.1, 5.2, 5.3 −→ Chap. 6, Sects. 6.1, 6.2, 6.3, 6.7, 6.8, 6.10 −→Chap. 7, Sects. 7.1, 7.2, 7.3, 7.4.

The author hopes that this course should be readable.

Contents

1 Probability Distributions and Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . 11.1 Probability Distributions and Characteristic Functions . . . . . . . . . . . . . . 11.2 Gaussian, Poisson and Infinitely Divisible Distributions . . . . . . . . . . . . 81.3 Random Fields and Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Wiener Processes, Poisson Random Measures and Lévy

Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Martingales and Backward Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.6 Quadratic Variations of Semi-martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.7 Markov Processes and Backward Markov Processes . . . . . . . . . . . . . . . . 371.8 Kolmogorov’s Criterion for the Continuity of Random Field . . . . . . . 41

2 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.1 Itô’s Stochastic Integrals by Continuous Martingale

and Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.2 Itô’s Formula and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.3 Regularity of Stochastic Integrals Relative to Parameters . . . . . . . . . . . 552.4 Fisk–Stratonovitch Symmetric Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.5 Stochastic Integrals with Respect to Poisson Random Measure . . . . 642.6 Jump Processes and Related Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.7 Backward Integrals and Backward Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 Stochastic Differential Equations and Stochastic Flows . . . . . . . . . . . . . . . . . 773.1 Geometric Property of Solutions I; Case of Continuous SDE . . . . . . . 773.2 Geometric Property of Solutions II; Case of SDE with Jumps . . . . . . 813.3 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.4 Lp-Estimates and Regularity of Solutions; C∞-Flows . . . . . . . . . . . . . . 963.5 Backward SDE, Backward Stochastic Flow. . . . . . . . . . . . . . . . . . . . . . . . . . 1003.6 Forward–Backward Calculus for Continuous C∞-Flows . . . . . . . . . . . 1013.7 Diffeomorphic Property and Inverse Flow for Continuous SDE . . . . 1043.8 Forward–Backward Calculus for C∞-Flows of Jumps . . . . . . . . . . . . . . 109

xv

xvi Contents

3.9 Diffeomorphic Property and Inverse Flow for SDE with Jumps . . . . 1163.10 Simple Expressions of Equations; Cases of Weak Drift

and Strong Drift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4 Diffusions, Jump-Diffusions and Heat Equations . . . . . . . . . . . . . . . . . . . . . . . . 1254.1 Continuous Stochastic Flows, Diffusion Processes

and Kolmogorov Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.2 Exponential Transformation and Backward Heat Equation . . . . . . . . . 1294.3 Backward Diffusions and Heat Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.4 Dual Semigroup, Inverse Flow and Backward Diffusion . . . . . . . . . . . . 1404.5 Jump-Diffusion and Heat Equation; Case of Smooth Jumps . . . . . . . . 1464.6 Dual Semigroup, Inverse Flow and Backward

Jump-Diffusion; Case of Diffeomorphic Jumps. . . . . . . . . . . . . . . . . . . . . . 1554.7 Volume-Preserving Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614.8 Jump-Diffusion on Subdomain of Euclidean Space . . . . . . . . . . . . . . . . . 164

5 Malliavin Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675.1 Derivative and Its Adjoint on Wiener Space . . . . . . . . . . . . . . . . . . . . . . . . . 1685.2 Sobolev Norms for Wiener Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1745.3 Nondegenerate Wiener Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.4 Difference Operator and Adjoint on Poisson Space. . . . . . . . . . . . . . . . . . 1905.5 Sobolev Norms for Poisson Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1965.6 Estimations of Two Poisson Functionals by Sobolev Norms . . . . . . . . 2015.7 Nondegenerate Poisson Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2095.8 Equivalence of Nondegenerate Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2145.9 Product of Wiener Space and Poisson Space . . . . . . . . . . . . . . . . . . . . . . . . . 2225.10 Sobolev Norms for Wiener–Poisson Functionals . . . . . . . . . . . . . . . . . . . . 2265.11 Nondegenerate Wiener–Poisson Functionals . . . . . . . . . . . . . . . . . . . . . . . . . 2335.12 Compositions with Generalized Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

6 Smooth Densities and Heat Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2456.1 H -Derivatives of Solutions of Continuous SDE . . . . . . . . . . . . . . . . . . . . . 2466.2 Nondegenerate Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2506.3 Density and Fundamental Solution for Nondegenerate Diffusion. . . 2536.4 Solutions of SDE on Wiener–Poisson Space . . . . . . . . . . . . . . . . . . . . . . . . . 2596.5 Nondegenerate Jump-Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2656.6 Density and Fundamental Solution for Nondegenerate

Jump-Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2736.7 Short-Time Estimates of Densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2776.8 Off-Diagonal Short-Time Estimates of Density Functions . . . . . . . . . . 2846.9 Densities for Processes with Big Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2886.10 Density and Fundamental Solution on Subdomain . . . . . . . . . . . . . . . . . . 295

7 Stochastic Flows and Their Densities on Manifolds . . . . . . . . . . . . . . . . . . . . . . 3037.1 SDE and Stochastic Flow on Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3037.2 Diffusion, Jump-Diffusion and Their Duals on Manifold . . . . . . . . . . . 3117.3 Brownian Motion, Lévy Process and Their Duals on Lie Group . . . . 317

Contents xvii

7.4 Smooth Density for Diffusion on Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . 3217.5 Density for Jump-Diffusion on Compact Manifold . . . . . . . . . . . . . . . . . . 328

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Chapter 1Probability Distributions and StochasticProcesses

Abstract We introduce some basic facts on probability distributions and stochasticprocesses. Probability distributions and their characteristic functions are discussedin Sect. 1.1. Criteria for smooth densities of distributions are given by theircharacteristic functions. In Sect. 1.2, we consider Gaussian, Poisson and infinitelydivisible distributions and give criteria for these distributions to have smoothdensities. Concerning the density problem of an infinitely divisible distribution, westudy the Lévy measure in detail. Regarding that the origin 0 is the center of theLévy measure, we will give criteria for its smooth density by means of ‘the ordercondition’ at the center of the Lévy measure.

Then we consider stochastic processes. In Sect. 1.4, we consider Wiener pro-cesses, Poisson processes, Poisson random measures and Lévy processes. Amongthem, Poisson random measures are exposed in detail. Next, in Sects. 1.5 and 1.6,we discuss martingales, semi-martingales and their quadratic variations. These arestandard tools for the Itô calculus. In Sect. 1.7, we define Markov processes. Thestrong Markov property will be discussed. In Sect. 1.8, we study Kolmogorov’scriterion for a random field with multi-dimensional parameter to have a continuousmodification.

1.1 Probability Distributions and Characteristic Functions

Let Ω be a set and let F be a family of subsets of Ω with the following threeproperties: (a) Ω ∈ F , (b) if A ∈ F , then its complement set Ac ∈ F , and (c)if An ∈ F , n = 1, 2, . . ., then

⋃n An ∈ F . The set F is called a σ -field of Ω .

An element of Ω is denoted by ω and is called a sample. Elements of F are calledevents. The pair (Ω,F) is called a measurable space. A map P : F → [0, 1] iscalled a measure if it satisfies the following: (a) P(A) ≥ 0 for any A ∈ F , and (b)if An, n = 1, 2, . . . are disjoint, then P(

⋃∞n=1 An) = ∑∞

n=1 P(An). Further, if (c)P(Ω) = 1 is satisfied, P is called a probability measure. The triple (Ω,F , P ) iscalled a probability space.

© Springer Nature Singapore Pte Ltd. 2019H. Kunita, Stochastic Flows and Jump-Diffusions, Probability Theoryand Stochastic Modelling 92, https://doi.org/10.1007/978-981-13-3801-4_1

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2 1 Probability Distributions and Stochastic Processes

Let (Ω,F , P ) be a probability space. A family of events {Aλ, λ ∈ Λ} iscalled independent, if any finite subset {Aλ1, . . . , Aλn} of {Aλ, λ ∈ Λ} satisfiesP(

⋂ni=1 Aλi ) =

∏ni=1 P(Aλi ). Next, let G be a subset of F . If G is a σ -field, it is

called a sub σ -field of F . Suppose that {Gλ, λ ∈ Λ} is a family of sub σ -fields ofF . It is called independent, if any collection of events {Aλ, λ ∈ Λ}, where each Aλ

runs in Gλ, is independent.Let A,B be events. Suppose that P(B) > 0. The conditional probability of A

given B is defined by P(A|B) = P(A∩B)P (B)

. Then P(A|B) = P(A) holds if and onlyif A and B are independent.

Let Rd be a d-dimensional Euclidean space. Its elements are denoted by x =(x1, . . . , xd). Let B(Rd) be the Borel field of R

d . A probability measure μ on(Rd ,B(Rd)) is called a probability distribution or simply a distribution. If integralsbi =

∫Rd xidμ(x) and aij = ∫

Rd (xi − bi)(xj − bj )μ(dx) exist, b = (b1, . . . , bd) iscalled the mean vector and A = (aij ) is called the covariance matrix. For a vectorb = (b1, . . . , bd), we set |b| = (

∑i=1 |bi |2)1/2 and for a matrix A = (aij ), we set

|A| = (∑

i,j a2ij )

1/2. When d = 1, R1 coincides with the space R of real numbers.b and A are called the mean and the variance, respectively.

Let S be a Hausdorff topological space with the second countability and let B(S)be the Borel field of S. A map X;Ω → S is called measurable (or G-measurable)if the set {ω; X(ω) ∈ E} belongs to F (or G) for any Borel subset E in S. Ameasurable map X;Ω → S is called an S-valued random variable. If S = R

d ,X is called a d-dimensional random variable. Further, if d = 1, it is a real (real-valued) random variable or simply a random variable. Let {Xλ; λ ∈ Λ} be a familyof S-valued random variables. We denote by σ({Xλ}) the smallest sub σ -field of Fwith respect to which {Xλ} are measurable. Let {Yλ′ ; λ′ ∈ Λ′} be another family ofS′-valued random variables and let σ({Yλ′ }) be the smallest σ field with respect towhich {Yλ′ } are measurable. Then two families of random variables {Xλ} and {Yλ′ }are called mutually independent if these sub σ -fields are independent.

Given an S-valued random variable X, we set μ(E) = P({X ∈ E}), E ∈ B(S).It is a probability measure on (S,B(S)), called the law of X.

The expectation of a real random variable X is defined as the Lebesgue integralby its law μ; E[X] = ∫

xμ(dx), if the integral is well defined. It is equal to theLebesgue integral of X by the measure P ; E[X] = ∫

ΩX(ω)P (dω). If E[|X|] is

finite, X is said to be an integrable random variable.Let A be an event. We denote by 1A the indicator function of the event A, i.e.,

1A(ω) is a function taking the value 1 if ω ∈ A and the value 0 if ω /∈ A. Theexpectation E[X1A] is often denoted by E[X;A].

Let {Xn, n = 1, 2, . . .} be a sequence of real random variables. Let X be anotherreal random variable. The sequence {Xn} is said to converge to X almost surely,simply written as Xn → X, a.s, if there exists Ω ∈ F such that P(Ω) = 1 andfor any ω ∈ Ω , {Xn(ω)} converges to X(ω). If P(|Xn − X| > ε) converges to0 as n → ∞ for any ε > 0, {Xn} is said to converge to X in probability. Letp ≥ 1. If E[|Xn|p] < ∞ holds for any n and E[|Xn − X|p] → 0 as n → ∞,{Xn} is said to converge to X in Lp. A sequence of random variables {Xn} iscalled uniformly integrable, if for any ε > 0 there is a positive constant c such

1.1 Probability Distributions and Characteristic Functions 3

that supn

∫|Xn|>c

|Xn| dP < ε. If {Xn} is a sequence of a uniformly integrable

random variables converging to X in probability, then {Xn} converges to X in L1,i.e., limn→∞ E[|Xn −X|] = 0.

Let μ1 and μ2 be distributions on Rd . For a Borel subset E of Rd , we set μ(E) =∫

Rd μ1(E − x)μ2(dx). Then μ is a distribution on Rd . We denote μ by μ1 ∗ μ2

and call it the convolution of two distributions μ1 and μ2. It is immediate to seeμ1 ∗ μ2 = μ2 ∗ μ1. Let X, Y be independent Rd -valued random variables and letμX and μY be their laws, respectively. Let μX+Y be the law of X + Y . Then itholds that μX+Y = μX ∗ μY . Let X and Y be independent random variables. ThenE[f (X)g(Y )] = E[f (X)]E[g(Y )] holds for any bounded continuous functionsf, g.

Let X be an S-valued random variable and let Z be a real integrable randomvariable. The weighted law of X with respect to Z is a bounded signed measure on(S,B(S)) defined by μZ(E) = E[Z; {X ∈ E}]. Its total variation measure |μZ|is given by |μZ|(E) = E[|Z|; {X ∈ E}]. If a bounded signed measure μ on R

d

satisfies∫Rd |x|n|μ|(dx) < ∞, where |μ| is the total variation of the measure μ,

then the measure μ is said to have an n-th moment.The characteristic function of a bounded signed measure μ on R

d is defined by

ψ(v) =∫

Rd

ei(v,x)μ(dx), v = (v1, . . . , vd) ∈ Rd , (1.1)

where i = √−1 and (v, x) = ∑dj=1 vjxj . Let Z be a real integrable random

variable, X = (X1, . . . , Xd) be an Rd -valued random variable and let μZ be its

weighted law. Let ψ(v) be its characteristic function. Then we have

E[ei(v,X)Z] =∫

Rd

ei(v,x)μZ(dx) = ψ(v).

For a differentiable function f (x) on Rd , the partial derivatives ∂

∂xif (x) are

denoted by ∂xi f . We set ∂f = ∂xf = (∂x1f, . . . , ∂xd f ). Further, if f is avector function (f 1, . . . , f d ′), we set ∂f = (∂xj f

k)j=1,...,d,k=1,...,d ′ . It is calledthe Jacobian matrix. When d = d ′, we will often denote it by ∇f .

The set of positive integers is denoted by N and the set of nonnegative integersis denoted by N. Let j be a multi-index of nonnegative integers of length d; j =(j1, . . . , jd), where j1, . . . , jd are nonnegative integers. We set |j| = j1 + · · · + jd .Let j′ = (j ′1, . . . , j ′d) be another multi-index of length d. If ji ≤ j ′i holds for anyi = 1, . . . , d, then we denote the relation by j ≤ j′. We define the higher orderdifferential operator ∂ j and product xj for x = (x1, . . . , xd) ∈ R

d by

∂ j = ∂ jx = ∂

j1x1 · · · ∂jdxd , xj = x

j11 · · · xjd

d ,


if |j| ≥ 1. If |j| = 0, i.e., j = (0, . . . , 0), we define ∂0f = f and x0 = 1.Let n0 ∈ N. If a function f (x) on R

d is n0-times continuously differentiable and|∂ jf (x)| are bounded for any j with |j| ≤ n0, f is said to be a function of Cn0

b -class,or a C

n0b -function.

Proposition 1.1.1 Let n0 ∈ N. If a bounded signed measure μ has n0-th moment,then its characteristic function ψ(v) is a C

n0b -function. Further, it holds that

∂ jvψ(v) = i|j|

∫

Rd

ei(v,x)xjμ(dx) (1.2)

for any j with |j| ≤ n0.

Proof We prove (1.2) by the induction of n0 ∈ N. The case n0 = 0 is trivial.Suppose it is valid for j with |j| ≤ n0 − 1. Set ψ(v) := ∫

ei(v,x)xjμ(dx). We havefor h �= 0

ψ(v + hej )− ψ(v)

h=

∫ {ei(v+hej ,x) − ei(v,x)

h

}xjμ(dx),

where ej , j = 1, . . . , d are unit vectors in Rd . Let h tend to 0 in the above. Since

∣∣∣ei(v+hej ,x) − ei(v,x)

h

∣∣∣ ≤ |xj |

holds for any h, the right-hand side converges to∫ei(v,x)(ixj )x

jdμ, by theLebesgue convergence theorem. Therefore ψ(v) is differentiable with respect tovj and we get

∂vj ψ(v) = i

∫ei(v,x)xj x

jμ(dx).

It holds that ∂vj ψ(v) = ∂ j+ejψ(v) and∫ei(v,x)xj x

jμ(dx) = ∫ei(v,x)xj+ejμ(dx).

Therefore we get the formula (1.2) for j + ej . ��From the above proposition, the moment of the measure μ can be computed as

∫xjμ(dx) = i−|j|∂ j

vψ(v)

∣∣∣v=0

.

A Borel subset E of Rd is called μ-continuous, if μ(∂E) = 0 holds. Here ∂E isthe boundary set of E defined by E − Eo.

Theorem 1.1.1 (Lévy’s inversion formula) The characteristic function deter-mines the measure μ uniquely: If a rectangular set E = ∏d

j=1(aj , bj ] isμ-continuous, then


μ(E) = limN→∞

( 1

2π

)d∫ N

−N

· · ·∫ N

−N

d∏

j=1

e−ivj bj − e−ivj aj

−ivjψ(v1, . . . , vd) dv1 · · · dvd .

(1.3)

Proof Using Fubini’s theorem, we have

∫ N

−N

· · ·∫ N

−N

d∏

j=1


−ivjψ(v1, . . . , vd) dv1 · · · dvd

=∫ N

−N

·∫ N

−N

d∏

j=1


−ivj

( ∫· · ·

∫ei(∑

j vj xj )μ(dx1 · · · dxd))dv1 · · · dvd

=∫· · ·

∫ ( ∫ N

−N

· · ·∫ N

−N

d∏

j=1

eivj (xj−bj ) − eivj (xj−aj )

−ivjdv1 · · · dvd

)μ(dx1 · · · dxd)

=∫

· · ·∫ ( d∏

j=1

∫ N

−N

eivj (xj−bj ) − eivj (xj−aj )

−ivjdvj

)μ(dx1 · · · dxd).

Set the integral∫ N

−N· · · dvj of the above by J (N, xj , aj , bj ). It holds that

J (N, x, a, b) = 2∫ N

0

sin(x − a)v

vdv − 2

∫ N

0

sin(x − b)v

vdv.

Since limN→∞∫ N

0sin tt

dt = π2 holds, we have

limN→∞ J (N, x, a, b) =

⎧⎨

⎩

0, if x < a or b < x,

π, if x = a or x = b,

2π, if a < x < b.

The above convergence is a bounded convergence. Therefore, by Lebesgue’sbounded convergence theorem,

limN→∞

∫ N

−N

· · ·∫ N

−N

d∏

j=1

J (N, xj , aj , bj )μ(dx1 · · · dxd) = (2π)dμ(E),

if μ(∂E) = 0. We have thus proved the inversion formula (1.3). ��Corollary 1.1.1 Let X1, . . . , Xd be real random variables. Denote by X its vector(X1, . . . , Xd). If

E[ei(v,X)] =d∏

j=1

E[eivjXj ] (1.4)


holds for all v, then X1, . . . , Xd are mutually independent.Conversely, if X1, . . . , Xd are mutually independent, (1.4) is valid. Further, the

characteristic function of Y = X1+· · ·+Xd is equal to the product of characteristicfunctions of Xj , j = 1, . . . , d, i.e.,

∏dj=1 E[eivXj ].

Proof Let μ be the law of X and let μj , j = 1, . . . , d be one-dimensional lawsof Xj , j = 1, . . . , j respectively. Suppose that (1.4) holds. Then equation (1.4) iswritten as

∫ei(v,x) dμ(x1, . . . , xd) =

∫ei(v,x) dμ1(x1) · · · dμ(xd).

Therefore we have μ = μ1 × · · · × μd by the above theorem. Hence X1, . . . , Xd

are independent.Conversely, if X1, . . . , Xd are independent, then eiv1X1 , . . . , eivdXd are also

independent. Therefore (1.4) holds. The formula implies that the characteristicfunction of Y is equal to

∏dj=1 E[eivXj ]. ��

We study the relation for the existence of the smooth density of the measure μ

and the decay property of the characteristic function ψ(v) as |v| → ∞.

Proposition 1.1.2 Let ψ(v) be the characteristic function of a bounded signedmeasure μ.

1. If∫ |ψ(v)| dv < ∞, then μ has a bounded continuous density function f (x).

Further, we have the inversion formula:

f (x) =( 1

2π

)d∫

Rd

e−i(v,x)ψ(v) dv. (1.5)

2. If∫ |ψ(v)||v|n0 dv < ∞ for some positive integer n0, then the density function

f (x) is of Cn0b -class. Further, for any multi-index j with |j| ≤ n0, we have

∂ jf (x) = (−i)|j|

(2π)d

∫

Rd

e−i(v,x)ψ(v)vj dv. (1.6)

3. Let m0 be a positive integer. If ψ(v) is of Cm0b -class and functions ∂ i

v(ψ(v)vj)

are integrable and converge to 0 as |v| → ∞ for any |j| ≤ n0 and |i| ≤ m0, thenthe density function is of Cn0

b -class and satisfies

(ix)i∂ jf (x) = (−i)|j|

(2π)d

∫

Rd

e−i(v,x)∂ iv(ψ(v)vj) dv (1.7)

for any |j| ≤ n0, |i| ≤ m0. Furthermore, there exists a positive constant c suchthat |∂ jf (x)| ≤ c

(1+|x|)m0 holds for all x.


Proof

1. If |ψ(v)| is integrable, we can rewrite the inversion formula (1.3) as

μ(E) =( 1

2π

)d∫ ∞

−∞· · ·

∫ ∞

−∞

d∏

j=1


−ivjψ(v1, . . . , vd) dv1 · · · dvd,

since the integrand is an integrable function on Rd with respect to the Lebesgue

measure dv1 · · · dvd . We consider the right-hand side. For fixed a1, . . . , ad , it isa function of (b1, . . . , bd), which we denote by F(b1, . . . , bd). Set

φ(vj , aj , bj ) = e−ivj aj − e−ivj bj

ivj.

It holds that |φ(vj , aj , bj )| ≤ |bj − aj | and |∂bj φ(aj , vj , bj )| ≤ 1 for anyvj , aj , bj . Then for any j = 1, . . . , d, we can change the order of the partialderivative ∂bj and the integral; we find that for any j , F(b1, . . . , bd) is partiallydifferentiable with respect to bj . Repeating this argument for j = 1, . . . , d, wefind that the function F is partially differentiable with respect to bj , j = 1, . . . , dand we get

∂b1 · · · ∂bd F (b1, . . . , bd)=( 1

2π

)d∫ ∞

−∞· · ·

∫ ∞

−∞e−i

∑dj=1 vj bj ψ(v1, . . . , vd) dv1 · · · dvd .

Denote the above function by f (b1, . . . , bd). It is a bounded continuous functionand we have

∫ b1a1

· · · ∫ bdad

f (x) dx = μ(E) if E is a μ-continuous rectangular

set Πdj=1(aj , bj ]. Now we can choose an increasing sequence of μ-continuous

rectangular sets En such that⋃

n En = Rd . Then we have μ(En) → 1. This

implies∫Rd f (x) dx = 1. Therefore, all rectangular sets should be μ-continuous

and the assertion is proved.2. If |ψ(v)||v|n0 is integrable on R

d , we can change the order of derivative operator∂ j, |j| ≤ n0 and the integral operator in the inversion formula (1.5). Therefore weget the assertion.

3. Suppose that ψ(v) satisfies conditions of 3. The density function f (x) is of Cn0b -

class by 2. By the formula of integration by parts, we have

∫

Rd

e−i(v,x)∂vk (ψ(v)vj) dv = ixk

∫

Rd

e−i(v,x)ψ(v)vj dv.

Repeating this argument, we get the formula (1.7) for any |j| ≤ n0 and |i| ≤ m0.Further, since the right-hand side of (1.7) is a bounded function, |x||i||∂ jf (x)| isa bounded function. Therefore |∂ jf (x)| ≤ c/(1 + |x|)m0 holds. ��Let f (x) be a (complex) Cn0 -function on R

d . It is called rapidly decreasing ifthe function |∂ jf (x)|(1 + |x|2)i converges to 0 as |x| → ∞ for any |j| ≤ n0 and


nonnegative integer i. A C∞-function f (x) on Rd is said to be rapidly decreasing,

if it is a rapidly decreasing Cn0 -function for any n0 ∈ N.

Theorem 1.1.2 Let ψ(v) be the characteristic function of a bounded signedmeasure μ. If ψ(v) is a rapidly decreasing continuous function, then the measure μ

has a C∞-density function f . Further, if ψ(v) is a rapidly decreasing C∞-function,f is a rapidly decreasing C∞-function.

Proof If ψ(v) is a rapidly decreasing continuous function, then |ψ(v)||v|n is anintegrable function for any n. Therefore μ has a C∞

b -density function. Further, ifψ(v) is a rapidly decreasing C∞-function, ∂ i

v(ψ(v)vj) are integrable and convergesto 0 as |v| → ∞ for any i and j. Then |xi||∂ jf (x)| is a bounded function for any iand j by (1.7). Therefore f (x) should be rapidly decreasing. ��

1.2 Gaussian, Poisson and Infinitely Divisible Distributions

We give some distributions and their characteristic functions which will be used inthis monograph.

1. Exponential distribution. Let μ be a distribution on R having the densityfunction f (x) given by f (x) = λe−λx if x > 0, and f (x) = 0 if x ≤ 0, whereλ > 0. Then μ is called the exponential distribution with parameter λ. Its mean is 1

λ

and the variance is 1λ2 . Further, its characteristic function is ψ(v) = λ

λ−iv.

2. Gamma distribution. Let μ be a distribution on R having the density function

f (x) given by f (x) = λkxk−1e−λx

Γ (k)if x > 0, and f (x) = 0 if x ≤ 0, where λ > 0,

k is a positive integer and Γ (k) = (k − 1)!. Then μ is called a Gamma distributionwith parameters λ and k. Its mean is k

λand variance is k

λ2 . Its characteristic function

is ψ(v) =(

λλ−iv

)k

. If k = 1, it coincides with the exponential distribution with

parameter λ.Let σ1, . . . , σk be mutually independent random variables with the same expo-

nential distribution with parameter λ. Then the law of τk := σ1 + · · · + σk isthe Gamma distribution with parameters λ and k. Indeed, since laws of σj areexponential distributions with the same parameter λ, their characteristic functionsare equal to λ

λ−iv. Sine σj are independent, the characteristic function of τk should

be the k-fold product of λλ−iv

, i.e., it is equal to(

λλ−iv

)k

(Corollary 1.1.1). Hence

the law of τk is equal to the Gamma distribution with parameters λ and k.3. Gaussian distribution. Let d ′ be a positive integer and let b = (b1, . . . , bd

′) be

a d ′-vector. Let A = (aij ) be a d ′ × d ′ positive definite symmetric matrix and letdetA be its determinant. Assuming that detA �= 0, we define

f (x) = 1

(2π)d2 | detA| 1

2

exp{− 1

2(x − b,A−1(x − b))

}. (1.8)

1.2 Gaussian, Poisson and Infinitely Divisible Distributions 9

Then μ(B) := ∫Bf (x) dx, B ∈ B, is a probability distribution on R

d ′ , called ad ′-dimensional Gaussian distribution. Its mean vector is equal to b and covariancematrix is equal to A. It is known that its characteristic function is equal to

ψ(v) = exp{i(v, b)− 1

2(v,Av)

}, v ∈ R

d ′ . (1.9)

Conversely, for a given d ′-vector b and a nonnegative definite symmetric d ′ × d ′-matrix A, there exists a distribution μ on R

d ′ , whose characteristic function is equalto (1.9). Such a distribution is unique. It is also called a Gaussian distribution. Itsmean and covariance are again given by b and A, respectively. If the matrix A ispositive definite, μ has a density function written as in (1.8). If the rank of A is lessthan d ′, the measure μ is singular with respect to the Lebesgue measure.

4. Poisson distribution. Let μ be a distribution on R having point masses on theset N := {0, 1, 2, . . .} and

μ({n}) = λn

n! e−λ, n = 0, 1, 2, . . . , (1.10)

where λ is a positive constant. It is called a Poisson distribution. Its mean and thevariance are equal to λ. λ is called the parameter of the Poisson distribution. Itscharacteristic function is computed as

ψ(v) = exp{λ(eiv − 1)

}, v ∈ R. (1.11)

5. Compound Poisson distribution. Let μ be a distribution on Rd ′ , whose

characteristic function is given by

ψ(v) = exp{ ∫

Rd′0

(ei(v,z) − 1)ν(dz)}, v ∈ R

d ′ . (1.12)

where ν is a finite measure on Rd ′0 = R

d ′ \ {0}. The distribution μ is called acompound Poisson distribution (associated with measure ν). A Poisson distributionis a one-dimensional compound Poisson distribution associated with ν = λδ1, whereδ1 is the delta measure at the point 1.

6. Infinitely divisible distribution. A distribution μ on Rd ′ is called infinitely

divisible, if the characteristic function of μ is represented by

ψ(v) = exp{i(v, b)− 1

2(v,Av)+

∫

Rd′0

(ei(v,z)−1−i(v, z)1D(z))ν(dz)}. (1.13)

Here, v, b ∈ Rd ′ , A is a matrix as in (1.9), D = {z ∈ R

d ′0 ; |z| ≤ 1} and ν is a measure

on Rd ′0 satisfying

∫R

d′0(|z|2∧1)ν(dz) < ∞. A measure ν on R

d ′0 satisfying the above

property is called a Lévy measure. The triple (A, ν, b) is called the characteristics


of the infinitely divisible distribution. Occasionally, we regard the Lévy measure ν

is a measure on Rd ′ satisfying ν({0}) = 0. We call the point 0 the center of the Lévy

measure.Let μ be a distribution on R

d ′ . Suppose that for any n ∈ N, there exists adistribution μn such that μ = μn

n, where μnn is the n-times convolution of the

distribution μn. It is known that its characteristic function ψ(v) is representedby (1.13). Hence μ is an infinitely divisible law. The formula (1.13) is called theLévy–Khintchine formula. Conversely, if the characteristic function of a distributionμ is written as (1.13), for any n ∈ N there are distributions μn such that μ = μn

n.This is the origin of the word ‘infinitely divisible’.

7. Stable distribution. An infinitely divisible distribution μ is called stable if forany a > 0, there are b > 0 and c ∈ R

d ′ such that its characteristic function ψ(v)

satisfies ψ(v)a = ψ(bv)ei(c,v). Further, if the above holds for any a > 0, b = a1/β

with some 0 < β ≤ 2 and c = 0, then μ is called an β-stable distribution. β iscalled the index of the stable distribution. If the drift vector b and the Lévy measureν are 0, the distribution μ satisfies the above property with β = 2. Hence a Gaussiandistribution with mean 0 is 2-stable. If μ is β-stable with 0 < β < 2, then A = 0.Further, using the polar coordinate, the Lévy measure ν is represented by

ν(B) =∫

Sd′−1

λ(dϑ)

∫ ∞

01B(rϑ)

dr

r1+β, (1.14)

where Sd ′−1 = {z ∈ Rd ′ ; |z| = 1} and λ is a finite measure on Sd ′−1.

In the one-dimensional case, S0 is equal to two points {−1,+1}. The Lévymeasure of the one-dimensional stable distribution is given by

ν(dz) ={c1z

−1−β dz on (0,∞),

c2|z|−1−β dz on (−∞, 0),(1.15)

with c1 ≥ 0, c2 ≥ 0, c1 + c2 > 0. In particular if the distribution μ is symmetric,then it holds that c1 = c2 > 0. Its characteristic function is given by

ψ(v) = exp{− c|v|β

(1 − iγ tan

πβ

2signv

)+ iτv

}, (1.16)

where c > 0, γ ∈ [−1, 1] and τ ∈ R, if β �= 1.If μ is a d ′-dimensional rotation invariant β-stable distribution, its characteristic

function is given by

ψ(v) = exp{−c|v|β}, v ∈ Rd ′ , (1.17)

where c > 0. For further details of infinitely divisible distributions and stabledistributions, we refer to Sato [99].


Let us consider whether or not an infinitely divisible distribution μ has a smoothdensity. If μ is rotation invariant and β-stable, its characteristic function is givenby (1.17). It is a rapidly decreasing function of v and ψ(v)|v|n is an integrablefunction for any positive integer n. Therefore the measure μ has C∞

b -densityfunction by Proposition 1.1.2.

We are interested in more general infinitely divisible distributions. We startwith a one-dimensional infinitely divisible distribution with the second moment;its characteristic function is represented by

ψ(v) = exp{ibv +

∫

R

(eivz − 1 − ivz

z2

)ν′(dz)

}, (1.18)

where ν′ is a bounded measure on R. (At z = 0, we identify eivz−1−ivzz2 with

− 12v

2.) Setting ν′({0}) = a and ν(dz) := 1z2 ν

′(dz) for z �= 0 (Lévy measure),we get the Lévy–Khintchine formula with characteristics (a, b, ν). Now, if a > 0,the distribution μ has a C∞-density, because μ is written as a convolution of twodistributions μ1, μ2, where μ1 is a Gaussian measure with mean b and variance a. Ifa = 0, we consider the order of the concentration of the mass of ν around the center0. Suppose that the mass of ν′ is highly concentrated near the center and satisfiesthe order condition

0 < lim infρ→0

ν′(B0(ρ))

ρα≤ lim sup

ρ→0

ν′(B0(ρ))

ρα< ∞, (1.19)

for some 0 < α < 2, where B0(ρ) = {z; |z| ≤ ρ}. Then the Lévy measure ν is saidto satisfy the order condition of exponent α at the center. If the Lévy measure ν isa finite measure, it does not satisfy the order condition, since lim inf of (1.19) is 0for any 0 < α < 2. It was shown by Orey that if the one-dimensional Lévy measuresatisfies the order condition of exponent 0 < α < 2 at the center, then the law μ hasa C∞-density.

Now, let μ be a d ′-dimensional infinitely divisible distribution with character-istics (A, ν, b). Suppose that the matrix A is nondegenerate. Then μ is written asthe convolution of two distributions μ1 and μ2, where μ1 is a Gaussian distributionwith mean b and covariance A, and μ2 is an infinitely divisible distribution withcharacteristics (0, ν, 0). Since μ1 has a C∞-density, the convolution μ1 ∗ μ2 hasalso a C∞-density.

Suppose next that the matrix A is degenerate. Let {A0(ρ); 0 < ρ < 1} be afamily of open neighborhoods of the center in R

d ′ , which satisfies the followingtwo properties for any 0 < ρ < 1:

(i) A0(ρ) ⊂ B0(ρ) = {z ∈ Rd ′ ; |z| < ρ}, (ii)

⋃

ρ′<ρ

A0(ρ′) = A0(ρ).


It is called a family of star-shaped neighborhoods of the center. We define a functionϕ0(ρ), 0 < ρ < 1 by

ϕ0(ρ) :=∫

A0(ρ)

|z|2ν(dz) = ν′(A0(ρ)), (1.20)

where ν is the Lévy measure of the infinitely divisible distribution and ν′(dz) =|z|2ν(dz). The function ϕ0(ρ), 0 < ρ < 1 is a left continuous increasing(nondecreasing) function of ρ. Let α be a real number such that 0 < α < 2. Thefunction ϕ0(ρ) is said to satisfy the order condition of exponent α at the center (withrespect to the family of star-shaped neighborhoods {A0(ρ)}), if there is a positiveconstant c satisfying

ϕ0(ρ) ≥ cρα, ∀0 < ρ < 1.

For each ρ, we define a matrix Γρ of the second moment by

Γρ =(∫

A0(ρ)zizj ν(dz)

ϕ0(ρ)

), (1.21)

where z = (z1, . . . , zd′) ∈ R

d ′ . The pair (A, ν) is called nondegenerate at thecenter (with respect to a family of star-shaped neighborhoods {A0(ρ)}), if matrices{A+ Γρ; 0 < ρ < 1} are uniformly positive definite. In particular, when the matrixA is equal to O, the Lévy measure ν is said to be nondegenerate at the center.

We define the order for symmetric d ′ ×d ′ matrices. Let A1 and A2 be symmetric(d ′ × d ′)-matrices. If (u,A1u) ≥ (u,A2u) holds for all u ∈ R

d ′ , we say that A1is greater than or equal to A2 and write this as A1 ≥ A2. Now if {A + Γρ} isuniformly positive definite, there exists a symmetric positive definite matrix B suchthat A + Γρ ≥ B holds for all ρ. B is called a lower bound of {A + Γρ}. Takinga positive constant c such that (u, (A + Γρ)u) ≥ c|u|2 holds for all u ∈ R

d ′ , thenB = cI is what we want. Here I is the identity matrix.

We will obtain an upper estimate of the characteristic function of a nondegenerateinfinitely divisible distribution. It is interesting to compare the next lemma with theequation (1.17) for rotation invariant stable distribution.

Lemma 1.2.1 Let μ be an infinitely divisible distribution with characteristics(A, ν, b), where the matrix A is degenerate. Let ψ be its characteristic function.If, at the center, the pair (A, ν) is nondegenerate and satisfies the order condition ofexponent α with respect to a suitable family of star-shaped neighborhoods {A0(ρ)},there exist positive constants c1, c2 such that

|ψ(v)| ≤ c2 exp{−c1|v|2−α}, ∀|v| ≥ 1. (1.22)


Proof From the Lévy–Khintchine formula, we have

|ψ(v)| = exp{− 1

2(v,Av)+

∫

Rd′0

R(ei(v,z)−1−i(v, z)1D(z)

)ν(dz)

},

where Rw means the real part of the complex number w. We have

−R(ei(v,z)−1−i(v, z)1D(z)

)= 1 − cos(v, z) = 2 sin2 (v, z)

2≥ 1

2(v, z)2,

if |(v, z)| ≤ 23π . Therefore,

−∫

Rd′0

R(ei(v,z) − 1 − i(v, z)1D(z)

)ν(dz) ≥ 1

2

∫

{|(v,z)|≤ 2π3 }

(v, z)2ν(dz).

Let Γρ the matrix defined by (1.21). Since A0(|v|−1) ⊂ {|(v, z)| ≤ 2π3 }, we have

∫

{|(v,z)|≤ 2π3 }

(v, z)2ν(dz) ≥∫

A0(|v|−1)

(v, z)2ν(dz) =∑

i,j

vivjΓij

|v|−1ϕ0(|v|−1).

Since ϕ0(|v|−1) ≥ c|v|−α holds, we have

1

2(v, Av)−

∫R

(ei(v,z)−1−i(v, z)1D(z)

)ν(dz) ≥ c ∧ 1

2

∑

i,j

vi

|v| α2vj

|v| α2 (A+Γ|v|−1)ij .

There exists a positive constant c′ such that A+ Γρ ≥ c′I holds for all 0 < ρ < 1.

Then the last term of the above dominates c′(c∧1)2 |v|2−α if |v| > 1. Consequently

there exists c1 > 0 such that |ψ(v)| ≤ exp{−c1|v|2−α} if |v| > 1. ��From the above lemma, ψ(z) is a rapidly decreasing function. Further, if the

Lévy measure ν has moments of any order, i.e.,∫ |z|nν(dz) < ∞ for any n ∈ N, the

characteristic function given by (1.13) is infinitely differentiable. Then the densityfunction is a rapidly decreasing C∞-function. Therefore we have the followingtheorem from Theorem 1.1.2.

Theorem 1.2.1 Let μ be an infinitely divisible distribution with characteristics(A, ν, b).

1. If the matrix A is nondegenerate, μ has a C∞b -density.

2. If at the center, the pair (A, ν) is nondegenerate and satisfies the order conditionof exponent 0 < α < 2 with respect to a suitable family of star-shapedneighborhoods, then μ has a C∞

b -density. Suppose further that the Lévy measurehas moments of any order. Then the density is a rapidly decreasing C∞-function.


We can check easily that the Lévy measure of a β-stable distribution satisfies theorder condition of exponent 2−β at the center, since its Lévy measure is representedby (1.14).

It should be noted that in the definition of the order condition at the centerof the Lévy measure, any regularity for the Lévy measure ν is not assumed. Itmay or may not be absolutely continuous. It may be a measure with point massonly. However, the characteristic function of the infinitely divisible distribution hasan upper estimate (1.22), which is compatible with the formula (1.17) for stabledistribution.

The nondegenerate condition at the center of the Lévy measure depends on thechoice of the family of neighborhoods {A0(ρ)}. We will consider an example. LetX1, . . . , Xd ′ be mutually independent one-dimensional random variables, whoselaws are stable with indices β1, . . . , βd ′ , respectively. Then the vector X =(X1, . . . , Xd ′) is subject to a d ′-dimensional infinitely divisible distribution. Further,the Lévy measure of the law of X is concentrated on axis {zej ; z ∈ R}, j =1, . . . , d ′, where ej are unit vectors in R

d ′ . On each axis {zej }, the Lévy measuretakes the form νj (dz) = cj

|z|1+βjdz.

If β1 = · · · = βd ′ = β, the Lévy measure is nondegenerate at the center point 0with respect to the family of open balls A0(ρ) = {|z| < ρ}. It holds that Γρ = I andϕ0(ρ) = cj

2−βρ2−β . If βj , j = 1, . . . , d ′ are distinct, the Lévy measure is no longer

nondegenerate at the center with respect to the family of open balls. We should thentake {A0(ρ)} as a family of star-shaped neighborhood such that

A0(ρ) ∩{zej ; z ∈ R

}=

{|z|

2−βjα ej ; |z| < ρ

}

holds for any j , where α = 2 − minj βj . Then the Lévy measure is nondegenerateat the center with respect to the above family of neighborhoods. This is a reason thatwe call A0(ρ) a star-shaped neighborhood. See [46].

1.3 Random Fields and Stochastic Processes

A family of S-valued random variables {Xλ; λ ∈ Λ}, where Λ is a topologicalspace, is called an S-valued random field (with parameter λ ∈ Λ). A random field{Xλ, λ ∈ Λ} is called measurable if Xλ(ω) is measurable with respect to the productσ -field B(Λ)×F , where B(Λ) is the topological Borel field of Λ. When S is Rd orR, it is called a d-dimensional or real-valued random field, respectively.

If a statement holds for ω ∈ Ω with P(Ω) = 1, we say that the statement holdsfor almost all ω or almost surely and it is written simply as a.s. Let X and X be S-valued random variables on the same state space S. If P({ω;X(ω) = X(ω)}) = 1holds, X is said to be equal to X almost surely and both are written as X = X a.s.Let {Xλ, λ ∈ Λ} and {Xλ, λ ∈ Λ} be two S-valued random fields with the same

1.4 Wiener Processes, Poisson Random Measures and LévyProcesses 15

parameter set Λ. These two random fields are said to be equivalent provided thatP(Xλ = Xλ) = 1 holds for all λ ∈ Λ. Further, {Xλ, λ ∈ Λ} is called a modificationof {Xλ, λ ∈ Λ}. If laws of (Xλ1 , . . . , Xλn) and (Xλ1 , . . . , Xλn) coincide for anyfinite subset λ1, . . . , λn of Λ, then {Xλ} and {Xλ} are said to be equivalent in law.

We denote by D the set of all maps x;Λ → S. The value of x at λ is denoted byxλ. For λ1, . . . , λn ∈ Λ and Borel subsets B1, . . . , Bn of S, we define a cylinder setof D by B = {x; xλ1 ∈ B1, . . . , xλn ∈ Bn}. Let B(D) be the smallest σ -field of D,which contains all cylinder sets, where λ1, . . . , λn run Λ and B1, . . . , Bn run Borelsets of S.

For a given random field {Xλ, λ ∈ Λ}, there is a unique probability measure Pon B(D) satisfying

P(B) = P(Xλ1 ∈ B1, . . . , Xλn ∈ Bn).

The measure P is called the law of the random field {Xλ}.If the parameter set Λ is a time interval T such as [T0, T1], [0,∞) etc., then

the random field {Xt, t ∈ T} is called an S-valued stochastic process. Stochasticprocesses are often written as {Xt }, or simply as Xt, t ∈ T, Xt etc. If Xt(ω) iscontinuous with respect to t for almost all ω, {Xt } is called a continuous process.Further, if Xt(ω) is right continuous with left-hand limits with respect to t for almostall ω, {Xt } is called a cadlag process. For a cadlag process {Xt }, we denote its leftlimit by Xt−, i.e., Xt− = limε↓0 Xt−ε . If Xt(ω) is left continuous with right-handlimits with respect to t for almost all ω, {Xt } is called a caglad process. For a cagladprocess {Xt }, we denote its right limit by Xt+, i.e., Xt+ = limε↓0 Xt+ε . Further, theprocess {Xt } is said to be continuous in probability if P(d(Xt+h,Xt ) > ε) → 0 ash → 0 for any t ≥ 0 and ε > 0, where d is the metric of S.

When we consider a stochastic process {Xt, t ∈ [T0, T1]}, we usually assume thatthe process {Xt } describes the time evolution in the forward (positive) direction; atthe initial time T0, it starts from X0 and moves to Xt at time T0 < t ≤ T1. Wecould call it a forward process. However, in later discussions, we will consider aprocess {Xt , t ∈ [T0, T1]} (with the same time parameter), which describes the timeevolution in the backward (negative) direction; at the initial time T1, it starts fromXT1 and moves to Xt at time T0 ≤ t < T1. We will call it a backward process.

1.4 Wiener Processes, Poisson Random Measures and LévyProcesses

An Rd ′ -valued cadlag stochastic process {Xt, t ∈ [0,∞)} is said to have inde-

pendent increments, if {Xtm+1 − Xtm,m = 1, 2, . . . , n} are independent for any0 ≤ t1 < t2 < · · · < tn+1 < ∞. It is called time homogeneous if the law ofXt+h − Xs+h does not depend on h > 0. Let {Xt, t ∈ [0,∞)} be a cadlag R

d ′ -valued process with X0 = 0. It is called a Lévy process if it is time homogeneous,


continuous in probability and has independent increments. If {Xt } is a Lévy process,the d ′-dimensional law of Xt is infinitely divisible for each t . Then the characteristicfunction of the law of Xt is given by the Lévy–Khintchine formula.

Let {Xt } and {Xt } be two d ′-dimensional Lévy processes. Let μt and μt be lawsof random variables Xt and Xt , respectively. If μt = μt holds for all t > 0, thesetwo Lévy processes are equivalent in law and hence the law P of the Lévy processis uniquely determined by {μt , t > 0}. Indeed, for any bounded continuous functionf on R

d ′n and 0 ≤ t1 < t2 < · · · < tn, E[f (Xt1 , . . . , Xtn)] is computed as

∫· · ·

∫f (x1, . . . , xn)μt1(dx1)μt2−t1(dx2 − x1) · · ·μtn−tn−1(dxn − xn−1),

since Xt is time homogeneous and has independent increments. Hence if μt =μt holds for any t , we have E[f (Xt1 , . . . , Xtn)] = E[f (Xt1 , . . . , Xtn)] for anybounded continuous function f . Then {Xt } and {Xt } are equivalent.

Let {Xt } be a d ′-dimensional continuous Lévy process. If the law of each Xt

is Gaussian, the process {Xt } is called a Brownian motion. Since {Xt } is timehomogeneous, the mean vector of Xt is tb and covariance matrix is tA, where b andA are the mean and covariance of X1. If b = 0 and A = I , then {Xt } is called a (d ′-dimensional) Wiener process or a standard Brownian motion. For Wiener processes{Xt }, laws of {Xt } are unique, since μt are Gaussian distributions with means 0 andcovariance matrices tI .

We will show the existence of a one-dimensional Wiener process, constructingit by the method of Wiener. We will take T = [0, 1]. Let {Zn; n = 1, 2, . . .} be anindependent sequence of Gaussian random variables with means 0 and variances 1.Let {φn(t), n = 1, 2, . . .} be a complete orthonormal system of the real L2([0, 1])-space. We set φn(t) =

∫ t

0 φn(s) ds = (1(0,t], φn). Consider the infinite sum

Xt =∞∑

n=1

φn(t)Zn. (1.23)

We show that it converges uniformly in t ∈ [0, 1] a.s., if we take a good orthonormalsystem and {φn} be a system of functions on [0, 1] defined by

φn(t) =

⎧⎪⎨

⎪⎩

−1, ifm

2n≤ t <

m+ 1

2n, m = 0, 2, 4, . . . ,

1, ifm

2n≤ t <

m+ 1

2n, m = 1, 3, 5, . . .

Then it is a complete orthonormal system and ‖φn‖ ≡ supt |φn(t)| ≤ 1/2n+1 holdsfor any n. The system is called the Rademacher system.


Proposition 1.4.1 If {φn} is the Rademacher system, then the infinite sum (1.23)converges uniformly in t almost surely. Let {Xt, t ∈ [0, 1]} be the limit. It is acontinuous process. For each t , Xt is Gaussian with mean 0 and covariance t .Further, {Xt, t ∈ [0, 1]} has independent increments and is time-homogeneous.

Proof Set Xnt = ∑n

m=1 φm(t)Zm. Then, since supt |Xnt − Xn−1

t | ≤ ‖φn‖|Zn|,we have

∞∑

n=1

E[supt|Xn

t −Xn−1t |] ≤

∞∑

n=1

‖φn‖E[|Zn|] ≤∞∑

n=1

‖φn‖ < ∞.

This shows∑∞

n=1 supt |Xnt − Xn−1

t | < ∞, so that Xnt converges uniformly to Xt

a.s. Therefore Xt is continuous in t a.s.Note that the law of n-vector (Z1, . . . , Zn) is n-dimensional Gaussian with mean

0 and covariance I . Then the linear sum∑

j cjZj is one-dimensional Gaussian with

mean 0 and variance∑

j c2j . Therefore for each t , Xn

t is Gaussian with mean 0

and variance∑n

j=1 φn(t)2. Then the limit Xt is also Gaussian. Further, we have by

Perseval’s equality

E[XtXs] =∞∑

n=1

φn(t)φn(s) =∞∑

n=1

(1(0,t], φn)(1(0,s], φn) = (1(0,t], 1(0,s]) = t ∧ s.

Therefore E[X2t ] = t holds for any t .

Let 0 ≤ 0 < t1 < · · · < tn ≤ 1. Then (Xt1 − X0, . . . , Xtn − Xtn−1) is ann-dimensional Gaussian random variable. It holds that

E[(Xtm+1 −Xtm)2] = ti+1 − tm − tm + tm = tm+1 − tm.

Further, if tl < tm, we have

E[(Xtm+1 −Xtm)(Xtl+1 −Xtl )]= E[Xtm+1Xtm+1] − E[Xtm+1Xtl ] − E[XtmXtl+1 ] + E[XtmXtl1 ]= tl+1 − tl − tl+1 + tl = 0.

Therefore its n× n-covariance matrix is diagonal. This means

E[ei∑

m vm(Xtm−Xtm−1 )] = exp{− 1

2

∑

m

(tm − tm−1)v2m

}

=n∏

m=1

E[eivm(Xtm−Xtm−1 )

].


Then Xtn − Xtn−1 , . . . , Xt1 − X0 are mutually independent (Corollary 1.1.1).Therefore {Xt } has independent increments and is time-homogeneous. ��

We can choose other orthonormal system {φn} and show that (1.23) convergeuniformly in t a.s. Take φ0(t) = 1, φn(t) = √

2 cos nπt, n = 1, 2, . . .. Then

φ0(t) = t, φn(t) =√

2nπ

sin nπt, n = 1, 2 . . .. It is known that the infinite sum (1.23)is also uniformly convergent. It is called Wiener’s construction.

Let {Z′n} be another independent sequence of Gaussian random variables with

mean 0 and covariance 1, which is independent of {Zn}. We define a stochasticprocess by X′

t =∑∞

n=1 φn(t)Z′n for t ∈ [0, 1]. Then {X′

t } and {Xt } are equivalentin law. Further, these two processes are independent. {X′

t } is called an independent

copy of {Xt }. Now, take a sequence of independent copies {X(n)t }, n = 1, . . . of

the process {Xt } of Proposition 1.4.1. Define another continuous process Xt by∑ni=1 X

(i)1 +X

(n+1)t−n if t ∈ (n, n+ 1]. Then {Xt, t ∈ [0,∞)} is a Wiener process.

Next, take an independent copy {X′t , t ∈ [0,∞)} of the above one dimensional

Wiener process {Xt, t ∈ [0,∞)}. Then the pair {(Xt ,X′t ), t ∈ [0,∞)} is a

2-dimensional Wiener process. Repeating this argument, we can also define d ′-dimensional Wiener process.

Next, let {Nt ; t ∈ [0,∞)} be a Lévy process with values in N. It is called aPoisson process with intensity λ, if the law of each Nt is a Poisson distribution withparameter tλ. The law of a Poisson process {Nt } with intensity λ is unique, since{Nt } is a Lévy process.

We shall construct a Poisson process with intensity λ. Let σn; n = 1, 2, . . .be mutually independent positive random variables with a common exponentialdistribution with parameter λ. We set τ0 = 0 and τn = σ1 + · · · + σn for n ≥ 1. Wedefine a stochastic process Nt, t ∈ [0,∞) with values in N by

Nt(ω) = n, if τn(ω) ≤ t < τn+1(ω). (1.24)

Then P(Nt = n) = P(τn ≤ t < τn + σn+1). Note that the law of τn is the Gammadistribution with parameters λ and n. Then P(Nt = n) is computed as

λn+1

(n− 1)!∫∫

B

xn−1e−λxe−λy dx dy = e−λt (λt)n

n! ,

where B = {(x, y); 0 ≤ x ≤ t < x + y}. Therefore the law of Nt is Poisson withparameter λt . The random time τn defined above is called the n-th jumping timeof the stochastic process Nt . The sequence of random times {τn} is called jumpingtimes of the stochastic process {Nt }.Proposition 1.4.2 {Nt } defined by (1.24) is a Poisson process with intensity λ.

Proof We want to show that {Nt } is time homogeneous and has independentincrements. Our discussion is close to Sato [99]. We first show that the law of(τn+1 − t, σn+2, . . . , σn+m) under the condition Nt = n is the same as the lawof (σ1, . . . , σm). Set a = P(Nt = n). Then we have


P(τn+1 − t > s1, σn+2 > s2, . . . , σn+m > sm|Nt = n)

= P(τn ≤ t, τn+1 − t > s1, σn+2 > s2, . . . , σn+m > sm)/a

= P(τn ≤ t, τn+1 − t > s1)P (σn+2 > s2, . . . , σm+n > sm)/a

= P(τn+1 − t > s1|Nt = n)P (σ2 > s2, . . . , σm > sm)

= P(σ1 > s1)P (σ2 > s2, . . . , σm > sm)

= P(σ1 > s1, σ2 > s2, . . . , σm > sm). (1.25)

Here we used the equality P(τn+1 − t > s|Nt = n) = e−λs = P(σ1 > s), which isshown as follows. Note

P(τn < t, τn+σn > t+s) = λn+1

(n− 1)!∫∫

B ′xn−1e−λxe−λy dx dy = e−λ(t+s) (λt)

n

n! ,

where B ′ = {(x, y); 0 < x < t + s < x + y}. Then we have

P(Nt = n, τn+1 > t + s)

P (Nt = n)= P(τn ≤ t, τn+1 > t + s)

P (Nt = n)= e−λs .

Now we will prove that {Nt } is time homogeneous and has independentincrements. We have

P(Nt1 −Nt0 = n1, . . . , Ntm −Ntm−1 = nm|Nt0 = n0)

= P(τn0+n1 ≤ t1 < τn0+n1+1, . . . , τn0+···+nm ≤ tm < τn0+···+nm+1|Nt0 = n0)

= P(τn1 ≤ t1 − t0 < τn1+1, . . . , τn1+···+nm ≤ tm − t0 < τn1+···+nm+1).

In the last equality, we applied (1.25). Then we get

P(Nt1 −Nt0 = n1, . . . , Ntm −Ntm−1 = nm|Nt0 = n0)

= P(Nt1−t0 = n1, . . . Ntm−t0 −Ntm−1−t0 = nm).

Repeating this argument, we get the equality

P(Nt0 = n0, Nt1 −Nt0 = n1, . . . , Ntm −Ntm−1 = nm)

= P(Nt0 = n0) · · ·P(Ntm−tm−1 = nm). (1.26)

We have in particular,

P(Nt0 = n0, Nt1 −Nt0 = n1) = P(Nt0 = n0)P (Nt1−t0 = n1).


Summing up the above for n0 = 0, 1, 2, . . .. Then we get

P(Nt1 −Nt0 = n1) = P(Nt1−t0 = n1).

This shows that Nt is time homogeneous. Further, we can rewrite (1.26) as

P(Nt0 = n0, Nt1 −Nt0 = n1, . . . , Ntm −Ntm−1 = nm)

= P(Nt0 = n0) · · ·P(Ntm −Ntm−1 = nm). (1.27)

This proves that {Nt } has independent increments. ��Now, let {N ′

t } be a Poisson process with intensity λ. Then its law coincides withthe law of {Nt } constructed above. Hence {N ′

t } is a cadlag nondecreasing processwith values in N such that N ′

t − N ′t− is 0 or 1, almost surely. Set τ ′0 = 0 and define

random times τ ′n, n = 1, 2, . . . by induction as

τ ′n ={

inf{t > τ ′n−1; N ′t > N ′

τ ′n−1}, if the set {· · · } is non-empty,

∞, otherwise.

Then τ ′n is subject to the Gamma distribution with parameters λ and n, if we restrictthe distribution to the interval [0,∞). The sequence {τ ′n} is called the jumping timesof the Poisson process {N ′

t }.Let T = [0,∞) and U = T × R

d ′0 . Elements of U are denoted by u = (r, z)

where r ∈ T, z ∈ Rd0 . Let (U,B(U), n) be a measure space, where n is a σ -finite

measure on U. A family of N-valued random variables N(B), B ∈ B(U) is called the(abstract) Poisson random measure on U with intensity measure n, if the followingshold:

1. For every ω, N(·, ω) is an N-valued measure on U.2. For each B, the law of N(B) is a Poisson distribution with parameter n(B).3. If B1, . . . , Bn are disjoint, N(B1), . . . , N(Bn) are independent.

We are interested in the Poisson random measure in the case that n is the productmeasure drν(dz), where dr is the Lebesgue measure on [0,∞) and ν is a σ -finitemeasure on R

d ′0 . We shall construct it. We first consider the case where ν is a finite

measure. Let {Sn; n = 1, 2, . . .} be a sequence of mutually independent randomvariables with values in R

d ′ with the common distribution μ(dz) = ν(dz)/λ, whereλ = ν(Rd ′

0 ). Let {Nt } be a Poisson process with intensity λ, which is independent

of {Sn}. For 0 ≤ s < t < ∞ and a Borel subset E of Rd ′0 , we set

N((s, t] × E) :=∑

Ns<n≤Nt

1E(Sn) =∑

k;s<τk≤t

1E(Sk), (1.28)

where τk, k = 1, 2, . . . are jumping times of the Poisson process {Nt }.


Proposition 1.4.3 The random measure N(dr dz) defined by (1.28) is a Poissonrandom measure on U = [0,∞)× R

d ′0 with the intensity n(dr dz) = drν(dz).

Proof Set Ns,t (E) = N((s, t] × E). Let Ek, k = 1, . . . , n be disjoint subsets ofR

d ′0 . The characteristic function of (Ns,t (E1), . . . , Ns,t (En)) is computed as

E[ei∑

k vkNs,t (Ek)] =∞∑

m=0

P(Nt −Ns = m)E[ei∑

k vk(∑

j≤m 1Ek(Sj ))]

=∞∑

m=0

e−λ(t−s) (λ(t − s))m

m! E[ei∑

k vk1Ek(S1)]m,

since S1, . . . , Sm are independent. We have

E[ei∑n

k=1 vk1Ek(S1)] =

∑

k

eivkμ(Ek)+ μ((⋃

k

Ek)c) =

n∑

k=1

(eivk − 1)μ(Ek)+ 1.

Therefore,

E[ei∑

k vkNs,t (Ek)] = e−λ(t−s)∞∑

m=0

(λ(t − s))m

m!( n∑

k=1

(eivk − 1)μ(Ek)+ 1)m

= e−λ(t−s) exp{ n∑

k=1

λ(t − s){(eivk − 1)μ(Ek)+ 1)}}

=n∏

k=1

exp{λ(t − s)(eivk − 1)μ(Ek)}

=n∏

k=1

E[eivkNs,t (Ek)].

Then Ns,t (E1), . . . , Ns,t (En) are mutually independent by Corollary 1.1.1. Further,each Ns,t (Ek) is Poisson distributed with parameter λ(t − s)μ(Ek) = (t − s)ν(Ek).

Next, consider N((0, t1] ×E1), . . . , N((tn−1, tn] ×En). These are independent,since Ntm − Ntm−1 ,m = 1, . . . , n are independent. Further, these are Poissondistributed. Consequently, N(B1), . . . , N(Bn) are independent if B1, . . . , Bn aredisjoint rectangular sets of the form Bm = (tm−1, tm]×Em. Then the independenceof N(B1), . . . , N(Bn) holds for any disjoint Borel sets B1, . . . , Bn of U. ��Theorem 1.4.1 For any given σ -finite measure ν on R

d ′0 and time interval [0,∞),

there exists a Poisson random measure N(dr dz) on U = [0,∞) × Rd ′0 with the

given intensity n(dr dz) = drν(dz). Further, laws of Poisson random measures onU with the given intensity are unique.


Conersely, let N(dr dz) be a Posson random measue on U with the intensitymeasure n(dr dz) = drν(dz), where ν is a finite measure. Then there exists aPoisson process {Nt } with intensity λ = ν(Rd ′

0 ) and an independent sequence of

Rd ′0 -valued random variables {Sn} with the common distribution μ(dz) = ν(dz)/λ,

and the Poisson random measure is written as (1.28).

Proof If ν is a finite measure, we construct a Poisson random measure on U withintensity dr dν as in Proposition 1.4.3. We will consider the case where ν is a σ -finite measure. Let Kn, n = 1, 2, . . . be a sequence of disjoint Borel sets of R

d ′0

such that⋃

n Kn = Rd ′0 and ν(Kn) < ∞ for any n. We decompose ν as a sum of

finite measures νn such that νn(dz) = ν(dz)1Kn . For each νn there exists a Poissonrandom measure Nn on U with intensity dt dνn. Then on a suitable probabilityspace, we can construct these Nn; n = 1, 2, . . . as independent Poisson randommeasures. We define N(B) = ∑

n Nn(B). Then N(B) is also a Poisson random

measure on U. Further, its intensity is dr d(∑

n νn) = dr dν. Therefore it is thedesired one.

Now, if B1, . . . , Bn are disjoint, the law of n-vector random variable defined byX = (N(B1), . . . , N(Bn)) is the product of laws of N(Bk), k = 1, . . . , n, sincethese are independent. Hence the law of the X is unique if Bk, k = 1, . . . , n aredisjoint. Then the law of X for any Borel sets B1, . . . , Bn is also unique. Thereforethe law of the Poisson random measure is unique.

We will prove the latter assertion of the theorem. Set Nt := N((0, t] × Rd ′0 ).

Then, Nt is a Poisson process with intensity λ := ν(Rd ′0 ). Let τ1 < · · · < τn <

· · · be jumping times of the Poisson process. We define a sequence of Rd ′ -valuedrandom variables by Sn = ∫

{τn}×Rd′0zN(dr dz). Then {Sn} are mutually independent

with common distribution μ(dz) = ν(dz)/λ. Further, N((s, t] × E) is representedby (1.28). ��

A d ′-dimensional Lévy process {Xt } is called a compound Poisson process withintensity ν, if the law of Xt is a compound Poisson distribution associated withmeasure tν for each t > 0.

Given a finite measure ν on Rd ′0 , we shall construct a compound Poisson process

{Xt } with intensity ν. Let {Nt } be a Poisson process with intensity λ = ν(Rd ′0 )

and let {Sn} be independent Rd ′ -valued random variables with common distributionμ = ν/λ. We set S0 = 0 and define an R

d ′ -valued cadlag process {Xt } by

Xt =∑

0≤n≤Nt

Sn =∫

(0,t]×Rd′0

zN(dr dz). (1.29)

Here, N is the Poisson random measure on [0,∞) × Rd ′0 , defined by (1.28) using

Poisson process {Nt } with intensity λ.


Proposition 1.4.4 The process {Xt } given by (1.29) is a compound Poisson processwith intensity ν.

Proof Since N is a Poisson random measure, the process {Xt } defined by (1.29)has independent increments. We shall compute the characteristic function of Xt . Wehave

E[ei(v,Xt )] =∞∑

n=0

E[e(v,∑

j≤n Sj )]P(Nt = n) = e−λt∞∑

n=0

(λt)n

n! μ(v)n,

where μ(v) is the characteristic function of the distribution μ. It is written as∫ei(v,z)μ(dz). Therefore the last term of the above is equal to

exp{−λt} exp{λt

∫ei(v,z)μ(dz)

}= exp

{t

∫(ei(v,z) − 1)ν(dz)

},

since λμ(dz) = ν(dz). Hence the law of Xt is a compound Poisson distributionassociated with measure tν. Then {Xt } is a compound Poisson process with intensityν. ��

Let N(dr dz) be a Poisson random measure on U with intensity measuren(dr dz) = drν(dz). We will define a compensated Poisson random measure by

N(dr dz) = N(dr dz)− n(dr dz). (1.30)

We will define the integral∫ t

0

∫f (z)N(dr dz) for a measurable function f (z)

such that∫ |f (z)|2ν(dz) < ∞. Suppose f (z) is a step function of the form

f = ∑k ck1Ek

, where Ek are disjoint Borel subsets of Rd ′0 . Then we define∫ t

0

∫f (z)N(dr dz) = ∑

k ckNt (Ek), where Nt (E) = N((0, t] × E). We haveE[Nt (Ek)Nt (El)] = 0 for k �= l, since Nt (Ek) and Nt (El) are independent.Therefore,

E[( ∫ t

0

∫f (z)N(dr dz)

)2] = t∑

k

c2kν(Ek) = t

∫f 2ν(dz). (1.31)

For a measurable function f (z) such that∫ |f |2ν(dz) < ∞, choose a sequence of

step functions fn satisfying∫ |f − fn|2ν(dz) → 0 as n → ∞. Then the sequence

of integrals∫ t

0

∫fn(z)N(dr dz) should converge in L2(P ). We denote the limit by∫ t

0

∫f (z)N(dr dz).

Let {Wt } be a Brownian motion with mean 0 and covariance At . Let N(ds dz)

be a Poisson random measure on U with intensity dtν(dz), which is indepen-dent of {Wt }. We will assume that ν is a Lévy measure. Then the integral∫ t

0

∫0<|z|≤1 zN(dr dz) is well defined. We set D = {z ∈ R

d ′0 ; |z| ≤ 1} and


Xt = bt +Wt +∫ t

0

∫

D

zN(dr dz)+∫ t

0

∫

Dc

zN(dr dz). (1.32)

Then {Xt } is a cadlag process and in fact a Lévy process.

Proposition 1.4.5 Let {Xt } be a Lévy process defined by (1.32). Then, the charac-teristic function of Xt is given by

exp t{i(v, b)− 1

2(v,Av)+

∫

Rd′0

(ei(v,z) − 1 − i(v, z)1D(z))ν(dz)}. (1.33)

Proof We know that bt + Wt is a Brownian motion with means bt and covariance

At . Then we have E[ei(v,bt+Wt )] = exp t{i(b, v) − 1

2 (v,Av)}. We shall consider

the characteristic function of

Yt =∫ t

0

∫

D

zN(dr dz)+∫ t

0

∫

Dc

zN(dr dz).

Set

Y εt =

∫ t

0

∫

|z|>ε

zN(dr dz)− t

∫

1>|z|>ε

zν(dz).

Then Y εt converges to Yt . Since

∫ t

0

∫|z|>ε

zN(ds dz) is a compound Poisson process

with Lévy measure ν1|z|>ε , E[ei(v,Y εt )] is computed as

E[

exp{i(v,

∫ t

0

∫

|z|>ε

zN(dr dz))}]

exp{− it (v,

∫

1≥|z|>ε

zν(dz)}

= exp t{ ∫

|z|>ε

(ei(v,z) − 1)ν(dz)}

exp{it (v,

∫

1≥|z|>ε

zν(dz)}

= exp t{ ∫

|z|>ε

(ei(v,z) − 1 − i(v, z)1D(z))ν(dz)}.

Let ε → 0, then we get the formula (1.33) for Yt . ��Let {X′

t } be a Lévy process and let μ′t be the law of X′

t . Since {X′t } is time

homogeneous and has independent increments, for any n, μ′t is equal to n-times

convolution of the law μ′tn

. Therefore μ′t is infinitely divisible. Then its characteristic

function is represented by (1.33) by the Lévy–Khinchin formula. Therefore, for anygiven Lévy process {X′

t }, the Lévy process {Xt } constructed by (1.32) is equivalentin law with {X′

t }. Therefore we have the following theorem.

1.5 Martingales and Backward Martingales 25

Theorem 1.4.2 Let {Xt, t ∈ [0,∞)} be a Lévy process. Then there exist a constantb, independent Brownian motion {Bt , t ∈ [0,∞)} and Poisson random measureN(dr dz), and Xt is represented by (1.32).

The formula (1.32) is called the Lévy–Itô decomposition or Lévy–Itô representationof a Lévy process.

In some problems, it is convenient to use a Poisson point process instead of aPoisson random measure. We will give its definition. By a point function on R

d ′0 ,

we mean a map q : Dq → Rd ′0 , where Dq is a countable set in [0,∞). A counting

measure of the point function is defined by

N(B, q) = �{t ∈ Dq; (t, q(t)) ∈ B}, (1.34)

where B are Borel sets in U = [0,∞) × Rd ′0 . A random variable q(ω) with values

in the space of point functions is called a point process. If the counting measure ofthe point process q(ω) is a Poisson random measure with intensity dt dν, the pointprocess is called a Poisson point process with intensity dtν(dz) and N of (1.34) iscalled the associated Poisson random measure.

Given a Lévy measure ν, there exists a Poisson point process with intensitydtν(dz). Indeed, if ν is a finite measure, consider the compound Poisson processwith intensity dtν(dz). It is represented as Xt(ω) = ∑

1≤n≤Nt (ω)Sn(ω) by (1.29).

We define a point process q = q(ω) by setting

Dq = {0 < τ1 < · · · < τn < ∞}, q(t) = (τi, Si), if τi = t.

Then it is a Poisson point process with intensity dtν(dz). We can show the existenceof the Poisson point process for any Lévy measure ν by an argument similar to theproof of Theorem 1.4.1.

1.5 Martingales and Backward Martingales

In this and the next section, we will study martingales. We will restrict our attentionsto three topics of martingales, which are often used in later stochastic calculus.Three topics are the optional sampling theorem, Doob’s inequality and quadraticvariation.

Let (Ω,F , P ) be a probability space. Let G be a sub σ -field of F and let X bean integrable random variable. A G-measurable and integrable random variable Y iscalled the conditional expectation of X given G, if it satisfies E[X1G] = E[Y1G]for any G ∈ G. Here, 1G is the indicator function of the set G.

The conditional expectation Y exists uniquely a.s. For the proof, consider asigned measure P (A) = E[X1A], A ∈ G. It is absolutely continuous with respect toP , i.e., P (A) = 0 holds for any A ∈ G with P(A) = 0. Then, in view of the Radon–Nikodym theorem, there exists an integrable G-measurable functional Y such that


P (A) = ∫AY dP holds for any A ∈ G. The uniqueness follows from the uniqueness

of the Radon–Nikodym density.The conditional expectation of X with respect to G is denoted by E[X|G].

Further, if A is an element of F , the conditional expectation E[1A|G] is denotedby P(A|G) and is called the conditional probability of event A given G.

Conditional expectations have following properties.

Proposition 1.5.1 Let X, Y be integrable random variables.

1. E[c1X + c2Y |G] = c1E[X|G] + c2E[Y |G] holds a.s. for any constants c1, c2.2. E[|X||G] ≥ |E[X|G]| holds a.s.3. If G1 ⊂ G2 ⊂ F , then E[E[X|G2]|G1] = E[X|G1] holds a.s.4. If σ(X) and G are independent, then E[X|G] = E[X] a.s.5. If XY is integrable and X is G-measurable, then E[XY |G] = XE[Y |G].6. Let f (x) be a convex function, bounded from below. If f (X) is integrable, we

have f (E[X|G]) ≤ E[f (X)|G] a.s.

Proposition 1.5.2 Let X1, . . . , Xn, . . . , X be integrable random variables.

1. (Fatou’s lemma) If Xn ≥ 0,

lim infn→∞ E[Xn|G] ≥ E[lim inf

n→∞ Xn|G], a.s.

2. Let p ≥ 1. If Xn converges to X in Lp, then E[Xn|G] converges to E[X|G] inLp.

Proofs of these propositions are straightforward. These are omitted.We will define a martingale with discrete time parameter. Let N be the set of all

positive integers. An increasing sequence {Fn, n ∈ N} of sub σ -fields of F is calleda filtration. Given a filtration {Fn}, we will define a martingale. Let {Xn; n ∈ N}be a sequence of real integrable random variables. We will assume that each Xn

is Fn-measurable. It is called a martingale with discrete time if E[Xn|Fm] = Xm

holds for any 1 ≤ m < n. We denote by F0 the trivial σ -field {∅,Ω}, where ∅ is theempty set.

A sequence of random variables fn, n ∈ N is called predictable if each fn isFn−1-measurable. Let {fn} be a predictable sequence. The martingale transformYn, n ∈ N is defined by

Yn =n∑

m=1

fm(Xm −Xm−1), where X0 = 0.

If E[|fm||Xm −Xm−1|] < ∞ holds for any m ≥ 1, {Yn} is a martingale, since Yn isFn-measurable and satisfies

E[Yn+1 − Yn|Fn] = fn+1E[Xn+1 −Xn|Fn] = 0.


Let τ be a random variable with values in N ∪ {∞}. It is called a stopping timeif {τ ≤ n} ∈ Fn holds for any n ∈ N. If τ is a stopping time, it is easily seen thatn∧ τ is also a stopping time for any positive integer n. For a stopping time τ , we set

Fτ ={B ∈ F; B ∩ {τ ≤ n} ∈ Fn for any n

}.

It is a sub σ -field of F .

Proposition 1.5.3 Let {Xn, n ∈ N} be a martingale and let τ be a stopping time.

1. The stopped process {Yn = Xn∧τ , n ∈ N} is again a martingale.2. Let σ be another stopping time. Then E[Xn∧τ |Fσ ] = Xn∧τ∧σ holds for any

n ∈ N.

Proof We set fn = 1τ≥n. It is a predictable sequence. Consider the martingaletransform by fn; Yn = ∑n

m=1 fm(Xm − Xm−1). It coincides with Xn∧τ . Therefore{Xn∧τ } is a martingale.

Let σ be another stopping time. If B ∈ Fσ , we have

E[Yn;B] =n∑

m=0

E[Yn;B ∩ {σ = m}] =n∑

m=0

E[Ym;B ∩ {σ = m}] = E[Yn∧σ ;B].

Further, Yn∧σ is Fσ -measurable, because

{Xn∧σ < a} ∩ {σ ≤ m} =⋃

l≤m

{Xl < a, σ = l} ∈ Fm

holds for any m. Therefore we get the equality E[Yn|Fσ ] = Yn∧σ a.s. SubstitutingYn = Xn∧τ , we get E[Xn∧τ |Fσ ] = Xn∧τ∧σ a.s. ��Proposition 1.5.4 Let {Xn, n ∈ N} be a martingale and let N ∈ N. Then we havefor any a > 0

aP ( supn≤N

|Xn| > a) ≤∫

supn≤N |Xn|>a

|XN | dP. (1.35)

Proof For a given a > 0, we set

τ = inf{n; |Xn| > a} (= ∞ if the set {· · · } is empty).

Then τ is a stopping time. Indeed, we have {τ ≤ k} = ⋃i≤k{|Xi | > a} ∈ Fk for

any k ∈ N. Then by Proposition 1.5.3, we have E[|XN ||Fτ ] ≥ |XN∧τ |. Therefore,

aP (τ ≤ N) ≤∫

τ≤N

|XN∧τ | dP ≤∫

τ≤N

|XN | dP.

Since {τ ≤ N} = {supn≤N |Xn| > a}, we get the inequality of the Proposition. ��


Proposition 1.5.5 Let {Xn, n ∈ N} be a martingale satisfying E[|Xn|p] < ∞ forany n, where p > 1. Set q = p

p−1 . Then we have for any N ∈ N,

E[

supn≤N

|Xn|p]≤ qpE[|XN |p]. (1.36)

Proof Set Y = supn≤N |Xn| and F(λ) = P(Y > λ). Then we have

E[Yp] = −∫ ∞

0λp dF(λ) =

∫ ∞

0F(λ) d(λp)− lim

λ→∞ λpF(λ)∣∣λ0 ≤

∫ ∞

0F(λ) d(λp).

Since λP (Y > λ) ≤ ∫Y>λ

|XN | dP holds by Proposition 1.5.4, we have

E[Yp] ≤∫ ∞

0

1

λ

( ∫

Y>λ

|XN |dP)d(λp) =

∫|XN |

( ∫ Y

0

1

λd(λp)

)dP

≤ p

p − 1

∫|XN |Yp−1 dP ≤ qE[|XN |p]

1p E[Y (p−1)q ] 1

q

≤ qE[|XN |p]1p E[Yp] 1

q .

Therefore we have the inequality of the proposition. ��We give the definition of a martingale with negative parameter. Let F−n, n ∈ N

be a sequence of sub σ -fields of F satisfying F−n ⊂ F−(n−1). Let {X−n, n ∈ N} bea sequence of integrable {F−n}-adapted random variables. It is called a martingaleif E[X−m|F−n] = X−n holds whenever −m > −n. A sub-martingale and a super-martingale are defined similarly.

We will next define a martingale with continuous parameter T = [0,∞). Let{Ft , t ∈ T} be a family of sub σ -fields of F satisfying Fs ⊂ Ft for any s < t .We assume further that Ft contains all null sets of F and satisfies Ft = ∧

ε>0 Ft+ε

(right continuous). Then the family {Ft , t ∈ T} is called a filtration. A randomvariable τ with values in T ∪ {∞} is called a stopping time if {ω; τ(ω) ≤ t} ∈ Ft

holds for any t ∈ T. For a stopping time τ , we set

Fτ ={B ∈ F; B ∩ {τ ≤ t} ∈ Ft for any t ∈ T

}.

It is a sub σ -field of F .Let {Xt } = {Xt, t ∈ T} be a stochastic process. {Xt } is called adapted for

{Ft } (or {Ft }-adapted) and {Ft } is called admissible for {Xt } (or {Xt }-admissible),if Xt is Ft -measurable for each t ∈ T. An {Ft }-adapted cadlag process {Xt } iscalled a martingale or {Ft }-martingale, if Xt is integrable for any t and equalitiesE[Xt |Fs] = Xs hold a.s. for any s < t . If equality signs are replaced by ≥ in theabove, it is called a sub-martingale. Further, if {−Xt } is a sub-martingale {Xt } iscalled a super-martingale.


Let p > 1. An {Ft }-adapted cadlag process {Xt } is called an Lp-martingale ifit is a martingale and satisfies E[|Xt |p] < ∞ for any t . An {Ft }-adapted cadlagprocess {Xt } is called a local martingale or local Lp-martingale, if there existsan increasing sequence of stopping times {τn} such that P(τn < T ) → 0 forany positive constant T and each stopped process X

(n)t := Xt∧τn , t ∈ T is a

martingale or Lp-martingale, respectively. A martingale is a local martingale andan Lp-martingale is a local Lp-martingale.

Theorem 1.5.1 (Doob) Let {Xt } be a martingale.

1. For any stopping time τ , the stopped process {Yt = Xt∧τ } is a martingale.2. For any two stopping times σ and τ , we have E[Xt∧σ |Fτ ] = Xt∧σ∧τ for any t .3. (Doob’s inequality) Suppose that {Xt } is an Lp-martingale for some p > 1. Set

q = pp−1 . Then we have

E[

sup0≤r≤t

|Xr |p]≤ qpE

[|Xt |p

], ∀t ∈ T. (1.37)

Proof We prove the first assertion. For n ∈ N, set t (n)m = m/2n,m = 1, . . . , 2nT .Let τ be a stopping time. For n ∈ N, we define

τn(ω) = t (n)m , if t(n)m−1 < τ(ω) ≤ t (n)m . (1.38)

Then τn, n = 1, 2, . . . is a sequence of stopping times decreasing to τ asn → ∞. We have E[Xt∧τn;B] = E[Xs∧τn;B] for any B ∈ Fs by Proposi-tion 1.5.3. Therefore if limn→∞ E[Xt∧τn;B] = E[Xt∧τ ;B] holds for any t , we getE[Xt∧τ ;B] = E[Xs∧τ ;B] and hence Xt∧τ is a martingale. In order to prove theabove convergence, we will show that the family of random variables {Xt∧τn−Xt∧τ }is uniformly integrable, i.e., for any ε > 0 there exists c > 0 such that

supn

∫

|Xt∧τn−Xt∧τ |>c

|Xt∧τn −Xt∧τ | dP < ε. (1.39)

Set Y−n = Xt∧τn − Xt∧τ and G−n = Ft∧τn . Then, in view of Proposition 1.5.3,2, Y−n, n ∈ N is a {G−n}-martingale with negative parameter. Therefore |Y−n| is asub-martingale. Since {|Y−n| > c} ∈ G−n, we have the inequality

∫

|Y−n|>c

|Y−n| dP ≤∫

|Y−n|>c

|Y−1| dP

for any n and c > 0. Set Q1(A) := ∫A|Y−1| dP on G−1. Then Q1 is absolutely

continuous with respect to P . Therefore for any ε > 0, there exists δ > 0 such thatfor any A ∈ G−1 with P(A) < δ, the inequality Q(A) < ε holds. Further, for thisδ, there exists c > 0 such that the inequality


P(|Y−n| > c) ≤ 1

cE[|Y−n|] ≤ 1

cE[|Y−1|] ≤ δ

holds for any n. We have thus shown that for any ε > 0, there exists c > 0 suchthat the inequality supn

∫|Y−n|>c

|Y−n| dP < ε holds. Therefore {Y−n} is uniformlyintegrable. Now we have

∫|Xt∧τn −Xt∧τ |1B dP ≤ ε +

∫

|Yn|≤c

|Yn|1B dP.

Since {Yn} converges to 0 a.s., the last term converges to 0 as n → ∞.Therefore we have limn→∞

∫ |Xt∧τn − Xt∧τ |1B dP = 0. This proves the equalitylimn→∞ E[Xt∧τn;B] = E[Xt∧τ ;B].

The second assertion can be verified similarly, using Proposition 1.5.3, 2 andapproximating both stopping times σ and τ from the above by sequences of stoppingtimes with discrete values.

Suppose next that {Xt } is an Lp-martingale for some p > 1. For n ∈ N, sett(n)m = m/2n ∧ t, m = 1, . . . , 2n(T + 1). Then E[supm<2n |Xt

(n)m|p] ≤ qpE[|Xt |p]

holds by Proposition 1.5.5. Since supr≤t |Xr | = limn→∞ supm |Xt(n)m| holds a.s., we

get the inequality (1.37). ��An {Ft }-adapted cadlag process At, t ∈ T is called an increasing process if At

is a nondecreasing function of t a.s. and satisfies A0 = 0 a.s. If At is continuousin t a.s., {At } is called a continuous increasing process. If At is written as A1

t − A2t

for any t , where A1t , A

2t , t ∈ T are increasing processes, {At } is called a process

of bounded variation. Further, if both {Ait }, i = 1, 2 are continuous, {At } is called

a continuous process of bounded variation. If a process {Xt } is written as the sumof a local martingale {Mt } and a continuous process of bounded variation {At }, itis called a semi-martingale. Further, if {Mt } is a continuous process, {Xt } is calleda continuous semi-martingale. If {Mt } is a local martingale with bounded jumps,{Xt } is called a semi-martingale with bounded jumps. The process {Mt } is calledthe martingale part of the semi-martingale {Xt }.

Let {Ft , t ∈ T} be a filtration and let τ be a stopping time with respect to thefiltration. A cadlag process {Xt, t ∈ [0, τ )} is called a local semi-martingale if (i)τ is accessible, i.e. there exists an increasing sequence of stopping times {τn} suchthat τn < τ and limn→∞ τn = τ hold a.s., and (ii) stopped processes X

(n)t = Xt∧τn

are semi-martingales for all n with respect to the filtration {Ft }.A family of sub σ -fields {Fs , s ∈ T} of F is called a backward filtration if

Ft ⊂ Fs for any s < t ,⋂

ε>0 Fs−ε = Fs for any s and each Fs contains null sets ofF . We will consider a backward semi-martingale. We consider left continuous ratherthan right continuous processes, since the time evolution is backward. Let {Fs} bea backward filtration. Let {Xs, s ∈ T} be an integrable caglad process, adapted to abackward filtration {Fs}. It is called a backward martingale if E[Xs |Ft ] = Xt holdsa.s. for any s < t . A random variable σ with values in T∪{−∞} is called a backwardstopping time, if {σ ≥ t} ∈ Ft holds for any t ∈ T. An {Fs}-adapted caglad process

1.6 Quadratic Variations of Semi-martingales 31

{Xs} is called a backward local martingale if there exists a decreasing sequence ofbackward stopping times σn such that limn→∞ P(σn > 0) = 0 and each stoppedprocess {Xs∨σn} is a backward martingale. Further, if Xs is written as the sum of abackward local martingale and a continuous process of bounded variation adaptedto the backward filtration {Fs}, {Xs} is called a backward semi-martingale.

Let {Fs,t , 0 ≤ s < t < ∞} be a family of sub σ -fields of F , each of whichcontains null sets of F . We assume that for any fixed s, F ′

t := Fs,t , t ≥ s is afiltration. We assume further that for any fixed t , F ′

s := Fs,t is decreasing withrespect to s, i.e., F ′

s ⊂ F ′s′ if s′ ≤ s and left continuous, i.e.,

⋂ε>0 F ′

s−ε = F ′s

holds for any s < t . Then {Fs,t } is called a two-sided filtration. A backward localsemi-martingale is defined similarly to a local semi-martingale.

Let {Xt, t ∈ [0,∞)} be a Wiener process and let N(dsdz) be a Poisson randommeasure on U = [0,∞)× R

d ′0 . Let

Fs,t = σ(Xu −Xv,N(dr dz); s ≤ u < v ≤ t, dr ⊂ [s, t]).

Then {Fs,t } is a two-sided filtration generated by Wiener process and Poissonrandom measure. Then {Xt } and {Yt } = {∫ t

0

∫R

d0f (z)N(dr dz)} are forward

martingales with respect to the filtration Ft = F0,t . Further, {Xs := XT − Xs} and{Ys := YT − Ys} are backward martingales with respect to the backward filtrationFs = Fs,T .

1.6 Quadratic Variations of Semi-martingales

In this section, we will restrict time parameter to the finite interval T = [0, T ].Martingales and stopping times with time parameter T are defined in the same wayas those with time parameter [0,∞). Let Π = {0 = t0 < t1 < · · · < tn = T } bea partition of T = [0, T ]. We set |Π | = maxm |tm − tm−1|. We define the quadraticvariation of a real cadlag process {Xt, t ∈ T} associated with the partition Π by

〈X〉Πt =n∑

m=1

(Xtm∧t −Xtm−1∧t )2, t ∈ T. (1.40)

We are interested in the limit of quadratic variations {〈X〉Πt } as |Π | → 0. If itconverges uniformly in t in probability to a process, the limit is denoted by 〈X〉t , t ∈T and is called the quadratic variation of {Xt }. The quadratic variation of {Xt } isoften written as {〈Xt 〉}.

In this section, we show that if {Xt, t ∈ T} is a semi-martingale with boundedjumps, the quadratic variation exists. We will first list properties of L2 martingales.


Proposition 1.6.1 Let {Xt } be an L2-martingale.

1. E[(Xt −Xs)2|Fs] = E[X2

t −X2s |Fs] holds for any 0 ≤ s < t ≤ T .

2. E[(Xv −Xu)(Xt −Xs)|Fs] = 0 holds for any 0 ≤ s < t ≤ u < v ≤ T .3. For any 0 ≤ s < t ≤ T , it holds that

F [(Xt −Xs)2|Fs] = E[〈X〉Πt − 〈X〉Πs |Fs]. (1.41)

Proof In view of Proposition 1.5.1, 1 and 5, we have

E[(Xt −Xs)2|Fs] = E[X2

t − 2XtXs +X2s |Fs]

= E[X2t |Fs] − 2E[Xt |Fs]Xs +X2

s

= E[X2t |Fs] −X2

s .

Then we get the equality of 1. Next, we have by Proposition 1.5.1, 3,

E[(Xv −Xu)(Xt −Xs)|Fs] = E[E[(Xv −Xu)|Fu](Xt −Xs)|Fs] = 0,

proving the orthogonal property of 2.We have

Xt −Xs =∑

m;s<tm−1

(Xtm∧t −Xtm−1∧t )+ (Xtl −Xs),

where tl−1 < s < tl . Then using the above orthogonal property, we have

E[(Xt −Xs)2|Fs] = E[

∑

m;s<tm−1

(Xtm∧t −Xtm−1∧t )2 + (Xtl −Xs)

2|Fs]

= E[〈X〉Πt − 〈X〉Πs |Fs].

��Let Πn = {0 = t

(n)0 < t

(n)1 < · · · < t

(n)tmn

= T }, n = 1, 2, . . . be a sequence ofpartitions of T such that |Πn| → 0 as n → ∞. We call {Πn} regular, if Πn ⊂ Πn+1holds for any n.

Lemma 1.6.1 Let {Xt } be an L4-martingale. Let {〈X〉Πnt } be the quadratic varia-

tion of {Xt } associated with the partition Πn. If {Πn} is regular and |Πn| convergesto 0, the sequence {〈X〉Πn

t } converges to an increasing process {〈X〉t } uniformly int ∈ T in L2. Further, {X2

t − 〈X〉t } is an L2-martingale.

Proof We will show first that {〈X〉Πn

T , n = 1, 2, . . .} are L2-bounded. By a directcalculation, we get the identity


X2t − 〈X〉Πn

t = 2( ∑

t(n)m ≤t

Xt(n)m−1

(Xt(n)m

−Xt(n)m−1

)+Xt(n)l

(Xt −Xt(n)l

)). (1.42)

Denote the right-hand side by 2MΠnt . We may regard that {MΠn

t , t ∈ Πn} is amartingale transform of the discrete martingale {Xt, t ∈ Πn}. It is an L2 martingalewith time parameter Πn. We have by Proposition 1.6.1 and Doob’s inequality

E[|MΠn

t |2] =∑

m

E[(X

t(n)m−1∧t

)2(Xt(n)m ∧t

−Xt(n)m−1∧t

)2]

≤ E[

supm

(Xt(n)m−1∧t

)2〈X〉Πnt

]

≤ 2aE[|Xt |4

]+ 1

2aE[(〈X〉Πn

t

)2].

Here we used the inequality |XY | ≤ a2 |X|2+ 1

2a Y2. Now, note the equality 〈X〉Πn

t =X2

t − 2MΠnt . Then we get

E[(〈X〉Πn

t )2] ≤ 2(E[X4

t

]+ 4E[(M

Πnt )2]) ≤ 2(1 + a)E

[X4

t

]+ 4

aE[(〈X〉Πn

t )2].

Taking a > 4, we find that supt∈T E[(〈X〉Πnt )2] is bounded with respect to n.

Since a bounded set in the L2-space is weakly compact, a subsequence of{〈X〉Πn

t , t ∈ Π∞} converges weakly to an increasing process {At, t ∈ Π∞} such thatsupt E[A2

t ] < ∞, where Π∞ = ⋃n Πn is a countable dense subset of T (it is the

set of dyadic rational numbers in T). It means that there exists a subsequence {n′} of

N such that for any random variable F with E[|F |2] < ∞, E[〈X〉Πn′t F ] converges

to E[AtF ] for all t ∈ Π∞ as n′ → ∞. At should be Ft -measurable for any t ∈ Π∞and nondecreasing with respect to t ∈ Π∞. Further, since {X2

t − 〈X〉Πn′t , t ∈ Πn′ }

are martingales, {X2t − At, t ∈ Π∞} should also be a martingale. Then we have

E[(Xt −Xs)2|Fs] = E[X2

t −X2s |Fs] = E[At − As |Fs], a.s.

if s < t and s, t ∈ Π∞.Using the above At , we will show that the sequence of martingale transforms

{MΠnt } converges to a martingale uniformly in t in L2. If n ≥ n′, these satisfy

MΠnt −M

Πn′t =

∑

m

(Xt(n)tm

∧t−X

t(n′)m′ ∧t

)(Xt(n)m+1∧t

−Xt(n)m ∧t

),

where t(n′)m′ ≤ t

(n)m . Since the equality

E[(X

t(n)m+1∧t

−Xt(n)m ∧t

)2∣∣∣F

t(n)m

]= E

[A

t(n)m+1∧t

− At(n)m ∧t

∣∣∣Ft(n)m

], a.s.


holds, we have

E[(M

Πnt −M

Πn′t )2

]=

∑

m

E[(X

t(n)tm

∧t−X

t(n′)m′ ∧t

)2(Xt(n)m+1∧t

−Xt(n)m ∧t

)2]

=∑

m

E[(X

t(n)tm

∧t−X

t(n′)m′ ∧t

)2(At(n)m+1∧t

− At(n)m ∧t

)]

= E[ ∫ t

0(Xφ(n)(s) −X

φ(n′)(s))2 dAs

],

where φ(n)(s) are step functions of s such that φ(n)(s) = t(n)m for t (n)m < s < t

(n)m+1.

Let n, n′ → ∞. Then the last term converges to 0. Therefore, because of Doob’sinequality, the sequence of martingales {M(n)

t } converges uniformly in t to a rightcontinuous L2-martingale Mt . This means that {〈X〉(n)t , t ∈ T} converges to anincreasing process {〈X〉t , t ∈ T} uniformly in t with respect to L2-norm. Further,we have X2

t − 〈X〉t = Mt for all t a.s. ��Theorem 1.6.1 Let {Xt } be a local martingale with bounded jumps. Then itsquadratic variation {〈X〉t } exists. It is an increasing process; further, {X2

t − 〈X〉t }is a local martingale.

Proof If Xt, t ∈ T is a local martingale with bounded jumps, we can choosea sequence of stopping times {τn} such that each stopped process is a boundedmartingale (hence L4-martingale). Indeed, let

σN = inf{t ∈ T; |Xt | > N} (= ∞ if the set {· · · } is empty).

It is called the first leaving time of the process Xt from the set {x; |x| ≤ N} orhitting time of the process Xt to the set {x; |x| > N}. It is a stopping time, since{σN ≤ t} = ⋃

0≤r≤t {|Xr | > N} ∈ Ft holds for any t . Here r runs rational numbers.Then Xt∧τn∧σN is a martingale by Theorem 1.5.1. Let M > 0 be a positive constantsuch that |ΔXt | ≤ M holds for any t a.s. Then |Xt∧τn∧σN | is bounded by N+M forall n. Therefore the limit limn→∞ Xt∧τn∧σN = Xt∧σN is also a martingale for anyN . Since P(σN < T ) → 0 ans N → ∞, {Xt } is a local martingale with respect tostopping times {σN,N = 1, 2, . . .}.

Set X(N)t = Xt∧σN . It is an L4-martingale. By the definition the quadratic

variation with respect to the partition Πn, it holds that 〈X(N)〉(n)t = 〈X〉(n)t∧σN. Further,

for any N , 〈X(N)〉(n)t converges to a cadlag increasing 〈X(N)〉t uniformly in t in L2-sense as n → ∞. Then if N ′ ≥ N we have 〈X(N ′)〉t∧σN = 〈X(N)〉t . Therefore thereexists a cadlag increasing process 〈X〉t such that 〈X〉t∧σN = 〈X(N)〉t . Consequently,

if {Πn} is regular, for any N 〈X〉(n)t converges to 〈X〉t uniformly for t ≤ τN in L2-sense. This shows that 〈X〉(n)t converges to 〈X〉t uniformly in t ∈ T in probability.Further, X2

t − 〈X〉t is a local martingale.


It remains to show that 〈X〉t does not depend on the choice of partitions {Πn}. Let

{Π ′n} be another regular sequence of partitions. Let {〈X〉Π ′

nt } be the sequence of the

associated quadratic variations and let 〈X〉′T be its limit as n → ∞. We can choosestill another regular sequence of partitions {Π ′′

n } such that for any Πn ∈ {Πn} andΠ ′

n′ ∈ {Π ′n}, there are Π ′′

n and Π ′′n′ in {Π ′′

n } such that Πn ⊂ Π ′′n and Π ′

n′ ⊂ Π ′′n′

holds. The sequence {〈X〉Π ′′n

t } converges to an increasing process 〈X〉′′t . It shouldcoincide with 〈X〉t and 〈X〉′t . Therefore the uniqueness of the quadratic variationfollows. ��Corollary 1.6.1 If {Xt } be a continuous local martingale, its quadratic variation{〈X〉t } is a continuous increasing process. Further, if {At } is a continuous increasingprocess such that {X2

t −At } is a local martingale, the equality At = 〈X〉t holds forany t ∈ T, a.s.

Proof If {Xt } is a continuous local martingale, its quadratic variation {〈X〉t } is acontinuous process, since {〈X〉Πt } are continuous processes. Suppose that {At } isa continuous increasing process such that {X2

t − At } is a local martingale. Then{Mt := 〈X〉t − At } is a continuous local martingale. Since Mt is a processof bounded variation, we have 〈M〉t = 0. Therefore M2

t is a continuous localmartingale and hence Mt = 0 for any t . Therefore 〈X〉t = At holds for ant t . ��

Let {Xt } and {Yt } be cadlag processes. For a partition Π , we define the quadraticcovariation of X, Y associated with Π by

〈X, Y 〉Πt =n∑

m=1

(Xtm∧t −Xtm−1∧t )(Ytm∧t − Ytm−1∧t ), t ∈ T. (1.43)

Theorem 1.6.2 Suppose that {Xt } and {Yt } are local martingales with boundedjumps. Then quadratic covariations {〈X, Y 〉Πt } converge to a process of boundedcovariation {〈X, Y 〉t } uniformly in t in probability. Further, {XtYt − 〈X, Y 〉t } is alocal martingale.

The process 〈X, Y 〉t is called the quadratic covariation of Xt and Yt .

Proof If Xt = Yt and it is a local martingale with bounded jumps, the existenceof the quadratic variation 〈X〉t is a direct consequence of Lemma 1.6.1. We shallconsider the case Xt �= Yt . It holds that

〈X, Y 〉Πt = 1

4

(〈X + Y 〉Πt − 〈X − Y 〉Πt

).

Let |Π | tend to 0. Then the right-hand side converges to 14 (〈X + Y 〉t − 〈X − Y 〉t ).

Therefore 〈X, Y 〉Πt converges uniformly in t in probability. Denote the limit by〈X, Y 〉t . It is a process of bounded variation. It satisfies


〈X, Y 〉t = 1

4

(〈X + Y 〉t − 〈X − Y 〉t

).

Then XtYt − 〈X, Y 〉t is a local martingale. ��Theorem 1.6.3 Let {Xt } be a continuous semimartingale and let {Yt } be semi-martingales with bounded jumps, decomposed as Xt = Mt +At and Yt = Nt + Bt

respectively, where {Mt,Nt } are local martingales and {At, Bt } are continuousprocesses of bounded variation. Then {〈X, Y 〉Πt } converges as |Π | → 0 to acontinuous process of bounded variation {〈X, Y 〉t , t ∈ T} uniformly in t inprobability. Further,

lim|Π |→0〈X, Y 〉Πt = 〈X, Y 〉t = 〈M,N〉t , t ∈ T. (1.44)

Proof It holds that

〈X, Y 〉Πt = 〈M,N〉Πt + 〈M,B〉Πt + 〈A,N〉Πt + 〈A,B〉Πt .

Let |Π | → 0. Then we have |〈M,B〉Πt | ≤ (〈M〉Πt )1/2(〈B〉Πt )1/2 → 0 and〈A,B〉Πt → 0 since 〈B〉Πt → 0. Similarly, |〈A,N〉Πt | → 0 holds. Since〈M,N〉Πt → 〈M,N〉t , (1.44) holds.

Next, note

|〈M,N〉t − 〈M,N〉s | ≤ (〈M〉t − 〈M〉s) 12 (〈N〉t − 〈N〉s) 1

2 .

The right-hand side converges to 0 uniformly a.s. as t → s, if 〈M〉t is continuous int a.s. Therefore 〈M,N〉t is a continuous process. ��

The quadratic covariation of Xt and Yt is often denoted by 〈Xt, Yt 〉, 〈Xt, Yt 〉t etc.Let [s, T ] be a sub-interval of [0, T ]. We shall consider the quadratic covariation

of two semi-martingales Xt, Yt on the interval [s, T ]. Let Π = {t0 < · · · < tm} be apartition of [s, T ]. We set t ′m = ttm∧t and define

〈X, Y 〉Πs,t =n∑

m=1

(Xt ′m −Xt ′m−1)(Yt ′m − Yt ′m−1

).

Assume the same conditions as in Theorem 1.6.2 for Xt and Yt . Then, as |Π | → 0,〈X, Y 〉Πs,t , t ∈ [s, T ] converges to a continuous process of bounded variation. Wedenote the limit by 〈X, Y 〉s,t and call it the quadratic covariation of Xt and Yt on theinterval [s, T ]. In particular, if both Xt, Yt are semi-martingales for t ∈ [0, T ], thenit holds that 〈X, Y 〉s,t = 〈X, Y 〉t − 〈X, Y 〉s almost surely for any 0 ≤ s < t ≤ T .

Note The quadratic covariation of two L2 martingales Xt, Yt was introduced byP.A. Meyer [85]. It is often denoted by [X, Y ]t . On the other hand, a bracketprocess 〈X, Y 〉t was introduced by Kunita–Watanabe [66], which is defined as a

1.7 Markov Processes and Backward Markov Processes 37

unique natural (predictable) process of bounded variation such that XtYt − 〈X, Y 〉tis a martingale. The existence of such bracket process is due to the Doob–Meyerdecomposition theorem of submartinales [84]. If one of Xt and Yt is continuous,say Xt is continuous, then both of [X, Y ]t and 〈X, Y 〉 are continuous processes andcoincide each other. In this monograph, we take 〈X, Y 〉t as the definition of thequadratic covariation of Xt and Yt .

Theorem 1.6.2 is discussed in Kunita [59], Karatzas–Shreve [55] for contin-uous semi-martingales. We discussed this again, without using the Doob–Meyerdecomposition of a sub-martingale. In this monograph, we do not follow difficultdiscussions concerning natural increasing processes and predictable increasingprocesses, which appear in the Doob–Meyer decomposition. See [55].

1.7 Markov Processes and Backward Markov Processes

Let S be a Hausdorff topological space with the second countability. Let B(S) beits topological Borel field. We denote by Bb(S) the set of all real bounded B(S)-measurable functions. Cb(S) is the set of all bounded continuous functions on S. Afunction P(x,E) of two variables x ∈ S,E ∈ B(S) is called a kernel on the space(S,B(S)) if for each x ∈ S, it is a finite measure in E and, for each E ∈ B(S), it isa B(S)-measurable function of x. We set for f ∈ Bb(S)

Pf (x) =∫

S

f (y)P (x, dy). (1.45)

It is a B(S)-measurable function. If it is a bounded function for any f ∈ Bb(S), thekernel is called bounded. A bounded kernel P defines a linear transformation fromBb(S) into itself.

Let T = [0,∞). Let {Ps,t (x, E); 0 ≤ s < t < ∞} be a family of boundedkernels satisfying the following properties.

1. Semigroup property: Ps,tPt,uf = Ps,uf holds for any s < t < u and f ∈ Bb(S).The semigroup property is equivalent to

∫

S

Ps,t (x, dy)Pt,u(y, E) = Ps,u(x, E), ∀x ∈ S, ∀E ∈ B(S).

The equation is called the Chapman–Kolmogorov equation.2. Continuity: limt↓s Ps,tf (x) = f (x) holds for all x for any f ∈ Cb(S).

Then {Ps,t (x, E)} is called a transition function. The family of linear transforma-tions {Ps,t } satisfying the above two properties is called a semi-group (of lineartransformations).


If Ps,t (x, S) = 1 holds for any x and s < t , the transition function is calledconservative and if Ps,t (x, S) ≤ 1 holds for all x and s < t , the transition functionis called Markovian. A conservative transition function is called a transitionprobability.

If Ps,t (x, E) is Markovian, we adjoin a cemetery ∞ to S as a one pointcompactification if S is non-compact and as an isolated point if S is compact.We set S′ = S ∪ {∞} and define a transition probability P ′

s,t (x, E) on S′ byP ′s,t (x, E) = Ps,t (x, E) if x ∈ S,E ⊂ S and

P ′s,t (x, {∞}) = 1 − Ps,t (x, S) if x ∈ S, = 1 if x = ∞. (1.46)

Then P ′s,t (x, E) is a transition probability. We denote by B∞(S) the set of f ∈

Bb(S′) such that f (∞) = 0. Further, C∞(S) is the set of all f ∈ Cb(S

′) such thatf (∞) = 0. Then it holds that P ′

s,t f (x) = Ps,tf (x) for any x ∈ S and s < t iff ∈ B∞(S).

Suppose that we are given a filtration {Ft ; t ∈ T} of sub σ -fields of F . Let t0 ∈ T.Let Xt, t ∈ [t0,∞) be an {Ft }-adapted S′-valued stochastic process, continuous inprobability. It is called a Markov process of initial state (Xt0 , t0) with transitionfunction {Ps,t (x, ·)} (or with semigroup {Ps,t }), if Xt0 ∈ S a.s. and

P(Xt ∈ E|Fs) = P ′s,t (Xs, E), a.s.

holds for any Borel subset E of S′ and for any t0 ≤ s < t < ∞. The Markovproperty is equivalent to that E[f (Xt )|Fs] = Ps,tf (Xs) holds for any f ∈ B∞(S)

and s < t . For each ω ∈ Ω , Xt(ω), t ∈ [t0,∞) is called a path of the Markovprocess. If paths of the Markov process are continuous a.s., the process is called acontinuous Markov process. If paths are right (or left) continuous a.s., it is calleda right continuous (or left continuous, respectively) Markov process. If Xt is anS-valued process, i.e., P(Xt ∈ S for all t) = 1, then Xt is called a conservativeMarkov process on S. The Markov process of the initial state (x, s) ∈ S×T is oftendenoted by X

x,st and the family {Xx,s

t } is called the system of Markov processeswith transition function {Ps,t (x, ·)}.

A transition function is time homogeneous if Ps+h,t+h(x,E) = Ps,t (x, E) holdsfor any 0 ≤ s < t < t + h < ∞), h > 0 and E. If it is time homogeneous, itholds that Ps+h,t+hf = Ps,tf for any f ∈ Bb(S). A time homogeneous transitionfunction {Ps,t (x, E)} is often denoted by {Pt−s(x, E)} and its semigroup {Ps,t } isoften denoted by {Pt−s}. Then it holds PsPtf = Ps+t f . A Markov process is timehomogeneous if the associated transition function is time homogeneous.

Let Xt, t ∈ [0,∞) be a d ′-dimensional Lévy process. Let Ft be the smallestσ -field containing all null sets of F , with respect to which Xu, 0 ≤ u ≤ t aremeasurable. For a Borel subset E of Rd ′ , we set

Ps,t (x, E) = P(Xt −Xs + x ∈ E).

Then, since Xt −Xs is independent of Fs , we have

1.7 Markov Processes and Backward Markov Processes 39

P(Xt ∈ E|Fs) = P(Xt −Xs +Xs ∈ E|Fs)

= P(Xt −Xs + y ∈ E)|y=Xs = Ps,t (Xs, E).

Therefore Xt is a time homogeneous Markov process with transition probability{Ps,t (x, E)} defined above.

In particular, if Xt = Wt, t ∈ [0,∞) is a Wiener process, it is a timehomogeneous Markov process with transition probability

Ps,t (x, E) = 1

(2π(t − s))d2

∫

E

e− 1

2(t−s)|x−y|2

dy.

If Xt = Nt is a Poisson process, it is a time homogeneous Markov process withstate space S = {0, 1, 2, . . .}. Its transition probability is given by

Ps,t (i, i + n) = (λ(t − s))n

n! e−λ(t−s), n = 0, 1, 2, . . .

Jumping times τn, n = 1, 2, . . . of the Poisson process Nt are stopping times. Weshow this by induction of n. Suppose that τn−1 is a stopping time. Then

{τn ≤ t} =⋃

r≤t

{Nr > Nτn−1 , τn−1 < t} =⋃

r≤t

{Nr > n− 1, τn−1 < t}.

Since {τn−1 < t} ∈ Ft holds, the above set belongs to Ft . Since this is valid for anyt , τn is a stopping time.

Let τ be a stopping time with respect to the filtration {Ft }. We set

Fτ = {A ∈ F; A ∩ {τ ≤ t} ∈ Ft holds for any t}. (1.47)

It is a sub σ -field of F . A right continuous Markov process Xt, t ∈ [t0,∞) is saidto have the strong Markov property, if the equality

P({Xτ+t ∈ E} ∩ {τ + t < ∞}|Fτ ) = P ′τ,τ+t (Xτ , E)1τ+t<∞ (1.48)

holds for all t > 0, E ∈ B(S) for any stopping time τ with τ ≥ t0. The propertyis equivalent to that the equality E[f (Xτ+t )1τ+t<∞|Fτ ] = Pτ,τ+t f (Xτ )1τ+t<∞holds for any f ∈ C∞(S).

Proposition 1.7.1 Assume that Ps,tf (x) is continuous in s, t, x for any f ∈C∞(S). Then the right continuous Markov process has the strong Markov property.

Proof Given a stopping time τ , we define for each positive integer n, a random timeτn by

τn = m

2n, if

m− 1

2n< τ ≤ m

2n, m = 1, 2, . . .


Then {τn, n = 1, 2, . . .} is a sequence of stopping times decreasing to the stoppingtime τ . Let B ∈ Fτ . Since τ ≤ τn, we have Fτ ⊂ Fτn . Therefore, for any B ∈ Fτ ,we have B ∩ {τn = m

2n } ∈ F m2n

. Then using the Markov property,

E[f (Xτn+t ); {τn + t < ∞} ∩ B]=

∑

m

E[f (X m

2n +t ); {τn = m

2n} ∩ B

]

=∑

m

E[P m

2n , m2n +t f (X m

2n); {τn = m

2n} ∩ B

]

= E[Pτn,τn+t f (Xτn); {τn + t < ∞} ∩ B].

Let n tend to ∞. Since Ps,tf (x) is continuous in s, t, x, we have

E[f (Xτ+t ); {τ + t < ∞} ∩ B] = E[Pτ,τ+t f (Xτ ); {τ + t < ∞} ∩ B].

This proves the strong Markov property. ��Let us consider again a Lévy process. Its semigroup Ps,tf (x) is written as

Ps,tf (x) =∫

Rd

f (x + y)μs,t (dy),

where μs,t (dy) = P(Xt − Xs ∈ dy). Therefore if f is a bounded continuousfunction of R

d , Ps,tf (x) is continuous in s, t, x. Therefore a Lévy process is astrong Markov process.

An S′-valued continuous Markov process with the strong Markov property iscalled a diffusion process.

Let {Ps,t (x, E); 0 ≤ s < t < ∞} be a family of bounded kernels on S satisfyingthe following properties.

1. Backward semigroup property: Pt,uPs,t f = Ps,uf holds for any s < t < u andf ∈ Bb(S),

2. Continuity: lims↑t Ps,t f (x) = f (x) holds for all x for any f ∈ Cb(S).

Then {Ps,t (x, E)} is called a backward transition function. The family of lineartransformations {Ps,t } satisfying the above two properties is called a backwardsemigroup (of linear transformations). Backward Markovian transition functionand backward transition probability are defined similarly to the forward case. IfPs,t (x, E) is Markovian, we define backward transition probability P ′

s,t (x, E) onS′ = S ∪∞ similarly to the forward case.

Suppose that we are given a backward filtration {Fs; s ∈ T} of sub σ -fields ofF . Let 0 < t1 < ∞. Let Xs, 0 ≤ s ≤ t1 be an {Fs}-adapted S′-valued cagladstochastic process. It is called a backward Markov process on S with backwardtransition function {Ps,t (x, ·)}, if P(Xs ∈ E|Ft ) = P ′

s,t (Xt , E) holds a.s. for any

1.8 Kolmogorov’s Criterion for the Continuity of Random Field 41

Borel subset E of S′ and 0 ≤ s < t ≤ t1. The backward Markov property isequivalent to that E[f (Xs)|Ft ] = Ps,t f (Xt ) holds for any f ∈ B∞(S) and s < t . Aleft continuous backward process Xs is called a backward strong Markov process, if

E[f (Xτ−s)1τ−s>0|Fτ ] = Pτ−s,τ f (Xτ )1τ−s>0

holds for any backward stopping time τ with τ ≤ t1 and any bounded continuousfunction f on S. If the backward semigroup Ps,t maps Cb(S) into itself and iscontinuous in s, t , then the left continuous backward Markov process is a backwardstrong Markov process.

1.8 Kolmogorov’s Criterion for the Continuity of RandomField

We shall introduce Kolmogorov’s criterion for a given random field to have amodification of a continuous random field. Let {X(x), x ∈ D} be a random fieldwith values in a Banach space S with the norm ‖ ‖, where D is a bounded domainof Rd . Elements of D are denoted by x = (x1, . . . , xd), y = (y1, . . . , yd) etc. Thedistance of x, y is defined by |x − y| = max1≤i≤d |xi − yi |.

The next theorem was shown first in the case d = 1 by Kolmogorov andChentzov [20] and then it was extended to the case d > 1 by Totoki [114].

Theorem 1.8.1 (Kolmogorov–Totoki) Let {X(x), x ∈ D} be a random field withvalues in a separable Banach space S with norm ‖ ‖, where D is a bounded domainin R

d . Assume that there exist positive constants γ,C and α > d satisfying

E[‖X(x)−X(y)‖γ ] ≤ C|x − y|α, ∀x, y ∈ D. (1.49)

Then there exists a continuous random field {X(x), x ∈ D} such that X(x) = X(x)

holds a.s. for any x ∈ D, where D is the closure of D.Further, if 0 ∈ D and E[‖X(0)‖γ ] < ∞, we have E[supx∈D ‖X(x)‖γ ] < ∞.

By a linear invertible transformation, the bounded domain D is transformed to abounded domain included in the cube [0, 1]d . So we will discuss the case where D

is included in the cube [0, 1]d .A real number 0 ≤ x ≤ 1 has a dyadic expansion x = ∑∞

k=1 ak2−k , whereak are 0 or 1. We set xn = ∑n

k=1 ak2−k and call it the dyadic rational of lengthn defined from x. Let x = (x1, . . . , xd) be an element of [0, 1]d . For x, we setxn = (xn

1 , . . . , xnd ). If x = xn holds for some positive integer n, x is called a dyadic

rational of length n. We denote by Δn the set of all dyadic rationals with length n in[0, 1]d and we set Δ := ⋃∞

n=1 Δn. It is a dense subset of [0, 1]d .


Let β be a positive number less than or equal to 1. Given a map f : D → S, wedefine, for each positive integer n, the variation of continuity and the variation ofβ-Hölder continuity of f by

Δn(f ) = maxx,y∈Δn∩D,|x−y|=2−n

‖f (x)− f (y)‖,

Δβn(f ) = 2βnΔn(f ).

Lemma 1.8.1 Suppose that a map f ;D → S satisfies∑∞

m=1 Δβm(f ) < ∞. Then

the inequality

‖f (x)− f (y)‖ ≤ 3( ∞∑

m=1

Δβm(f )

)|x − y|β (1.50)

holds for any x, y ∈ D ∩ Δ. Further, the map f ;D ∩ Δ → S is extended to aβ-Hölder continuous map f ; D → S such that f (x) = f (x) holds for x ∈ D ∩Δ.

Proof For a given map f (x), x ∈ D and a positive integer n, define a step functionfn(x), x ∈ D by fn(x) = f (xn) if xn ∈ D and fn(x) = f (x) if xn /∈ D, where xn

is the dyadic rational of length n defined from x. Then it holds that

‖fn(x)− fn−1(x)‖ = ‖f (xn)− f (xn−1)‖ ≤ Δn(f ) ≤ Δβn(f )

for any x ∈ D. Therefore if n > k we have

‖fn(x)− fk(x)‖ ≤n∑

m=k+1

‖fm(x)− fm−1(x)‖ ≤n∑

m=k+1

Δβm(f )

for any x. Hence the sequence of functions {fn} converges uniformly. Denote thelimit function by f . Then it holds that f (x) = f (x) for any x ∈ Δ = ⋃

n Δn.Now take any two points x, y ∈ Δ. There exists k ∈ N such that 2−(k+1) ≤ |x −

y| < 2−k . Further, there exists n ∈ N such that n > k and x = xn, y = yn ∈ Δn.Then we have

‖f (x)− fk(x)‖ = ‖f (xn)− f (xk)‖ ≤n∑

m=k+1

‖f (xm)− f (xm−1)‖

≤n∑

m=k+1

Δm(f ) ≤n∑

m=k+1

2−βmΔβm(f )

≤( ∞∑

m=1

Δβm(f )

)|x − y|β.

1.8 Kolmogorov’s Criterion for the Continuity of Random Field 43

‖f (y)− fk(y)‖ is estimated in the same way. Since ‖fk(x)− fk(y)‖ ≤ Δk(f ), weget

‖f (x)− f (y)‖ ≤ ‖f (x)− fk(x)‖ + ‖fk(x)− fk(y)‖ + ‖fk(y)− f (y)‖

≤ 3( ∞∑

m=1

Δβm(f )

)|x − y|β.

Therefore f satisfies the inequality (1.50) for any x, y ∈ Δ. Further f is β-Höldercontinuous by the above inequality. Since f (x) = f (x) holds for x ∈ D ∩ Δ,f (x), x ∈ D is a β-Hölder continuous extension of f (x), x ∈ D∩Δ. Hence we getthe assertion of the lemma. ��

We shall apply the above lemma to the random field X(x). Observe that for eachω, X(·, ω) restricting x to D ∩Δ can be regarded as a map form D ∩Δ to S. Thenwe have

‖X(x, ω)−X(y, ω)‖ ≤ 3

( ∞∑

n=1

Δβn(X(ω))

)|x − y|β, (1.51)

for any x, y ∈ D ∩Δ.

Lemma 1.8.2 Let β be a positive number satisfying βγ < α − d. Then,

E[( ∞∑

n=1

Δβn(X)

)γ ] 1γ ≤

( ∞∑

n=1

(2−

α−d−βγγ

)n)· (2dC)

1γ < ∞, (1.52)

where C is the positive constant in the inequality (1.49).

Proof We will consider the case γ ≥ 1 only. Observe the inequality

Δβn(X)γ ≤

(sup

x,y∈Πn∩D,|x−y|= 12n

‖X(x)−X(y)‖2nβ)γ

≤∑

(‖X(x)−X(y)‖2nβ)γ ,

where the summation is taken for all x, y ∈ D ∩ Δn such that |x − y| = 12n . Then

the number of summations is at most 2(n+1)d . Therefore

E[Δβn(X)γ ] ≤ 2(n+1)d+nβγ E[‖X(x)−X(y)‖γ ]

≤ 2n(d+βγ−α)2dC.

In the last inequality, we applied the inequality (1.49). Therefore we get


E[( ∞∑

n=1

Δβn(X)

)γ ] 1γ ≤

∞∑

n=1

E[Δβn(X)γ ] 1

γ

≤( ∞∑

n=1

2−nα−d−βγ

γ

)· (2dC)

1γ < ∞. (1.53)

��Proof of Theorem 1.8.1 The random field X(x) restricting x on D ∩Δ satisfies theinequality (1.50), where

∑∞n=1 Δ

βn(X) < ∞ a.s. Therefore {X(x), x ∈ D ∩ Δ}

is uniformly β-Hölder continuous a.s. Then there exists a uniformly continuousrandom field {X(x), x ∈ D} such that X(x) = X(x) holds a.s. for any x ∈ D ∩ Δ.Since X(x) is continuous in probability, the equality X(x) = X(x) holds a.s for anyx ∈ D. Consequently, the random field {X(x), x ∈ D} is a continuous modificationof the random field {X(x), x ∈ D}.

Now we have the inequality

supx∈D

‖X‖ ≤ ‖X(0)‖ + 3(∑

n

Δβn(X))

by (1.51). Therefore we have E[supx∈D ‖X(x)‖γ ] < ∞. ��

Chapter 2Stochastic Integrals

Abstract We discuss Itô’s stochastic calculus, which will be applied in laterdiscussions. In Sects. 2.1, 2.2, 2.3, and 2.4, we discuss stochastic calculus related tointegrals by Wiener processes and continuous martingales. In Sect. 2.1, we definestochastic integrals based on Wiener processes and continuous martingales. InSect. 2.2, we establish Itô’s formula. It will be applied for proving Lp-estimates ofstochastic integrals, called the Burkholder–Davis–Gundy inequality, and Girsanov’stheorem. The smoothness of the stochastic integral with respect to parameter willbe discussed in Sect. 2.3. Fisk–Stratonovitch symmetric integrals will be discussedin Sect. 2.4.

In Sects. 2.5 and 2.6, we discuss stochastic calculus based on Poisson randommeasures. Stochastic integrals are defined in Sect. 2.5. The chain rules formula forjump processes and Lp-estimates of jump integrals will be discussed in Sect. 2.6.In Sect. 2.7, we discuss the backward processes and backward integrals. Thesetopics are related to dual processes or inverse processes, which will be discussedin Chaps. 3 and 4.

2.1 Itô’s Stochastic Integrals by Continuous Martingaleand Wiener Process

Let (Ω,F , P ) be a probability space equipped with a filtration {Ft } of sub-σ -fieldsof F . In this chapter, we will restrict the time parameter to a finite interval T =[0, T ]. Let Xt, t ∈ T be a continuous local martingale with respect to the filtration{Ft } and let φ(r), r ∈ T be a real-valued process. Let 0 ≤ s < t ≤ T be a fixedtime. We want to define the stochastic integral of φ(r) based on dXr , written as∫ t

sφ(r) dXr . It is not a usual Lebesgue integral, since Xt is not a process of bounded

variation. At the beginning, we will introduce the class of integrand φ(r), for whichthe integrals can be defined.

The predictable σ -field P (with respect to the filtration {Ft }) is the σ -fieldon T × Ω generated by left continuous {Ft }-adapted processes. A P-measurablestochastic process ϕ(t), t ∈ [0, T ] is called a predictable process (with respect tothe filtration {Ft }).


45


https://doi.org/10.1007/978-981-13-3801-4_2

46 2 Stochastic Integrals

Let Xt, t ∈ T be a continuous L2-martingale adapted to the filtration {Ft } andlet 〈X〉t , t ∈ T be its quadratic variation. It is a unique continuous increasingprocess such that X2

t − 〈X〉t , t ∈ T is a martingale (Sect. 1.6). Let φ(r), r ∈ T

be a predictable process. We define its norm by

‖φ‖ = E[ ∫ T

0|φ(r)|2 d〈X〉r

] 12.

A stochastic process φ(r), r ∈ T is called a simple predictable process if it is writtenas φ(r) = ∑n

m=0 φm1(tm−1,tm](r), where 0 = t0 < t1 < · · · < tn = T and φm arebounded and Ftm−1 -measurable.

We shall define the stochastic integral based on a continuous L2-martingaleXt, t ∈ T. Let φ(r) be a simple predictable process. We define

Mt =n∑

m=1

φm(Xtm∧t −Xtm−1∧t ), t ∈ T

and call it the stochastic integral of φ(r) by dXr and denote it by∫ t

0 φ(r) dXr .Then, Mt, t ∈ T may be regarded as a martingale transform with discrete timeΠ = {0 = t0 < · · · < tn = T }. Therefore we have the equality E[Mt−Mr |Fr ] = 0a.s. for r ∈ Π . Let 0 ≤ s < t ≤ T . We may assume that s, t are adjoined in thepartition Π . Then, using Propositions 1.5.1 and 1.6.1, we have

E[(Mt −Ms)2|Fs] =

∑

m;tm−1≥s

E[(Mtm −Mtm−1)2|Fs]

= E[∑

m

φ2m(Xtm −Xtm−1)

2|Fs]

=∑

m

E[φ2mE[(Xtm −Xtm−1)

2|Ftm−1]|Fs]

=∑

m

E[φ2mE[〈X〉tm − 〈X〉tm−1 |Ftm−1]|Fs]

= E[∑

m

φ2m(〈X〉tm − 〈X〉tm−1)|Fs]

= E[ ∫ t

s

φ(r)2 d〈X〉r∣∣∣Fs

], a.s.

Since the above holds for any s < t , M2t − ∫ t

0 φ(r)2 d〈X〉r , t ∈ T is a martingale.Then the quadratic variation of Mt, t ∈ T is given by

∫ t

0 φ(r)2 d〈X〉r , t ∈ T in viewof Corollary 1.6.1. We have thus shown

2.1 Itô’s Stochastic Integrals by Continuous Martingale and Wiener Process 47

⟨ ∫ t

0φ(r) dXr

⟩=

∫ t

0|φ(r)|2 d〈X〉r , a.s. (2.1)

for any 0 < t ≤ T . Taking expectations of both sides, we get

E[∣∣∣

∫ t

0φ(r) dXr

∣∣∣2] = E

[ ∫ t

0|φ(r)|2 d〈X〉r

]≤ ‖φ‖2

for any t . Now let φ be a predictable process such that ‖φ‖ < ∞. Then there exists asequence of simple predictable processes {φn} such that ‖φ − φn‖ → 0 as n → ∞.Then, using Doob’s inequality for a martingale (Theorem 1.5.1), we have

E[

supt∈T

∣∣∣∫ t

0φn(r) dXr−

∫ t

0φn′(r) dXr

∣∣∣2] ≤4E

[∣∣∣∫ T

0(φn(r)−φn′(r)) dXr

∣∣∣2]

= 4‖φn − φn′ ‖2 → 0,

as n, n′ → ∞. Hence limn→∞∫ t

0 φn(r) dXr exists uniformly in t . We denote it by∫ t

0 φ(r) dXr and call it the stochastic integral or Itô integral of φ(r) by dXr . Thestochastic integral is a continuous L2-martingale. It satisfies (2.1).

We will extend the stochastic integral for a continuous local martingale Xt, t ∈ T.Let 〈X〉t , t ∈ T be its quadratic variation. Let L(〈X〉) be the set of all predictableprocesses φ(t), t ∈ T satisfying

∫ T

0 |φ(r)|2 d〈X〉r < ∞ a.s. Let φ ∈ L(〈X〉). ForN > 0, define a stopping time by

τN = inf{t ∈ T; |Xt | +∫ t

0|φ(r)|2 d〈X〉r > N},

= ∞, if the set {· · · } is empty.

Then the stopped process X(N)t = Xt∧τN , t ∈ T is a bounded continuous

martingale. See the proof of Theorem 1.6.1. It holds that 〈X(N)〉t = 〈X〉t∧τN .

Since E[∫ t

0 |φ(r)|2 d〈X(N)〉r ] < ∞, the family of stochastic integral M(N)t ≡∫ t

0 φ(r) dX(N)r , t ∈ T is well defined as an L2-martingale. Let N ′ > N and consider

M(N ′)t ≡ ∫ t

0 φ(r) dX(N ′)r . Then it holds that M(N ′)

t∧τN= M

(N)t . Then there exists a

continuous local martingale Mt, t ∈ T such that Mt∧τN = ∫ t

0 φ(r) dX(N)r holds for

any N = 1, 2, . . .. We set Mt =∫ t

0 φ(r) dXr and call it the stochastic integral or Itôintegral of φ(r) by dXr . It holds that 〈∫ t

0 φ(r) dXr 〉 =∫ t

0 φ(r)2 d〈X〉r a.s. for any0 < t ≤ T .

Let Xt, t ∈ T and Yt , t ∈ T be continuous local martingales and let φ ∈ L(〈X〉)and ϕ ∈ L(〈Y 〉). In view of (2.1), we have the relation

⟨ ∫ t

0φ dX,

∫ t

0ϕ dY

⟩=

∫ t

0φ(r)ϕ(r) d〈X, Y 〉r , a.s. (2.2)

for any 0 < t ≤ T .


Let φ1, φ2 ∈ L(〈X〉) and let c1, c2 be constants. Then we have

∫ t

0(c1φ1 + c2φ2)(r) dXr = c1

∫ t

0φ1(r) dXr + c2

∫ t

0φ2(r) dXr, a.s.

for any 0 < t ≤ T . Indeed, the equality holds if both φ1, φ2 are simple predictableprocesses. Then the equality can be extended to arbitrary predictable processesφ1, φ2 belonging to L(〈X〉).

For 0 ≤ s < t ≤ T , the integral∫ t

sφ(r) dXr is defined as

∫ t

s

φ(r) dXr :=∫ t

0φ(r)1[s,T ](r) dXr =

∫ t

0φ(r) dXr −

∫ s

0φ(r) dXr, a.s.

Let Xt, t ∈ T be a continuous semi-martingale decomposed as Xt = Mt + At .Let |A|t be the process of the total variation of At . Let φ(r) be a predictable processsatisfying

∫ T

0 |φ(r)|2 d〈X〉r < ∞ and∫ T

0 |φ(r)| d|A|r < ∞, a.s. We define theintegral

∫ t

0φ(r) dXr :=

∫ t

0φ(r) dMr +

∫ t

0φ(r) dAr,

where∫ t

0 φ(r, ω) dA(r, ω) is the Lebesgue–Stieltjes integrals for each ω.Now, we shall consider stochastic integrals by a Wiener process. Let d ′ be a

positive integer. Let Wt = (W 1t , . . . ,W

d ′t ), t ∈ [0,∞) be a d ′-dimensional Wiener

process. Suppose that Wt is adapted to a given filtration {Ft } and that Wu − Wt isindependent of Ft for any u > t . Then Wt, t ∈ [0,∞) is called an {Ft }-Wienerprocess. Components Wk

t , k = 1, . . . , d ′ of the {Ft }-Wiener process Wt are L2-martingales with respect to {Ft }. Indeed, since Wk

t −Wkr are independent of Fr , we

have E[Wkt −Wk

r |Fr ] = E[Wkt −Wk

r ] = 0 a.s. Further,

E[(Wkt −Wk

r )(Wlt −Wl

r )|Fr ]= E[(Wk

t −Wkr )(W

lt −Wl

r )] = δkl(t − r), a.s.

Therefore we have 〈Wk,Wl〉t = δkl(t − 0). We define a space of d ′-dimensionalpredictable processes by

LT ={φ(r) = (φ1(r), . . . , φd ′(r)); predictable and

∫ T

0|φ(r)|2 dr < ∞

},

where |φ(r)|2 = ∑d ′k=1 |φk(r)|2. Then for any φ(r) ∈ LT, integrals

∫ t

0 φk(r)dWkr ,

k = 1, . . . , d ′ are well defined for any 0 < t ≤ T . These are local martingales andsatisfy

2.2 Itô’s Formula and Applications 49

⟨ ∫φk dWk,

∫φl dWl

⟩

t= δkl

∫ t

0φk(r)φl(r) dr, a.s. (2.3)

We set

It (φ) =∫ t

0(φ(r), dWr) =

d ′∑

k=1

∫ t

0φk(r) dWk

r . (2.4)

We next consider the stochastic integral where the integrand depends on param-eter. Let Λ be a parameter set. We assume that Λ is an e-dimensional Euclideanspace R

e or its unit ball {λ ∈ Re; |λ| < 1}. We denote by LT(Λ) the set of all

d ′-dimensional measurable random fields φλ(t), (t, λ) ∈ T×Λ such that for any λ,φλ(t) is predictable and satisfies

∫Λ

∫T|φλ(r)|2 dr dλ < ∞ a.s. For φλ(t) ∈ LT(Λ),

stochastic integrals∫ t

0 (φλ(r), dWr) are well defined as continuous martingales foralmost all λ. Further, the family of integrals has a modification such that it is (t, λ)-measurable. We give a Fubini theorem for changing the order of integrals by dWr

and dλ.

Proposition 2.1.1 Let φλ(r) ∈ LT(Λ) and g(λ), λ ∈ Λ be a bounded measurablefunction of compact supports. Then we have for any t ∈ T,

∫

Λ

( ∫ t

0(φλ(r), dWr)

)g(λ) dλ =

∫ t

0

( ∫

Λ

φλ(r)g(λ) dλ, dWr

), a.s. (2.5)

Proof If φλ(t) is a simple functional written as∑

m φλ(m)1(tm−1,tm](t) with boundedFtm−1×B(Λ)-measurable functional φλ(m), the equality (2.5) can be shown directly.Since φλ(t) ∈ LT(Λ) can be approximated by a sequence of the above simplefunctionals, equality (2.5) is valid for any φλ(t) ∈ LT(Λ). ��

2.2 Itô’s Formula and Applications

A d-dimensional continuous stochastic process Xt = (X1t , . . . , X

dt ), t ∈ T is

called a d-dimensional continuous semi-martingale, if all Xit , t ∈ T; i = 1, . . . , d

are continuous semi-martingales. Let f (x1, . . . , xd , t) be a smooth function. Weconsider the differential rule of the composite functional f (Xt , t) with respect to t .The rule is called the formula of the change of variables or Itô’s formula.

Theorem 2.2.1 (Itô’s formula) Let f (x, t) be a C2,1-function on Rd × T and let

Xt, t ∈ T be a d-dimensional continuous semi-martingale. Then f (Xt , t), t ∈ T isa continuous semi-martingale. Further, we have


f (Xt , t) = f (Xs, s)+∫ t

s

∂f

∂t(Xr, r) dr (2.6)

+d∑

i=1

∫ t

s

∂f

∂xi(Xr, r) dX

ir +

1

2

d∑

i,j=1

∫ t

s

∂2f

∂xi∂xj(Xr, r) d〈Xi,Xj 〉r ,

a.s. for any 0 ≤ s < t ≤ T .

Proof It is sufficient to prove the case s = 0. Let Πn = {t0 < t1 < · · · < tn} be apartition of [0, t]. Then we have by the Taylor expansion of f (x, t),

f (Xt , t)−f (X0, 0)

=n∑

m=1

{f (Xtm, tm)− f (Xtm−1 , tm−1)

}

=∑

m

{f (Xtm, tm)− f (Xtm, tm−1)

}

+d∑

i=1

{∑

m

∂f

∂xi(Xtm−1 , tm−1)(X

itm−Xi

tm−1)}

+ 1

2

d∑

i,j=1

{∑

m

∂2f

∂xi∂xj(ξm, tm−1)(X

itm−Xi

tm−1)(X

jtm−X

jtm−1

)}

= I1 + I2 + I3,

where ξm are random variables such that |ξm − Xtm−1 | ≤ |Xtm − Xtm−1 |. Let |Πn|tend to 0. Then we get

limn→∞ I1 =

∫ t

0

∂f

∂t(Xr, r) dr, a.s.

limn→∞ I2 =

∑

i

∫ t

0

∂f

∂xi(Xr, r) dX

ir , a.s.

immediately from the definition of the stochastic integral. Further, applyingTheorem 1.6.2, we get

limn→∞ I3 = 1

2

∑

i,j

∫ t

0

∂2f

∂xi∂xj(Xr, r) d〈Xi,Xj 〉r , a.s.

Summing up these computations, we get the formula of the theorem. ��


We set

LT ={υ(r); predictable and

∫ T

0|υ(r)| dr < ∞

}.

Let φ(r) = (φik(r)) be a d × d ′-dimensional process such that (φi·(r)) ∈ LT

for any i and let υ(r) = (υ1(r), . . . , υd(r)) be a d-dimensional process such thatυi(r) ∈ LT for any i. We consider a d-dimensional semi-martingale represented by

Xt = X0 +∫ t

0φ(r) dWr +

∫ t

0υ(r) dr, (2.7)

where

∫ t

0φ(r) dWr =

( d ′∑

k=1

∫ t

0φik(r) dWk

r , i = 1, . . . , d).

Corollary 2.2.1 Let Xt = (X1t , . . . , X

dt ) be a continuous semi-martingale repre-

sented by (2.7). Then the formula (2.6) is rewritten as

f (Xt , t) = f (Xs, s)+∫ t

s

∂f

∂t(Xr, r) dr

+d∑

i=1

d ′∑

k=1

∫ t

s

∂f

∂xi(Xr, r)φ

ik(r) dWkr +

d∑

i=1

∫ t

s

∂f

∂xi(Xr, r)υ

i(r) dr

+ 1

2

d∑

i,j=1

∫ t

s

∂2f

∂xi∂xj(Xr, r)(

d ′∑

k=1

φik(r)φjk(r)) dr. (2.8)

Proof We have

〈Xi,Xj 〉t =∑

k,l

⟨ ∫φik(r) dWk

r ,

∫φjl(r) dWl

r

⟩

t=

∑

k

∫ t

0φik(r)φjk(r) dr

by (2.3). Therefore (2.6) implies (2.8). ��We will discuss three problems by applying the above Itô’s formula. We first give

Lévy’s characterization of a Wiener process.

Proposition 2.2.1 Let X1t , . . . , X

dt be continuous {Ft }-martingales satisfying

〈Xi,Xj 〉t = δij t, i, j = 1, . . . , d, a.s.


for any 0 < t < T . Then Xt = (X1t , . . . , X

dt ), t ∈ T is a d-dimensional {Ft }-

Wiener process.

Proof We apply Itô’s formula (2.6) to the function f (x, t) = e√−1(v,x). We have

e√−1(v,Xt−Xs)

= 1 +√−1∑

j

∫ t

s

e√−1(v,Xr−Xs)vj dX

jr − 1

2|v|2

∫ t

s

e√−1(v,Xr−Xs) dr.

Let A ∈ Fs . Multiply the functional 1A to each term of the above equation and then

take the expectation. Since E[ ∫ t

se√−1(v,Xr−Xs)vj dX

jr

∣∣∣Fs

]= 0 holds, we have

E[e√−1(v,Xt−Xs)1A] = P(A)− 1

2|v|2

∫ t

s

E[e√−1(v,Xr−Xs)1A] dr.

Differentiating the above with respect to t , ϕt = E[e√−1(v,Xt−Xs)1A] satis-

fies the differential equation, dϕt

dt= − 1

2 |v|2ϕt . Integrating it we obtain ϕt =e− 1

2 |v|2(t−s)P (A), or equivalently,

E[e√−1(v,Xt−Xs)1A] = e−

12 |v|2(t−s)P (A).

Consequently, the law of Xt −Xs is Gaussian with mean 0 and covariance (t − s)I .Further, Xt−Xs is independent of Fs . This proves that Xt is an {Ft }-Wiener process.

��We next study Lp-estimations of stochastic integrals. Let p ≥ 2. We set

Lp

T=

{φ ∈ LT; ‖φ‖p < ∞

}, L

p

T=

{φ ∈ LT; ‖φ‖p < ∞

},

where ‖φ‖p = E[ ∫ T

0 |φ(r)|p dr] 1p .

Proposition 2.2.2 (Burkholder–Davis–Gundy inequality) Let p ≥ 2.Suppose that φ(r) ∈ Lp

T. Then the integral It (φ) = ∫ t

0 (φ(r), dWr) is p-thintegrable. There exists a positive constant Cp such that for any 0 < t ≤ T , theinequality

E[|It (φ)|p]1p ≤ Cpt

12− 1

p ‖φ‖p (2.9)

holds for all φ ∈ Lp

T.

Proof We shall apply Itô’s formula to the function f (x, t) = |x|p and Xt = Mt :=∫ t

0 (φ(r), dWr). Then,


|Mt |p = p

∫ t

0|Mr |p−1sign(Mr) dMr + 1

2p(p − 1)

∫ t

0|Mr |p−2 d〈M〉r ,

where sign(Mr) is equal to 1 if Mr ≥ 0 and is equal to −1 if Mr < 0. The first termof the right-hand side is a martingale with mean 0. Therefore, taking the expectationfor each term, we have

E[|Mt |p] = 1

2p(p − 1)E

[ ∫ t

0|Mr |p−2 d〈M〉r

]

≤ 1

2p(p − 1)E

[supr<t

|Mr |p] p−2

pE[〈M〉

p2t

] 2p

≤ 1

2p(p − 1)qp−2E[|Mt |p]1−

2p E[〈M〉

p2t ]

2p .

Here we used Hölder’s inequality and Doob’s inequality for martingale. Thereforethere exists a positive constant Cp such that

E[|Mt |p] ≤ CpE[〈M〉p2t ], ∀0 < t < T .

Since 〈M〉t = ∫ t

0 |φ(r)|2 dr , we have E[〈M〉p2t ] ≤ E[∫ t

0 |φ(r)|p dr]t p2 −1 by

Hölder’s inequality. Therefore we get the assertion of the proposition. ��Set L∞− = ⋂

p≥2 Lp and L∞−

T= ⋂

p≥2 Lp

T. The proposition tells us that if

φ ∈ L∞−T

, then the stochastic integral∫ t

0 (φ, dW) belongs to L∞−.Next, we will consider exponential functionals of stochastic integrals given by

Zt = exp{ ∫ t

0(φ(r), dWr)− 1

2

∫ t

0|φ(r)|2 dr

}, t ∈ T, (2.10)

where φ ∈ LT. Apply Itô’s formula to the exponential function f (x, t) = ex . Thenwe get Zt = 1 + ∫ t

0 Zr dMr, where Mt is a local martingale written as Mt =∫ t

0 (φ(r), dWr). Therefore, Zt , t ∈ T is a positive local martingale. We will showthat Zt , t ∈ T is a super-martingale. Set τn = inf{0 < t < T ; Zt > n} (= ∞if {· · · } is empty). Then the stopped process Zt∧τn , t ∈ T is a positive martingale.Therefore the equality E[Zt∧τn |Fr ] = Zs∧τn holds for any s < t and n, a.s. Then,using Fatou’s lemma for conditional measure P(·|Fr ) (Proposition 1.5.2), we have

E[Zt |Fs] ≤ lim infn→∞ E[Zt∧τn |Fs] = lim inf

n→∞ Zs∧τn = Zs. (2.11)

Therefore Zt , t ∈ T is a super-martingale.We give a sufficient condition that Zt , t ∈ T is an Lp-martingale.


Proposition 2.2.3 Let p > 1. If exp∫ T

0 |φ(r)|2 dr is in L2p2−p, then Zt , t ∈ T

defined by (2.10) is an Lp-martingale. Further, if exp∫ T

0 |φ(r)|2 dr belongs toL∞−, Zt , t ∈ T is an L∞−-martingale.

Proof We shall rewrite Zpt as

Zpt = exp

{pMt − p2〈M〉t

}exp

{1

2(2p2 − p)〈M〉t

}. (2.12)

By the Schwartz inequality, E[Zpt ] is dominated by

E[(

exp{pMt − p2〈M〉t

})2] 12E[(

exp{1

2(2p2 − p)〈M〉t

})2] 12.

Since

(exp

{pMt − p2〈M〉t

})2 = exp{

2pMt − 1

2(2p)2〈M〉t

},

it is a positive local martingale and is a positive super-martingale, whose mean isless than or equal to 1. Therefore we have

E[Zpt ] ≤ E

[exp

{(2p2−p)〈M〉t

}] 12 =E

[exp

{(2p2−p)

∫ t

0|φ(r)|2 dr

}] 12< ∞.

It remains to show that Zt , t ∈ T is a martingale. Let {τn} be the sequence ofstopping times defined above. By a similar argument, we have

E[Zpt∧τn

] ≤ E[

exp{(2p2 − p)

∫ t∧τn

0|φ(r)|2 dr

}] 12.

Therefore supn E[Zpt∧τn

] < ∞. Consequently, the family of random variables{Zt∧τn , n = 1, 2, . . .} is uniformly integrable. Then the sequence Zt∧τn convergesto Zt in L1. Therefore, the inequality (2.11) can be replaced by the equality. Thismeans that Zt , t ∈ T is a martingale. The second assertion of the proposition isimmediate from the first assertion. ��

Now, suppose Zt , t ∈ T of (2.10) is a positive martingale. Define

Q(B) = E[ZT 1B ], ∀B ∈ FT . (2.13)

Then Q is a probability measure on (Ω,FT ). We denote it by ZT · P . SinceQ is equivalent (mutually absolutely continuous) to P , stochastic processes on(Ω,FT , P ) can be regarded as stochastic processes on (Ω,FT ,Q).

2.3 Regularity of Stochastic Integrals Relative to Parameters 55

Theorem 2.2.2 (Girsanov’s theorem) Wφt := Wt −

∫ t

0 φ(r) dr, t ∈ T is an {Ft }-Wiener process with respect to Q.

Proof We have the equality

Zt exp{i(v,Wφt −Wφ

s )}

= Zs exp{ ∫ t

s

(φ(r), dWr)− 1

2

∫ t

s

|φ(r)|2 dr + i(v,Wt −Ws)−∫ t

s

φ(r) dr}

= Zs exp{ ∫ t

s

((φ + iv), dWr)− 1

2

∫ t

s

|φ + iv|2 dr}

exp{− 1

2|v|2(t − s)

}.

Take A ∈ Fs and multiply both sides of the above by 1A and then takeexpectations. Since E[exp{∫ t

s(φ + iv) dWr − 1

2

∫ t

s|φ + iv|2 dr}|Fs] = 1 holds

by Proposition 2.2.3, we get for any A ∈ Fs ,

E[Zt exp{i(v,Wφt −Wφ

s )}1A] = exp{− 1

2|v|2(t − s)

}E[Zs1A].

Rewriting the above using the measure dQ = ZT dP , we get

EQ[exp{i(v,Wφt −Wφ

s )}1A] = exp{− 1

2|v|2(t − s)

}Q(A).

Therefore, Wφt is an {Ft }-Wiener process with respect to Q. ��

2.3 Regularity of Stochastic Integrals Relative to Parameters

In this section, we consider the continuity and the differentiability of stochasticintegrals with respect to parameters. We assume that the parameter set Λ is equal toR

e or a unit ball Be1 = {λ ∈ R

e; |λ| < 1}. We denote the partial derivative ∂∂λi

f

by ∂λi f . We set ∂λf = (∂λ1f, . . . , ∂λef ). For multi-index of nonnegative integersi = (i1, . . . , ie) we set ∂ i

λf = ∂i1λ1· · · ∂ieλef .

Let p > e ∨ 2. Let φλ(r), r ∈ T, λ ∈ Λ be a d ′-dimensional measurable randomfield belonging to LT(Λ). If it satisfies

supλ

E[ ∫ T

0|φλ(r)|p dr

]< ∞,

E[ ∫ T

0|φλ(r)− φλ′(r)|p dr

]≤ cp|λ− λ′|p (2.14)


for any λ, λ′ ∈ Λ with a positive constant cp, then φλ(r) is said to belong to

the space LLip,pT

(Λ) . Further, φλ(r) is said to belong to the space Ln+Lip,pT

(Λ),if it is n-times continuously differentiable with respect to λ for any s a.s. and forany |i| ≤ n, derivatives φ′

λ(r) = ∂ iλφλ(r) satisfy (2.14). If φλ(r) ∈ LLip,p

T(Λ),

the stochastic integral∫ t

0 (φλ(r), dWr) is well defined a.s. for each fixed λ. We areinterested in the problem of finding a modification of a family of random variables{∫ t

0 (φλ(s), dWs), λ ∈ Λ} which is continuous in (t, λ) a.s. and differentiable withrespect to λ a.s.

Proposition 2.3.1

1. Suppose that {φλ(r)} is a random field belonging to the class LLip,pT

(Λ) wherep > e ∨ 2. Then the family of Itô’s stochastic integrals {∫ t

0 (φλ(r), dWr)} has amodification which is continuous in (t, λ).

2. Suppose further that {φλ(r)} belongs to the class Ln+Lip,pT

(Λ) for some positiveinteger n and p > e ∨ 2. Then the above family of stochastic integrals has amodification, which is n-times continuously differentiable with respect to λ a.s.and the derivatives are continuous in (t, λ) a.s. Further, for any |i| ≤ n, we have

∂ iλ

∫ t

0(φλ(r), dWr) =

∫ t

0(∂ i

λφλ(r), dWr), ∀(t, λ) a.s. (2.15)

Proof

1. For a real continuous function f (t), t ∈ [0, T ], we define its norm by ‖f ‖ =sup0≤t≤T |f (t)|. Set Xλ

t = ∫ t

0 (φλ(r), dWr), t ∈ T. Since it is a martingale, wehave by Doob’s inequality and the Burkholder–Davis–Gundy inequality

E[‖Xλ −Xλ′ ‖p

]≤ cE

[ ∫ T

0|φλ(r)− φλ′(r)|p dr

]≤ c′|λ− λ′|.

Apply the Kolmogorov–Totoki theorem (Theorem 1.8.1). Then we find that Xλt

has a modification which is continuous in λ with respect to the norm ‖ ‖. Thenthe modification is continuous in (t, λ) a.s.

2. Suppose φλ ∈ L1+Lip,pT

(Λ). For ε ∈ Θ = (−1, 0) ∪ (0, 1) and a positive integer

1 ≤ i ≤ e, we set Yλ,εt = 1

ε(X

λ+εit − Xλ

t ), where εi = εei and ei, i = 1, . . . , eare unit vectors in R

e. It is a stochastic process with parameter (λ, ε) ∈ Λ×Θ .It holds by the mean value theorem that

Yλ,εt =

∫ t

0

(φλ+εi (r)− φλ(r)

ε, dWr

)

=∫ t

0

( ∫ 1

0∂λi φλ+θεi (r) dθ, dWr

), (2.16)

2.3 Regularity of Stochastic Integrals Relative to Parameters 57

where |θ | ≤ 1. Therefore

E[‖Yλ,ε − Yλ′,ε′ ‖p] ≤ cE[ ∫ T

0

( ∫ 1

0|∂λi φλ+θεi (r)− ∂λi φλ′+θε′i (r)| dθ

)p

dr]

≤ C{|λ− λ′|p + |ε − ε′|p}. (2.17)

Consequently, Yλ,εt has a modification which is continuous in t, λ, ε and is

continuously extended at ε = 0 ∈ Θ , by the Kolmogorov–Totoki theorem. Thismeans that Xλ

t is continuously differentiable with respect to λ for any t and thederivative ∂λiX

λt coincides with Y

λ,0t . It is continuous in t, λ a.s.

We have as ε → 0,

E[ ∫ T

0

∣∣∣∫ 1

0∂λi φλ+θεi (r) dθ − ∂λi φλ(r)

∣∣∣p

dr]→ 0.

Then we get the equality ∂λi∫ t

0 (φλ(r), dWr) =∫ t

0 (∂λi φλ(r), dWr) from (2.16).

Finally, if φλ(r) is of the class Ln+Lip,pT

(Λ), we can repeat the above argumentinductively. Then we find that the stochastic integral is n-times differentiable withrespect to the parameter and we get the equality (2.15) for any |i| ≤ n. ��For one dimensional predictable processes with parameter Λ, spaces LT(Λ),

LLip,pT

(Λ), Ln+Lip,pT

(Λ) are defined similarly. When the parameter is a spatialparameter x, we will often write φx(r) as φ(x, r).

Let (f k(x, r), k = 1, . . . , d ′) be a d ′-dimensional predictable process withspatial parameter x, belonging to the space L2+Lip,p

T(Rd) and let f 0(x, r) be a

predictable process with spatial parameter x belonging to L2+Lip,pT

(Rd). We set

F(x, t) = F(x, 0)+d ′∑

k=0

∫ t

0f k(x, r) dWk

r , ∀(x, t) a.s., (2.18)

where F(x, 0) is a C2-function, W 0t = t and

∫ t

0 f 0(x, r) dW 0r = ∫ t

0 f 0(x, r) dr .Then F(x, t) has a modification of C2,0-function of x, t by Proposition 2.3.1.

We shall obtain a differential rule for the composite of the above stochasticprocess F(x, t) with spatial parameter x and a continuous semi-martingale Xt .The equation (2.19) below is called a generalized Itô’s formula or the Itô–Wentzellformula.

Theorem 2.3.1 Let Xt be a d-dimensional continuous semi-martingale. Then thecomposite F(Xt , t) is a continuous semi-martingale. Further, for any s < t , wehave


F(Xt , t) = F(Xs, s)+d ′∑

k=0

∫ t

s

f k(Xr, r) dWkr

+d∑

i=1

∫ t

s

∂F

∂xi(Xr, r) dX

ir +

d∑

i=1

d ′∑

k=1

∫ t

s

∂f k

∂xi(Xr, r) d〈Wk,Xi〉r

+1

2

d∑

i,j=1

∫ t

s

∂2F

∂xi∂xj(Xr, r) d〈Xi,Xj 〉r . (2.19)

Proof It suffices to prove (2.19) in the case s = 0. Let Π = {t0 < t1 < · · · < tn} bea partition of [0, t]. Then we have

F(Xt , t)−F(X0, 0) =n∑

m=1

{F(Xtm−1 , tm)−F(Xtm−1 , tm−1)

}

+n∑

m=1

{F(Xtm, tm)−F(Xtm−1 , tm)

}

almost surely. Let |Π | → 0. Then we have

n∑

m=1

{F(Xtm−1 , tm)− F(Xtm−1 , tm−1)

}=

d ′∑

k=0

{ n∑

m=1

∫ tm

tm−1

f k(Xtm−1 , r) dWkr

}

→d ′∑

k=0

∫ t

0f k(Xr, r) dW

kr .

Further, by the Taylor expansion of F(tm, x), we have

n∑

m=1

{F(Xtm, tm)− F(Xtm−1 , tm)

}

=d∑

i=1

{ n∑

m=1

∂F

∂xi(Xtm−1 , tm−1)(X

itm−Xi

tm−1)}

+d∑

i=1

{ n∑

m=1

( ∂F

∂xi(Xtm−1 , tm)−

∂F

∂xi(Xtm−1 , tm−1)

)(Xi

tm−Xi

tm−1)}

+ 1

2

d∑

i,j=1

{ n∑

m=1

∂2F

∂xi∂xj(ξm, tm)(X

itm−Xi

tm−1)(X

jtm−X

jtm−1

)}

= I1 + I2 + I3,

2.4 Fisk–Stratonovitch Symmetric Integrals 59

where ξm are random variables such that |ξm − Xtm−1 | ≤ |Xtm − Xtm−1 |. Let |Π |tend to 0. Then we get

lim|Π |→0I1 =

d∑

i=1

∫ t

0

∂F

∂xi(Xr, r) dX

ir

immediately from the definition of the stochastic integral. Further, noting Theo-rem 1.6.2, we get

lim|Π |→0I2 =

d∑

i=1

d ′∑

k=0

lim|Π |→0

{ n∑

m=1

( ∫ tm

tm−1

∂f k

∂xi(Xtm−1 , r) dW

kr

)(Xi

tm−Xi

tm−1)}

=d∑

i=1

d ′∑

k=1

∫ t

0

∂f k

∂xi(Xr, r) d〈Wk,Xi〉r , a.s.

lim|Π |→0I3 = 1

2

d∑

i,j=1

∫ t

0

∂2F

∂xi∂xj(Xr, r) d〈Xi,Xj 〉r , a.s.

Summing up these computations, we get the formula of the theorem. ��

2.4 Fisk–Stratonovitch Symmetric Integrals

We will define the Fisk–Stratonovitch symmetric integral or simply the symmetricintegral of φ(r) by a continuous semi-martingale Xt, t ∈ T by

∫ t

0φ(r) ◦ dXr = lim|Π |→0

n∑

m=1

1

2

{φ(t ′m)+ φ(t ′m−1)

}(Xt ′m −Xt ′m−1

), (2.20)

if the right-hand side exists. Here Π = {t0 < · · · < tn} are partitions of [0, T ] andt ′m = tm ∧ t . It holds that

∑

m

1

2

{φ(t ′m)+φ(t ′m−1)

}(Xt ′m −Xt ′m−1

) =∑

m

φ(t ′m−1)(Xt ′m −Xt ′m−1)+ 1

2〈φ,X〉Πt ,

where 〈φ,X〉Πt is the quadratic covariation of processes φ and X associated withthe partition Π . If φ(t) is a semi-martingale with bounded jumps, then 〈φ,X〉Πt , t ∈[0, T ] converges to a continuous process of bounded variation, which we denote by〈φ,X〉t (see Sect. 1.6). Then we have:


Proposition 2.4.1 Let {φ(r)} be a semi-martingale with bounded jumps. Then, thesymmetric integral

∫ t

0 φ(r) ◦ dXr is well defined for any t ∈ T. Further, we have

∫ t

0φ(r) ◦ dXr =

∫ t

0φ(r) dXr + 1

2〈φ,X〉t , a.s., (2.21)

where∫ t

0 φ(r) dXr is the Itô integral.

For 0 ≤ s < t ≤ T , we define the symmetric integral by

∫ t

s

φ(r) ◦ dXr =∫ t

0φ(r) ◦ dXr −

∫ s

0φ(r) ◦ dXr .

Then it holds that∫ t

s

φ(r) ◦ dXr =∫ t

s

φ(r) ◦ dXt + 1

2(〈φ,X〉t − 〈φ,X〉s).

Remark It is known that the symmetric integral is well defined if the martingale partof φ(r) is a local L2-martingale. In such a case, the quadratic covariation 〈φ,X〉texists and the equality (2.21) holds. For the proof, we need additional arguments forthe quadratic covariation. See Protter [96].

A feature of the symmetric integral is that it works similarly as the usualdifferential calculus. Let Xt = (X1

t , . . . , Xdt ), t ∈ T be a d-dimensional continuous

semi-martingale represented by

Xit =

d ′∑

k=1

∫ t

0φik(r) ◦ dWk

r +∫ t

0υi(r) dr. (2.22)

Here, (φik(r)) is a d ′-dimensional continuous semi-martingale written as

φik(r) = φik(0)+d ′∑

l=1

∫ r

0φikl

1 (s) dWls +

∫ r

0φik

2 (s) ds,

where φikl1 (r) ∈ LT, and φik

2 (r) ∈ LT. Then, using Itô integral, Xt is written as

Xit =

d ′∑

k=1

∫ t

0φik(r) dWk

r +∫ t

0υi(r) dr + 1

2

∫ t

0φi

1(r) dr,

where φi1(r) = ∑d ′

k=1 φikk1 (r), i = 1, . . . , d. Set φ(r) = (φij (r)) and υ(r) =

(υi(r)). Using vector and matrix notations, we rewrite (2.22) as

Xt =∫ t

0φ(r) ◦ dWr +

∫ t

0υ(r) dr.


We shall reconsider the differential rule of the composite functional f (Xt , t),when Xt, t ∈ T is a semi-martingale represented by the symmetric integral. It willbe seen that the differential rule for the symmetric integral is similar to the usualdifferential rule.

Theorem 2.4.1 Let {Xt } be a d-dimensional continuous semi-martingale repre-sented by (2.22). Let f (x, t) be a C3,1-function on R

d × T. Then we have for any0 ≤ s < t ≤ T ,

f (Xt , t) = f (Xs, s)+∫ t

s

∂f

∂t(Xr, r) dr (2.23)

+d∑

i=1

d ′∑

k=1

∫ t

s

∂f

∂xi(Xr, r)φ

ik(r) ◦ dWkr +

d∑

i=1

∫ t

s

∂f

∂xi(Xr, r)υ

i(r) dr.

Proof It is sufficient to prove the case s = 0. We shall rewrite the term expressed bysymmetric integral, using Itô integral. If f is a C1,3-function, Theorem 2.2.1 tellsus that ∂f

∂xi(Xt , t) is a continuous semi-martingale and it satisfies

∂f

∂xi(Xt , t) = ∂f

∂xi(X0, 0)+

d∑

j=1

∫ t

0

∂2f

∂xi∂xj(Xr, r) dX

jr + Ct ,

where Ct is a continuous process of bounded variation. Then the martingale part ofthe semi-martingale ∂f

∂xi(Xt , t)φ

ik(t) is

d∑

j=1

d ′∑

l=1

∫ t

0

∂2f

∂xi∂xj(Xr, r)φ

ik(r)φjl(r) dWlr +

d ′∑

l=1

∫ t

0

∂f

∂xi(Xr, r)φ

ikl1 (r) dWl

r .

Therefore

⟨ ∂f∂xi

(Xt , t)φik(t),Wk

t

⟩

t

=d∑

j=1

∫ t

0

∂2f

∂xi∂xj(Xr, r)φ

ik(r)φjk(r) dr +∫ t

0

∂f

∂xi(Xr, r)φ

ikk1 (r) dr.

Then∑d ′

k=1

∫ t

0∂f∂xi

(Xr, r)φik(r) ◦ dWk

r is equal to

d ′∑

k=1

∫ t

0

∂f

∂xi(Xr, r)φ

ik(r) dWkr+

1

2

d∑

j=1

∫ t

0

∂2f

∂xi∂xj(Xr, r)

( d ′∑

k=1

φik(r)φjk(r))dr

+ 1

2

∫ t

0

∂f

∂xi(Xr, r)φ

i1(r) dr.


Therefore the right-hand side of (2.23) is equal to the right-hand side of (2.8) if wereplace υ(r) in (2.8) by υ(r)+ 1

2 φ1(r). Therefore the formula (2.23) holds. ��Let Λ be the e-dimensional parameter set. Let {φλ(r), λ ∈ Λ, t ∈ T} be a

measurable random field such that for any λ, φλ(t) is a continuous semi-martingale.We assume

∫

Λ

∫

T

(|φλ(r)|2 + |〈φλ(r),Xr 〉|

)dλ dr < ∞. (2.24)

We give a Fubini theorem for the change of the order of the symmetric integrals◦dWr and dλ.

Proposition 2.4.2 Let {φλ(r), λ ∈ Λ, r ∈ T} be a measurable random field suchthat for any λ, φλ(r), r ∈ T is a continuous semi-martingales satisfying (2.24). Thenwe have for any t ∈ T

∫

Λ

( ∫ t

0φλ(r) ◦ dWr

)g(λ) dλ =

∫ t

0

( ∫

Λ

φλ(r)g(λ) dλ)◦ dWr, a.s. (2.25)

for any bounded measurable function g(λ) with compact supports.

The proof is straightforward from Proposition 2.1.1.We shall study the regularity of the symmetric integrals with respect to parame-

ters.

Proposition 2.4.3 Let φλ(r) = (φ1λ(r), . . . , φ

d ′λ (r)), λ ∈ Λ, r ∈ T be a d ′-

dimensional measurable random field such that for any λ, φλ(r) is a continuoussemi-martingale written as

φλ(r) =∫ r

0φ1,λ(s) dWs +

∫ r

0φ2,λ(s) ds, (2.26)

where φiλ(r), φ

i1,λ(r) ∈ LLip,p

T(Λ) for i = 1, . . . , d for some p > e ∨ 2.

1. Then the symmetric integral∫ t

0 φλ(r) ◦ dWr has a modification which iscontinuous in (t, λ).

2. Assume further that φiλ(r), φ

i1,λ(r), i = 1, . . . , d belong to Ln+Lip,p

T(Λ) for

some positive integer n and p > e ∨ 2. Then the symmetric integral is n-timescontinuously differentiable with respect to λ a.s. and the derivative is continuousin (t, λ) a.s. Further, we have for any |i| ≤ n,

∂ iλ

∫ t

0φλ(r) ◦ dWr =

∫ t

0∂ iλφλ(r) ◦ dWr, ∀(t, λ) a.s. (2.27)


Proof

1. The symmetric stochastic integral∫ t

0 φλ(r) ◦ dWr is equal to the Itô integral∫ t

0 φλ(r) dWr+ 12

∫ t

0 φ1,λ(r) dr. If φλ(r) satisfies (2.14), the first term of the right-hand side has a modification which is continuous in (t, λ) by Propositions 2.3.1.The second term of the right-hand side is continuous in (t, λ) obviously.Therefore the symmetric integral has a modification which is continuous in (t, λ).

2. Under the assumption, integrals∫ t

0 φλ(r) dWr and 12

∫ t

0 φ1,λ(r) dr are continu-ously differentiable with respect to λ and we can change the order of ∂λ and

∫,

in view of Proposition 2.3.1. Then we get the formula (2.27). ��We shall rewrite the generalized Itô’s formula (Theorem 2.3.1) in the previous

section using symmetric integrals. We will assume slightly stronger conditionsfor f k(x, t). Let f k(x, t), k = 1, . . . , d ′ be predictable processes with spatialparameter x represented by f k(x, t) = ∫ t

0 (fk1 (x, r), dWr)+

∫ t

0 f k2 (x, r) dr, where

f k1 (x, r) ∈ L3+Lip,p

T(Rd) and f k

2 (x, r) ∈ L3+Lip,pT

(Rd) for p > 2 ∨ d. Let

f 0(x, r) ∈ L3+Lip,pT

(Rd). Let F(x, t), t ∈ T be a stochastic process with a spatialparameter x ∈ R

d defined by

F(x, t) = F(x, 0)+d ′∑

k=0

∫ t

0f k(x, r) ◦ dWk

r , (2.28)

where F(x, 0) is a C3-function of x. Then F(x, t) is a C3,0-function of x, t a.s. andit is a continuous semi-martingale for any x by Proposition 2.4.3.

Theorem 2.4.2 For any d-dimensional continuous semi-martingale Xt, t ∈ T, wehave

F(Xt , t) = F(Xs, s) (2.29)

+d ′∑

k=0

∫ t

s

f k(Xr, r) ◦ dWkr +

d∑

i=1

∫ t

s

∂F

∂xi(Xr, r) ◦ dXi

r ,

for any 0 ≤ s < t ≤ T .

Proof It is sufficient to prove it in the case s = 0. Since f k(Xt , t) is a continuoussemi-martingale, we have, similarly to the proof of Theorem 2.4.1,

∫ t

0f k(Xr, r) ◦ dWk

r =∫ t

0f k(Xr, r) dW

kr + 1

2

d∑

i=1

∫ t

0

∂f k

∂xi(Xr, r) d〈Wk,Xi〉r ,

if k = 1, . . . , d ′. Next, ∂F∂xi

(Xt , t) is a continuous semi-martingale represented by


∂F

∂xi(Xt , t)− ∂F

∂xi(X0, 0)

=d ′∑

k=0

∫ t

0

∂f k

∂xi(Xr, r) dW

kr +

d∑

j=1

∫ t

0

∂2F

∂xi∂xj(Xr, r) dX

jr + Ac

t ,

where Act is a continuous process of bounded variation, in view of Theorem 2.3.1.

Therefore,

∫ t

0

∂F

∂xi(Xr, r) ◦ dXi

r=∫ t

0

∂F

∂xi(Xr, r) dX

ir +

1

2

d ′∑

k=1

∫ t

0

∂f k

∂xi(Xr, r) d〈Wk,Xi〉r

+ 1

2

d∑

j=1

∫ t

0

∂2F

∂xi∂xj(Xr, r) d〈Xj ,Xi〉r .

Then (2.19) implies (2.29). ��

2.5 Stochastic Integrals with Respect to Poisson RandomMeasure

We will define stochastic integrals based on Poisson random measures and com-pensated Poisson random measures. Let N(du) ≡ N(dr dz) be a Poisson randommeasure on U = T × R

d ′0 with intensity n(du) = n(dr dz) = dr ν(dz), where ν

is a Lévy measure. Here, elements of T are denoted by r and elements of Rd ′0 are

denoted by z = (z1, . . . , zd′). We define the compensated Poisson random measure

N by

N(du) = N(du)− n(du). (2.30)

Let {Ft , t ∈ T} be a filtration such that for any 0 ≤ s < t < t ′ ≤ T , N((s, t] × E)

is {Ft }-adapted and N((t, t ′] × E) are independent of Ft . Then N(du) is called an{Ft }-Poisson random measure. Let ψ(u) = ψ(r, z) be a measurable random fieldwith parameter u ∈ U. It is called predictable, if for any z, ψ(r, z) is a predictableprocess with respect to the filtration {Ft }. We set

L2U=

{ψ(u); predictable and E

[ ∫

U

|ψ(u)|2n(du)]< ∞

}. (2.31)

For 0 < t ≤ T , we set Ut = {u = (r, z) ∈ U; 0 ≤ r ≤ t, z ∈ Rd ′0 }. It is a

subdomain of U. We will define the stochastic integral∫Ut

ψ(u)N(du) for ψ ∈ L2U

.

2.5 Stochastic Integrals with Respect to Poisson Random Measure 65

We first consider the integral for a simple predictable random field ψ(r, z) written as

ψ(r, z) =n∑

m=1

ψm(z)1(tm−1,tm](r),

where 0 = t0 < · · · < tn = T and ψm(z) are Ftm−1×B(Rd ′0 )-measurable functionals

such that E[∫ (|ψm(z)| + |ψm(z)|2)ν(dz)] < ∞. We define

Mt :=∫

Ut

ψ(u)N(du) =n∑

m=1

∫

Rd′0

ψm(z)N((t ′m−1, t′m], dz), (2.32)

where t ′m = tm ∧ t and N((s, t], A) = N((s, t] × A) − (t − s)ν(A) are signedmeasures on R

d ′0 for almost all ω.

Lemma 2.5.1 Mt, t ∈ T defined by (2.32) is an L2-martingale. Further,

M2t −

∫

Ut

|ψ(u)|2n(du), t ∈ T (2.33)

is also a martingale.

Proof We denote∫ψm(z)N((tm−1, tm], dz) by Zm. We want to show that for any

m

E[Zm|Ftm−1] = 0, (2.34)

E[Z2m|Ftm−1] = (tm − tm−1)E

[ ∫ψm(z)

2ν(dz)

∣∣∣Ftm−1

].

Suppose that ψm(z) is a deterministic step function ψm(z) = ∑i ci1Ei

(z), whereci are constants and Ei are disjoint subsets of Rd ′

0 with ν(Ei) < ∞. Since Zm andFtm−1 are independent, we have

E[Zm|Ftm−1] = E[Zm] =∑

i

ciE[N((tm−1, tm] × Ei)] = 0,

E[Z2m|Ftm−1] = E[Z2

m] =∑

i,j

cicjE[N((tm−1, tm] × Ei)N((tm−1, tm] × Ej)].

Since the law of N((tm−1, tm] × Ei) is Poisson with parameter (tm − tm−1)ν(Ei),its variance E[N((tm−1, tm], Ei)

2] is equal to (tm − tm−1)ν(Ei). Further, if i �= j ,N((tm−1, tm], Ei) and N((tm−1, tm], Ej ) are independent, so that the expectation ofthe product of these two random variables is equal to 0. Consequently, we get


E[Z2m|Ftm−1] = (tm − tm−1)

∑

i

c2i ν(Ei) = (tm − tm−1)

∫

Rd′0

ψm(z)2ν(dz).

Then, equalities (2.34) hold for any deterministic function ψm(z) satisfying∫(|ψm(z)| + |ψm(z)|2)ν(dz) < ∞. Next, if ψm(z) is Ftm−1 -measurable, we may

regard that it is a deterministic function under the conditional measure P(·|Ftm−1)

a.s. Therefore equalities (2.34) hold a.s.Further, if m > l, E[ZmZl |Ftl−1] = E[E[Zm|Ftm−1]Zl |Ftl−1] = 0. Then, since

Mt = ∑m;tm≤t Zm, Mt −Ms satisfies

E[Mt −Ms

∣∣∣Fs

]= E

[ ∑

m;s<tm≤t

Zm

∣∣∣Fs

]= 0, a.s., (2.35)

E[(Mt −Ms)

2|Fs

]= E

[( ∑

m;s<tm≤t

Zm

)2∣∣∣Fs

]

= E[ ∫ t

s

∫

Rd′0

∣∣∣ψ(u)|2n(du)∣∣∣Fs

], a.s. (2.36)

Equation (2.35) shows that Mt is a martingale and (2.36) shows that M2t is integrable

and M2t −

∫Ut|ψ(u)|2n(du) is a martingale. ��

For ψ(u) ∈ L2U

, there exists a sequence ψn(u) of simple predictable random

fields such that E[ ∫ |ψ(u) − ψn(u)|2n(du)

]→ 0. Set Mn

t = ∫Ut

ψn(u)N(du).

Then we have by Doob’s inequality and (2.36)

E[ sup0≤t≤T

|Mnt −Mn′

t |2] ≤ 4E[|MnT −Mn′

T |2]

≤ 4E[ ∫

U

|ψn(u)− ψn′(u)|2n(du)]→ 0,

as n, n′ → ∞. Therefore Mnt converges to an L2-martingale Mt uniformly in

0 < t < T in L2-norm. We denote it by∫Ut

ψ(u)N(du). It is a right continuous

L2-martingale. It satisfies (2.35) and (2.36). The equality (2.35) shows that Mt =∫Ut

ψ(u)N(du) is an L2- martingale and (2.36) shows that M2t −

∫Ut|ψ(u)|2n(du)

is an L1-martingale. For 0 < s < t ≤ T and a Borel subset B of Rd ′0 , we define the

stochastic integral based on the compensated Poisson random measure by

∫ t

s

∫

B

ψ(r, z)N(dr dz) :=∫

Ut

ψ1T×BN(du)−∫

Us

ψ1T×BN(du).

Set

2.6 Jump Processes and Related Calculus 67

LU ={ψ(u); predictable and

∫

U

|ψ(u)|2n(du) < ∞, a.s.}. (2.37)

We can extend the stochastic integral for ψ belonging to LU. In fact, if ψ ∈ LU, thestochastic integral

∫ t

s

∫R

d′0ψ(r, z)N(dr dz) can be defined as a local L2-martingale.

Since the discussion is similar to that for continuous martingales, it is omitted.The integral is called the stochastic integral based on compensated Poisson randommeasure N . We have the following.

Theorem 2.5.1 For any ψ ∈ LU, Mt :=∫ t

0

∫R

d′0ψ(r, z)N(dr dz), t ∈ T is defined

as a local L2-martingale. Further, M2t − ∫ t

0

∫R

d′0ψ(r, z)2n(dr dz), t ∈ T is a local

martingale.

2.6 Jump Processes and Related Calculus

Suppose that the Wiener process Wt = (W 1t , . . . ,W

d ′t ), t ∈ T and Poisson random

measure N(dr dz), (r, z) ∈ T × Rd ′0 are defined on the same probability space

(Ω,F , P ) with the filtration {Ft }, are independent of each other, and that Wt isan {Ft }-Wiener process and N(dr dz) is an {Ft }-Poisson random measure. Thenfunctional spaces of predictable processes LT, LU are defined on the probabilityspace.

Let D be a Borel subset of Rd ′0 including the set {z ∈ R

d ′0 ; |z| ≤ 1}. Let ψ(u) be a

predictable random field such that |ψ(u)| < ∞ a.e. n(du)P (dω) and ψ(u)1D(z) ∈LU. We define the integral of ψ by ND(du) by

∫ t

s

∫

B

ψ(r, z)ND(dr dz) =∫ t

s

∫

D∩Bψ(r, z)N(dr dz)+

∫ t

s

∫

Dc∩Bψ(r, z)N(dr dz),

where the last integral by dN is the Lebesgue–Stieltjes integral. Let Xt =(X1

t , . . . , Xdt ), t ∈ T be a d-dimensional cadlag process represented by

Xit = Xi

0+∫ t

0(φi(r), dWr)+

∫ t

0υi(r) dr+

∫ t

0

∫

Rd′0

ψi(r, z)ND(dr dz). (2.38)

Here φi ∈ LT, υi ∈ LT and ψi(·, z)1D(z) ∈ LU. We call it an Itô process. Thesecond, the third and the fourth term of the right-hand side of (2.38) are called,the diffusion part, the drift part and the jump part of the Itô process Xt . AnyLévy process is an Itô process, since a Lévy process has a Lévy–Itô decomposition(Theorem 1.4.2).

Let Xt be an Itô process represented above. Let ϕ(r) = (ϕij (r)) be a predictableprocess such that components of ϕ(r)φ(r), ϕ(r)υ(r), ϕ(r)ψ(r, z)1D(z) belong toLT, LT,LU, respectively. Then the stochastic integral


∫ t

0ϕ(r) dXr :=

∫ t

0ϕ(r)φ(r) dWr+

∫ t

0ϕ(r)υ(r) dr+

∫ t

0

∫

Rd′0

ϕ(r)ψ(r, z)ND(dr dz)

is well defined. It is again an Itô process.We give the differential rule for the above Itô process (Itô’s formula).

Theorem 2.6.1 ([66]) Let f (x1, . . . , xd , t) be a function of C2,1-class. Let Xt bean Itô process given by (2.38). Then, f (Xt , t), t ∈ [0, T ] is an Itô process andsatisfies

f (Xt , t) = f (Xs, s)+∫ t

s

∂f

∂t(Xr, r) dr (2.39)

+d∑

i=1

d ′∑

k=1

∫ t

s

∂f

∂xi(Xr−, r)φik(r) dWk

r +d∑

i=1

∫ t

s

∂f

∂xi(Xr, r)υ

i(r) dr

+ 1

2

d∑

i,j=1

∫ t

s

∂2f

∂xi∂xj(Xr, r)

( d ′∑

k=1

φik(r)φjk(r))dr

+∫ t

s

∫

D

{f (Xr+ψ(r, z), r)−f (Xr, r)−

∑

i

∂f

∂xi(Xr, r)ψ

i(r, z)}n(dr dz)

+∫ t

s

∫

Rd′0

{f (Xr− + ψ(r, z), r)− f (Xr−, r)

}ND(dr dz), a.s.

for any 0 ≤ s < t ≤ T , Here Xr− := limε↓0 Xr−ε is a predictable process.

Proof We will prove the formula (2.39) in the case d = 1, d ′ = 1 only. We firstassume that ψ(r, z) satisfies ψ(r, z) = ψ(r, z)1|z|≥ε a.s., where ε > 0. Let Jt =∫ t

0

∫|z|≥ε

zND(dr dz). Its jumping times are defined by τ0 = 0 and for n ≥ 1 byinduction as

τn = inf{t; T > t > τn−1; Jt − Jτn−1 > 0},= ∞, otherwise.

Then Xt has possible jumps at these stopping times. We have

f (Xt , t)− f (X0, 0) =∑

n;τn≤t

{f (Xτn−, τn)− f (Xτn−1 , τn−1)}

+∑

n;τn≤t

{f (Xτn, τn)− f (Xτn−, τn)}

= I1 + I2.


We shall calculate I1. If τn−1 ≤ t , we set

X′t = Xτn−1 +

∫ t

τn−1

φ(r) dWr +∫ t

τn−1

υ ′(r) dr,

where υ ′ = υ − ∫Dψν(dz). Then it holds that Xt = X′

t for τn−1 ≤ t < τn and ithas no jumps for t ≥ τn. We apply Itô’s formula for the continuous Itô process X′

t .We have

f (Xτn−, τn)− f (Xτn−1 , τn−1) =∫ τn

τn−1

∂f

∂t(Xr, r) dr

+∫ τn

τn−1

∂f

∂x(Xr−, r)φ(r) dWr + 1

2

∫ τn

τn−1

∂2f

(∂x)2 (Xr, r)φ(r)2 dr

+∫ τn

τn−1

∂f

∂x(Xr, r)υ(r) dr −

∫ τn

τn−1

∫

D

∂f

∂x(Xr, r)ψ(r, z)n(dr dz).

Summing up these for n such that τn ≤ t , we get

I1 =∫ t

0

∂f

∂t(Xr, r) dr +

∫ t

0

∂f

∂x(Xr−, r)φ(r) dWr + 1

2

∫ t

0

∂2f

(∂x)2(Xr, r)φ(r)

2 dr

+∫ t

0

∂f

∂x(Xr, r)υ(r) dr −

∫ t

0

∫

D

∂f

∂x(Xr, r)ψ(r, z)n(dr dz).

Next, we have

I2 =∑

0<r≤t,ΔXr �=0

{f (Xr, r)− f (Xr−, r)}

=∫ t

0

∫

Rd′0

{f (Xr− + ψ(r, z), r)− f (Xr−, r)

}N(dr dz)

=∫ t

0

∫

Rd′0

{f (Xr− + ψ(r, z), r)− f (Xr−, r)

}ND(dr dz)

+∫ t

0

∫

D

{f (Xr + ψ(r, z), r)− f (Xr, r)

}n(dr dz),

almost surely, since Xr = Xr− holds a.e. dn dP . From these two computations ofI1 and I2, we get (2.39).

We next consider the general case. In the definition of Itô process, we replaceψ(r, z) by ψε(r, z) = ψ(r, z)1|z|>ε . Let Xε

t be the Itô process associated with α, υ

and ψε . Then Xεt has finite jumps, so that Itô’s formula is valid for Xε

t . Further, eachterm of (2.39) converges as ε → 0. Hence we get (2.39) for Xt . ��


We will give Lp-estimations of the integral∫Ut

ψ(u)N(du). We transform theintensity measure n on U to a bounded measure m on U by setting

m(du) = γ (u)2n(du), where γ (u) = |z| ∧ 1 for u = (r, z).

For p ≥ 2, we define

Lp

U=

{ψ ∈ LU; ‖ψ‖p < ∞

}, (2.40)

where ‖ψ‖p = E[ ∫

U

∣∣∣ψ(u)γ (u)

∣∣∣p

m(du)] 1p . In the case p = 2, the above L2

Ucoincides

with L2U

defined by (2.31). We set L∞−U

= ⋂p>2 Lp

U.

Proposition 2.6.1 Let p ≥ 2. Let ψ(u) be an element of Lp

U. Then the stochastic

integral∫Ut

ψ(u)N(du) ≡ ∫ t

0

∫R

d′0ψ(r, z)N(dr dz) is p-th integrable. Further,

there exists a positive constant Cp such that

E[∣∣∣

∫

Ut

ψ(u)N(du)

∣∣∣p] 1

p ≤ Cp‖ψ‖p (2.41)

holds for any 0 < t ≤ T and ψ ∈ Lp

U.

The equation (2.41) is rewritten as

E[∣∣∣

∫ t

0

∫

Rd′ψ(r, z)N(dr dz)

∣∣∣p] ≤ CpE

[ ∫ t

0

∫

Rd′

∣∣∣ψ(r, z)

γ (z)

∣∣∣p

drμ(dz)],

where γ (z) = |z| ∧ 1 and μ(dz) = γ (z)2ν(dz).

Proof We consider the process Yt =∫ t

0

∫ψN(dr dz). We apply the formula (2.39)

for f (x, t) = |x|p and D = Rd ′0 . Then,

|Yt |p =∫ t

0

∫ {|Yr− + ψ |p − |Yr−|p}N(dr dz) (2.42)

+∫ t

0

∫ {|Yr + ψ |p − |Yr |p − p|Yr |p−2Yrψ}n(dr dz).

Denote the first term of the right-hand side by Zt . Then Zt is a local martingale. Wemay choose an increasing sequence of stopping times τn such that P(τn < T ) → 0as n → ∞, Zt∧τn is a martingale with mean 0 and the last term in (2.42) stopped atτn is integrable. To make the notation simple, we denote t ∧ τn by t ′. Since

|Yr + ψ |p − |Yr |p − p|Yr |p−2Yrψ = 1

2p(p − 1)|Yr + θψ |p−2ψ2

≤ c1|Yr |p−2ψ2 + c2|ψ |p


holds for some |θ | < 1, we get from (2.42),

E[|Yt ′ |p] ≤ c1E[ ∫ t ′

0

∫|Yr |p−2ψ2n(dr dz)

]+ c2E

[ ∫ t ′

0

∫|ψ |pn(dr dz)

].

(2.43)We shall compute the first term of the right-hand side. It holds that

∫ t ′

0

∫|Yr |p−2ψ2n(dr dz) ≤ sup

r|Yr |p−2

∫ t ′

0

∫ψ2n(dr dz)

≤ c3 supr

|Yr |p + c4

( ∫ t ′

0

∫ψ2n(dr dz)

) p2,

where c3 = p−2p

λp

p−2 , c4 = 2pλ−

p2 and λ is a positive constant. In the last inequality,

we used the inequality ab ≤ (λa)p′

p′ + (b/λ)q′

q ′ , where a, b > 0, λ > 0 p′, q ′ > 1 and1p′ + 1

q ′ = 1. Therefore,

E[ ∫ t ′

0

∫|Yr |p−2ψ2n(dr dz)

]≤ c3E

[sup

0<r≤t ′|Yr |p

]+ c4E

[( ∫ t ′

0

∫ψ2n(dr dz)

) p2].

Substitute the above to right-hand side of (2.43). Then, using Doob’s inequality,we get

E[

sup0<r≤t ′

|Yr |p]≤ qpc1c3E

[sup

0<r≤t ′|Yr |p

]

+ qpc1c4E[( ∫ t ′

0

∫|ψ |2n(dr dz)

) p2]+ qpc2E

[ ∫ t ′

0

∫|ψ |pn(dr dz)

].

We can choose λ such that qpc1c3 < 1. Then move qpc1c3E[sup0<r≤t ′ |Yr |p] tothe left-hand side. Then we get the inequality

E[

sup0<r≤t ′

|Yr |p]

(2.44)

≤ c5E[( ∫ t ′

0

∫|ψ |2n(dr dz)

) p2]+ c6E

[ ∫ t ′

0

∫|ψ |pn(dr dz)

].

So far we have chosen constants c5, c6 not depending on the form of martingales Yt

and stopping times τn. Therefore the inequality (2.44) holds for any t ′ = t ∧ τn withthe common constants c5, c6. Now let n tend to infinity. Then we find that (2.44)holds for any t .


Further, we have for p ≥ 2,

E[( ∫ t

0

∫|ψ(r, z)|2n(dr dz)

) p2]≤ cE

[ ∫

Ut

∣∣∣ψ(u)

γ (u)

∣∣∣p

m(du)],

E[ ∫ t

0

∫|ψ(r, z)|pn(dr dz)

]≤ E

[ ∫

Ut

∣∣∣ψ(u)

γ (u)

∣∣∣p

m(du)].

Therefore the assertion of the proposition follows. ��The above proposition is an analogue of the Burkholder–Davis–Gundy inequality

(Proposition 2.2.2). The proposition tells us that if ψ ∈ L∞−U

, then the stochasticintegral

∫ t

0 ψN(dr dz) belongs to L∞−.We will study the regularity of stochastic integrals based on Poisson random

measures with respect to parameter λ ∈ Λ, by applying the above proposition. LetΛ be R

e or its unit ball. Let ψλ(r, z), r ∈ T, z ∈ Rd ′ , λ ∈ Λ be a measurable

random field such that for any λ, z, ψλ(r, z) is a predictable process. We assumethat it is continuously differentiable in z and satisfies ψλ(r, 0) = 0. Suppose that forsome p > e ∨ 2, there exists a positive constant c such that

supλ,r

E[ ∫ ∣∣∣

ψλ(r, z)

γ (z)

∣∣∣p

μ(dz)]< ∞, (2.45)

supr

E[ ∫ ∣∣∣

ψλ(r, z)

γ (z)− ψλ′(r, z)

γ (z)

∣∣∣p

μ(dz)]≤ cp|λ− λ′|p (2.46)

for any λ, λ′ ∈ Λ with a positive constant cp. Then ψλ(r, z) is said to belong to

the space LLip,pU

(Λ). Assume further that ψλ(r, z) is n-times differentiable withrespect to λ for any r, z a.e. and the derivatives ψλ(r, z) = ∂ i

λψλ(r, z) satisfy (2.45)and (2.46) for any multi-index i with |i| ≤ n, then ψλ is said to belong to the classLn+Lip,pU

(Λ).

Proposition 2.6.2 If ψλ ∈ LLip,pU

(Λ) for some p > e ∨ 2, then the stochasticintegral

∫ t

0

∫R

d′0ψλ(r, z)N(dr dz) has a modification which is continuous in λ and

cadlag in t . If ψλ ∈ Ln+Lip,pU

(Λ), then the integral has a modification which isn-times continuously differentiable in λ a.s. Further, for any i with |i| ≤ n, we have

∂ iλ

∫ t

0

∫

Rd′0

ψλ(r, z)N(dr dz) =∫ t

0

∫

Rd′0

∂ iλψλ(r, z)N(dr dz), ∀t, a.s. (2.47)

The proof of the above proposition can be carried out similarly to the proof ofProposition 2.3.1, making use of Proposition 2.6.1 instead of the Burkholder–Davis–Gundy inequality.

Note Stochastic integrals by Wiener processes and continuous martingales arediscussed in many books. See Ikeda–Watanabe [41], Karatzas–Shreve [55], Revuz–

2.7 Backward Integrals and Backward Calculus 73

Yor [97], Rogers–Williams [98] and Oksendall [90]. Generalized Itô’s formula orthe Itô–Wentzell formula (Theorem 2.2.1) is taken from Kunita [59].

Stochastic integrals by Poisson random measures are discussed in Ikeda–Watanabe [40], Applebaum [1] and Kunita [60]. Lp-estimates of the integrals givenin Proposition 2.6.2 is an improvement of Lp-estimates given in [1] and [60]. TheseLp-estimates are needed for the study of the stochastic differential equation andthe application of the Malliavin calculus to the solution. These problems will bediscussed in Chaps. 3, 4 and 6.

2.7 Backward Integrals and Backward Calculus

We defined the stochastic integral by a Wiener process in Sect. 2.1. We could regardit as the forward Itô integral. We will define the backward Itô integral. We first showthat a Wiener process is a backward martingale with respect to a two-sided filtration.Let Wt, t ∈ T = [0, T ] be a Wiener process. It is called an {Fs,t }-Wiener process iffor any 0 ≤ s < t ≤ T , Wt−Ws is Fs,t -measurable and the following three σ -fieldsare mutually independent:

F0,s , σ (Wu −Wv; s ≤ u < v ≤ t), Ft,T .

An example of two-sided filtration is given by Fs,t = σ(Wu −Wv; s ≤ u < v ≤ t).Let Wt = (W 1

t , . . . ,Wd ′t ), t ∈ T be an {Fs,t }-Wiener process. We consider a

Backward process Wt = (W 1t , . . . ,W

d ′t ), t ∈ T, where

W kt := Wk

t −WkT , t ∈ T.

Then W kt , t ∈ T, k = 1, . . . , d ′ are backward martingale with respect to the

backward filtration {Ft,T }. i.e., W kt satisfy E[W k

s |Ft,T ] = W kt , for s < t , since

W kt − W k

s = (Wkt −Wk

s ) is independent of Ft,T .A backward predicable σ -field is the σ -field generated by right continuous

{Ft,T }-adapted processes. Let LT be the set of all backward predictable process

φ(r) = (φ1(r), . . . , φd ′(r)), r ∈ T satisfying∫ T

0 |φ(r)|2 dr < ∞, a.s. We can

define backward Itô integrals of φ based on the backward martingales Wt . In fact, ifφk(r), r ∈ T is a continuous process, we define

∫ T

s

φk(r) dWkr : = lim|Π |→0

n∑

m=1

φk(tm−1)(Wktm−1

− W ktm)

= lim|Π |→0

n∑

m=1

φk(tm−1)(Wktm−1

−Wktm),


where Π = {s = tn < · · · < t0 = T } are partitions of the interval [s, T ]. Itis a backward local L2-martingale with respect to s ∈ [0, T ]. The backward Itôintegral coincides with the forward Itô integral

∫ T

sφk(r) dWk

r , provided that φk(r)

is a forward–backward predictable process of bounded variation. The backward Itôintegral can be extended for any φ(r) of LT. It is a continuous backward localmartingale. We define

∫ t

sφk(r) dWk

r = ∫ T

sφk(r) dWk

r − ∫ T

tφk(r) dWk

r . Further,

we set∫ t

s(φ(r), dWr) = ∑d ′

k=1

∫ t

sφk(r) dWk

r .A backward symmetric integral is well defined for a backward semi-martingale

φ(r) with bounded jumps. In fact, the following limit exists:

lim|Π |→0

∑

m

1

2

{φk(tm)+ φk(tm−1)

}(Wk

tm−1−Wk

tm)

= lim|Π |→0

{∑

m

φ(tm−1)(Wktm−1

−Wktm)− 1

2(〈φ,Wk〉ΠT − 〈φ,Wk〉Πs )

}.

We denote it by∫ T

sφk(r)◦dWk

r and call it the backward symmetric integral of φk(r)

by dWkt . For 0 ≤ s < t ≤ T , the backward symmetric integral

∫ t

sφk(r) ◦ dWk

r is

defined by∫ T

sφk(r) ◦ dWk

r − ∫ T

tφk(r) ◦ dWk

r . Then it holds that

∫ t

s

φk(r) ◦ dWkr =

∫ t

s

φ(r) dWkr − 1

2

(〈φ,Wk〉t − 〈φ,Wk〉s

). (2.48)

Further, we set∫ t

s(φ(r), ◦dWr) = ∑d ′

k=1

∫ t

sφk(r) ◦ dWk

r .We next consider the backward integral by Poisson random measure. Let {Fs,t }

be a two sided filtration. Let N(dr dz) be a Poisson random measure with theintensity measure n(dr dz) = drν(dz). We call it an {Fs,t }-Poisson randommeasure if (1) for any s < t ,

Fs,t ⊃ σ(N((u, v] × E); s ≤ u ≤ v ≤ t),

and (2) for any s < t , the following three σ -fields are independent;

F0,s , σ (N((u, v] × E); s ≤ u ≤ v ≤ t), Ft,T .

Let N be an {Fs,t }-Poisson random measure with the intensity measuren(dr dz) = drν(dz). Let LU be the set of backward predictable processes ψ(r, z)

satisfying∫ T

0

∫ |ψ(r, z)|2n(dr dz) < ∞ a.s. For ψ ∈ LU, we will define the

backward integral of ψ based on the compensated Poisson random measure. InSect. 1.5, we defined a backward martingale as a caglad (left continuous withright-hand limit) process. We can define the backward integral of ψ by thecompensated Poisson random measure N as a caglad backward martingale, inthe same way as the forward integral discussed in Sect. 2.5. Denote it by Xs . Set

2.7 Backward Integrals and Backward Calculus 75

Xs+ = limε↓0 Xs+ε . Then it holds that Xs = Xs+ a.s. for any s, since the Poissonrandom measure satisfies N({s} × R

d ′0 ) = 0 almost surely for any s. Then Xs+ is a

cadlag backward martingale. We denote it as∫ T

s

∫R

d′0ψ(r, z)N(dr dz). We can also

define the backward integrals∫ T

s

∫Bψ(r, z)ND(dr dz) as cadlag processes, where

D and B are Borel subsets of Rd ′0 as in the previous section.

Let Xs = (X1s , . . . , X

ds ), s ∈ [0, T ] be a backward cadlag process such that

Xis = XT+

∫ T

s

(φi(r), dWr)+∫ T

s

υi(r) dr+∫ T

s

∫

Rd′0

ψ i(r, z)ND(dr dz) (2.49)

holds for any 0 ≤ s < T , where XT is a constant, φi (r) ∈ LT, υi (r) ∈ LT andψ i(r, z)1D(z) ∈ LU. Here LT is the set of backward predictable processes υ(r)

satisfying∫ T

0 |υ(r)| dr < ∞ a.s. We call it a backward Itô process.Below, we give a backward differential rule for the backward Itô Process.

Since the proof is parallel to the proof of Itô’s formula for a forward Itô process(Theorem 2.6.1), it will be omitted.

Theorem 2.7.1 Let Xs = (X1s , . . . , X

ds ), s ∈ [0, T ] be a backward Itô pro-

cess given by (2.49). Let f (x1, . . . , xd , t) be a function of C2,1-class. Then,f (Xs, s), s ∈ [0, T ] is a one-dimensional backward Itô process. Further, for any0 ≤ s < t ≤ T , it is written as

f (Xs, s) = f (Xt , t)−∫ t

s

∂f

∂t(Xr , r) dr (2.50)

+d∑

i=1

d ′∑

k=1

∫ t

s

∂f

∂xi(Xr , r)φ

ik(r) dWkr +

d∑

i=1

∫ t

s

∂f

∂xi(Xr , r)υ

i(r) dr

+ 1

2

d∑

i,j=1

∫ t

s

∂2f

∂xi∂xj(Xr , r)

( d ′∑

k=1

φik(r)φjk(r))dr

+∫ t

s

∫

D

{f (Xr+ψ(r, z), r)−f (Xr , r)−

d∑

i=1

∂f

∂xi(Xr , r)ψ

i(r, z)}n(dr dz)

+∫ t

s

∫

Rd′0

{f (Xr + ψ(r, z), r)− f (Xr , r)}ND(dr dz), a.s.

Chapter 3Stochastic Differential Equationsand Stochastic Flows

Abstract In this chapter, we show that solutions of a continuous symmetricstochastic differential equation (SDE) on a Euclidean space define a continuousstochastic flow of diffeomorphisms and that solutions of an SDE with diffeomorphicjumps define a right continuous stochastic flow of diffeomorphisms. Sections 3.1and 3.2 are introductions. Definitions of these SDEs and stochastic flows willbe given and the geometric property of solutions will be explained as well asbasic facts. Rigorous proof of these facts will be given in Sects. 3.3, 3.4, 3.5,3.6, 3.7, 3.8 and 3.9. In Sect. 3.3, we study another Itô SDE with parameter,called the master equation. Applying results of Sect. 3.3, we show in Sect. 3.4 thatsolutions of the original SDE define a stochastic flow of C∞-maps. For the proofof the diffeomorphic property, we need further arguments. In Sect. 3.5 we considerbackward SDE and backward stochastic flow of C∞-maps. Further, the forward–backward calculus for stochastic flow will be discussed in Sects. 3.5, 3.6 and 3.8.These facts will be applied in Sects. 3.7 and 3.9 for proving the diffeomorphicproperty of solutions.

3.1 Geometric Property of Solutions I; Case of ContinuousSDE

Let T = [0,∞). Let d be a positive integer and l, m be nonnegative integers. Letus introduce some classes of functions on R

d × T. A function f (x, t) on Rd × T is

said to be of Cl,m-class (or Cl,m-function), if it is l-times continuously differentiablewith respect to x, and m-times continuously differentiable with respect to t . Further,if f and its derivatives are all bounded continuous functions of x, t , the function f

is said to be of Cl,mb -class (or Cl,m

b -function). Further, if f is of class Cl,mb for any

positive integer l, f is said to be of C∞,mb -class. Let d ′ be another positive integer.

A function f (x, t, z) on Rd × T × R

d ′ is said to be of Cl,m,nb -class if it is l-times

continuously differentiable with respect to x, m-times continuously differentiablewith respect to t , and n-times continuously differentiable with respect to z, and f


77


https://doi.org/10.1007/978-981-13-3801-4_3

78 3 Stochastic Differential Equations and Stochastic Flows

and its derivatives are bounded continuous functions of x, t, z. Further, if f is ofclass C

l,m,nb for any positive integer l, f is said to be of C∞,m,n

b -class.Let φ be a map from R

d into itself. Suppose that the map φ is ‘one to one’and ‘onto’. Then the inverse map φ−1;Rd → R

d is well defined. If both maps φ

and φ−1 are C∞-maps, φ is called a diffeomorphism of Rd or a diffeomorphic mapon R

d .Let Wt = (W 1

t , . . . ,Wd ′t ), t ∈ T be a d ′-dimensional Wiener process. For 0 ≤

s < t < ∞, we denote by Fs,t the smallest σ -field containing all null sets of F withrespect to which random variables Wu − Wv, s ≤ u, v ≤ t are measurable. Then{Fs,t , 0 ≤ s < t < ∞} is called a two-sided filtration generated by the Wienerprocess. We set Ft = F0,t . Then {Ft } is a filtration, with respect to which Wi

t are{Ft }-Wiener processes.

Let Vk(x, t) = (V 1k (x, t), . . . , V

dk (x, t)), k = 0, 1, . . . , d ′ be a family of vector

fields on Rd with parameter t ∈ T of C

∞,1b -class. We consider a continuous

symmetric SDE on Rd with coefficients Vk(x, t), k = 0, 1, . . . , d ′. Suppose we are

given t0 ∈ T and a Ft0 -measurable random variable X0 with values in Rd . Suppose

that there exists a continuous Rd -valued {Ft }-semi-martingale Xt = (X1t , . . . , X

dt ),

t ≥ t0 satisfying

Xit = Xi

0 +d ′∑

k=1

∫ t

t0

V ik (Xr, r) ◦ dWk

r +∫ t

t0

V i0 (Xr, r) dr, i = 1, . . . , d

or in vector notation,

Xt = X0 +d ′∑

k=1

∫ t

t0

Vk(Xr, r) ◦ dWkr +

∫ t

t0

V0(Xr, r) dr (3.1)

for all t ≥ t0 almost surely. Then the semi-martingale Xt, t ≥ t0 is called a solutionof the symmetric SDE with coefficients Vk(x, t), k = 0, . . . , d ′, starting from X0 attime t0.

The equation is said to have a (path-wise) unique solution if two solutions Xt, t ≥t0 and X′

t , t ≥ t0 of equation (3.1) having the same initial condition should satisfyXt = X′

t for all t ≥ t0 a.s.We will discuss the existence and the uniqueness of the solution of the equation.

However, instead of this equation, it is often convenient to rewrite it as Itô’sstochastic differential equation. Suppose that Xt is a solution of equation (3.1). ByProposition 2.4.1, the symmetric integral in equation (3.1) can be rewritten as

∫ t

t0

Vk(Xr, r) ◦ dWkr =

∫ t

t0

Vk(Xr, r) dWkr + 1

2〈Vk(Xt , t),W

kt 〉t0,t . (3.2)

Here Vk(Xt , t) is a continuous semi-martingale and it is represented by Itô’s formula(Corollary 2.2.1) as

3.1 Geometric Property of Solutions I; Case of Continuous SDE 79

Vk(Xt , t)−Vk(X0, t0)

=∑

i,l

∫ t

t0

∂Vk

∂xi(Xr, r)V

il (Xr, r) dW

lr +

∫ t

t0

∂Vk

∂t(Xr, r) dr

+ 1

2

∑

i,j

∫ t

t0

∂2Vk

∂xi∂xj(Xr, r)(

∑

l

V il (Xr, r)V

jl (Xr, r)) dr.

Then quadratic covariation of Vk(Xt , t) and Wkt on the interval [t0, t] is computed as

〈Vk(Xt , t),Wkt 〉t0,t =

∑

l

⟨ ∫ t

t0

Vl(r)Vl(Xr, r) dWlr ,W

kt

⟩

=∫ t

t0

Vk(r)Vk(Xr, r) dr,

where Vk(r)Vk(x, r) = ∑i V

ik (x, r)

∂∂xi

Vk(x, r). Consequently, setting

V ′0(x, r) = V0(x, r)+ 1

2

d ′∑

k=1

Vk(r)Vk(x, r), (3.3)

a solution Xt, t ≥ t0 of equation (3.1) satisfies the following continuous Itô SDE:

Xt = X0 +d ′∑

k=1

∫ t

t0

Vk(Xr, r) dWkr +

∫ t

t0

V ′0(Xr, r)dr. (3.4)

Conversely, if Xt, t ≥ t0 is a solution of Itô’s equation (3.4), Xt, t ≥ t0 is acontinuous semi-martingale and satisfies the symmetric equation (3.1). It is knownthat the equation (3.4) has a path-wise unique solution, if coefficients V ′

0, Vi, i =1, . . . , d ′ are Lipschitz continuous with respect to x. Therefore, equation (3.1) hasalso a unique solution.

For a given x ∈ Rd and s ∈ T, let Xx,s

t , t ≥ s be the solution of equation (3.2)starting from x at time s. We are interested in the geometric property of the familyof solutions {Xx,s

t , t ≥ s}. For this, we need the definition of the stochastic flow. LetC be the set of continuous maps from R

d into itself. Let {Φs,t ; 0 ≤ s < t < ∞} (orsimply denoted by {Φs,t }) be a family of C-valued random variables. It is called astochastic flow of C∞-maps if it satisfies:

(1) Maps Φs,t ;Rd → Rd are C∞ a.s. for any s < t .

(2) Φt,u ◦ Φs,t = Φs,u a.s. for any s < t < u. Here φ ◦ ψ is the composite of twomaps φ,ψ of Rd into itself.

Further, if it satisfies the following (1′) and the above (2), then it is called astochastic flow of diffeomorphisms.


(1′) Maps Φs,t ;Rd → Rd are diffeomorpshims a.s. for any s < t .

A stochastic flow {Φs,t } is called continuous if Φs,t (x), x ∈ Rd and its

derivatives ∂ iΦs,t (x) are continuous with respect to (x, s, t) (s < t) a.s. for anymulti-index i.

Let {Φs,t } be a continuous stochastic flow of diffeomorphisms. If for any x, s,X

x,st := Φs,t (x) is a solution of equation (3.1) starting from x at time s, it is said

that the stochastic flow is generated by the SDE, or the SDE defines the stochasticflow.

We want to show the following two facts:

1. Solutions of equation (3.1) generate a continuous stochastic flow {Φs,t } ofdiffeomorphisms.

2. Let Ψs,t be the inverse map of Φs,t . Then for any x, t , Xs = Ψs,t (x), s < t isa backward semi-martingale with respect to the two-sided filtration {Fs,t } andsatisfies the backward continuous symmetric stochastic differential equation:

Ψs,t (x) = x −d ′∑

k=1

∫ t

s

Vk(Ψr,t (x), r) ◦ dWkr −

∫ t

s

V0(Ψr,t (x), r) dr. (3.5)

{Ψs,t } is called the inverse flow of the stochastic flow {Φs,t }. The fact thatequation (3.1) defines a continuous stochastic flow of C∞-maps will be provedin Sect. 3.4. The proof of the diffeomorphic property is difficult. It will be givenin Sect. 3.7, after the discussion of the backward symmetric stochastic differentialequation in Sect. 3.5.

It should be noted that equation (3.1) for the flow {Φs,t } and equation (3.5) forthe inverse flow {Ψs,t } are symmetric, because these two equations are written byusing symmetric integrals. We write W 0

t = t conventionally and we will write thesymmetric equation (3.1) for the solution Φs,t (x) by

Φs,t (x) = x +d ′∑

k=0

∫ t

s

Vk(Φs,r (x), r) ◦ dWkr . (3.6)

Then the inverse flow satisfies the backward symmetric SDE.

Ψs,t (x) = x −d ′∑

k=0

∫ t

s

Vk(Ψr,t (x), r) ◦ dWkr . (3.7)

Using Itô integrals, the symmetry will not be valid. See the discussion afterTheorem 3.7.1. Another reason that we consider the symmetric equation is that theextension of the definition of (3.1) to manifolds is direct forward. It will be discussedin Chap. 7.

3.2 Geometric Property of Solutions II; Case of SDE with Jumps 81

We are also interested in the following two properties. One is the Lp-estimate ofthe flow {Φs,t }. The other is the property of independent increments.

3. For any p ≥ 2, s < t and |i| > 0, we have

supx

E[|Φs,t (x)|p](1 + |x|)p < ∞, sup

xE[|∂ iΦs,t (x)|p] < ∞. (3.8)

supx

E[|Ψs,t (x)|p](1 + |x|)p < ∞, sup

xE[|∂ iΨs,t (x)|p] < ∞. (3.9)

4. The flow {Φs,t } has independent increments; for any 0 ≤ t0 < · · · tn ≤ T ,Φti−1,ti , i = 1, . . . , n are independent.

Let G be the set of all diffeomorphic maps on Rd . It is a group by the

multiplication φψ := φ ◦ ψ and the inverse φ−1 := inverse map of φ. We define ametric d on G by d(φ,ψ) = ∑∞

N=11

2NdN (φ,ψ)

1+dN (φ,ψ), where

dN(φ,ψ) =∑

|i|≤N

(sup|x|≤N

|∂ i(φ(x)− ψ(x))| + sup|x|≤N

|∂ i(φ−1(x)− ψ−1(x))|).

Then G is a complete metric space. The flow {Φs,t } may be regarded as a G-valuedrandom field, satisfying Φs,t = Φtn,tΦtn−1,tn · · ·Φt1,s for any t0 < t1 < · · · < tn.Hence it has independent increments with respect to the multiplication. Then thecontinuous stochastic flow {Φs,t } may be considered as a (time inhomogeneous)Brownian motion (Wiener process) on the group G.

3.2 Geometric Property of Solutions II; Case of SDEwith Jumps

Discussions for the flow property are often parallel to those for a more general classof SDE having jumps. We shall introduce SDEs on R

d governed by a Wiener processand a Poisson random measure. These SDEs may have jumps and can be regardedas an extension of a continuous SDE.

Let Rd ′0 = R

d ′ \ {0}. Its elements are denoted by z = (z1, . . . , zd′). Let

N(dr dz) be a Poisson random measure on U = T×Rd ′0 with the intensity measure

n(dr dz) := drν(dz), where ν is a Lévy measure on Rd ′0 . We assume that it is

independent of the Wiener process Wt . For 0 ≤ s < t < ∞, we denote by Fs,t thesmallest σ -field containing all null sets of F with respect to which random variablesWu − Wv, s ≤ u, v ≤ t and N((u, v] × E), s ≤ u < v ≤ t, E ∈ B(Rd ′

0 ) aremeasurable. Then {Fs,t } is a two-sided filtration. It is called a two-sided filtrationgenerated by the Wiener process Wt and the Poisson random measure N(dr dz). Weset Ft = F0,t . Then {Ft } is a filtration of sub σ -fields of F .


Let Vk(x, t), k = 1, . . . , d ′ be a family of vector fields of C∞,1b -class defined on

Rd . Let g(x, t, z) be a map from R

d × T × Rd ′ to R

d satisfying the following twoconditions.

Condition (J.1). Components gi(x, t, z), i = 1, . . . , d of g(x, t, z) are functionsof C

∞,1,2b -class on R

d × T× Rd ′ . Further, g(x, t, 0) = 0 for any x, t .

Condition (J.2). Set φt,z(x) = g(x, t, z)+x. For any t, z, the map φt,z;Rd → Rd

is a diffeomorphism of Rd . Further, h(x, t, z) := x−φ−1t,z (x) satisfy Condition (J.1).

Let z = (z1, . . . , zd′) ∈ R

d ′ . We set

bkε =∫

ε≤|z|≤1zkν(dz), Vk(x, t) = ∂zkφt,z(x)

∣∣∣z=0

, k = 1, . . . , d ′.

Vk(x, t), k = 1, . . . , d ′ are called tangent vectors of maps φt.z(x) at z = 0.Suppose we are given time 0 ≤ t0 < ∞ and Ft0 -adapted R

d -valued randomvariable X0. Suppose that there exists an R

d -valued semi-martingale Xt, t ≥ t0(with bounded jumps) satisfying

Xt = X0 +d ′∑

k=0

∫ t

t0

Vk(Xr, r) ◦ dWkr (3.10)

+ limε→0

{ ∫ t

t0

∫

|z|≥ε

g(Xr−, r, z)N(dr dz)−d ′∑

k=1

bkε

∫ t

t0

Vk(Xr−, r) dr}.

Then Xt, t ≥ t0 is called a solution of a symmetric SDE with characteristics(Vk(x, t), k = 0, . . . , d ′, g(x, t, z), ν), starting from X0 at time t0. We should checkthat the last term is well defined for any semi-martingale Xt, t ≥ t0. It holds that

∫ t

t0

∫

|z|≥ε


k=1

bkε

∫ t

t0

Vk(Xr−, r) dr

=∫ t

t0

∫

|z|≥ε

g(Xr−, r, z)N(dr dz)

+∫ t

t0

∫

|z|≥ε

{g(Xr−, r, z)− 1D(z)

d ′∑

k=1

zkVk(Xr, r)}n(dr dz), (3.11)

where D = {z ∈ Rd ′0 ; |z| ≤ 1}. Let ε → 0. The first term of the right-hand

side converges to an L2-martingale, since E[∫∫ |g(Xr, r, z)|2n(dr dz)] < ∞. Sinceg(x, t, z) − ∑

k zkVk(x, t) = O(|z|2) holds by the Taylor expansion, the second

term converges as ε → 0. Consequently, for any cadlag process Xt, t ≥ t0, theleft-hand side of the above converges in L2 as ε → 0.


In order to solve the equation, we will consider the Itô SDE with jumps

Xt = X0 +∫ t

t0

α(Xr−, r) dWr +∫ t

t0

β(Xr, r) dr

+∫ t

t0

∫χ(Xr−, r, z)N(dr dz), (3.12)

where α(x, r) = (V ik (x, r)),

β(x, r) = V0(x, r)+ 1

2

d ′∑

k=1

Vk(r)Vk(x, r) (3.13)

+ limε→0

∫

|z|≥ε

{g(x, r, z)− 1D(z)(

d ′∑

k=1

zkVk(x, r))}dν,

and χ(x, r, z) = g(x, r, z).It will be shown in Sect. 3.4 that the solution of the Itô equation (3.12) exists

uniquely. Further, for the solution Xt , the symmetric integral of equation (3.10) iswell defined and in fact is the unique solution of equation (3.10).

Let {Φs,t } be a stochastic flow of C∞-maps (or of diffeomorphisms). If Xx,st :=

Φs,t (x), t ≥ s is a solution of equation (3.10) starting from x at time s for any x, s,the stochastic flow is said to be generated by SDE (3.10) and SDE (3.10) definesthe stochastic flow {Φs,t }. One of main subjects of this chapter is to prove thatunder Conditions (J.1) and (J.2), SDE (3.10) generates a stochastic flow {Φs,t } ofdiffeomorphisms.

We begin with some preliminary observations for solutions of some specialequations. If g(x, t, z) ≡ 0, equation has no jump part. It is a continuous SDEwritten as (3.1). We next consider an SDE which does not contain a diffusionpart. We assume that the Lévy measure is a finite measure. Then the jump partof equation (3.10) is written simply as

∫ t

t0

∫

Rd′g(Xr−, r, z)N(dr dz)−

∑

k

bk0

∫ t

t0

Vk(Xr−, r) dr,

where bk0 = limε→0 bkε . Hence setting V0(x, t) = V0(x, t) − ∑

k bk0Vk(x, t), the

equation is written as

Xt = X0 +∫ t

t0

V0(Xr, r) dr +∫ t

t0

∫

Rd′0

g(Xr−, r, z)N(dr dz). (3.14)

Suppose first that V0(x, t) = 0. The above Poisson random measure N(dr dz)

is constructed from a Poisson process Nt with the intensity λ = ν(Rd ′0 ) and a


sequence of i.i.d. random variables {Sn} with the common law μ(E) = ν(E)/λ. Infact, N((t0, t], E) := ∑

0<n≤Nt−Nt01E(Sn) is a Poisson random measure on U =

[t0,∞) × Rd0 with intensity drν(dz) (Proposition 1.4.3). Now, let t0 ≡ τ0 < τ1 <

· · · τn < · · · be a sequence of jumping times of the process Nt − Nt0 , t ∈ [t0,∞).We set

Φt0,t (x) ={x, if t < τ1,

φτn,Sn ◦ · · · ◦ φτ1,S1(x), if τn ≤ t < τn+1, n = 1, 2, . . .(3.15)

Here, φτn,Sn ◦ · · · ◦φτ1,S1 are composites of maps φτm,Sm;Rd → Rd ,m = 1, . . . , n.

Then the map Φt0,t ;Rd → Rd is a diffeomorphism a.s., since φτm,Sm,m = i, . . . , n

are diffeomorphic. Further, the stochastic process Xt := Φt0,t (X0), t ≥ t0 satisfies

Xt = X0 +∑

m;t0<τm≤t

{φτm,Sm(Xτm−1)−Xτm−1

}

= X0 +∑

m;t0<τm≤t

g(Xτm−1 , τm, Sm)

= X0 +∫ t

t0

∫

Rd′0

g(Xr−, r, z)N(dr dz). (3.16)

Therefore, {Φt0,t } defined by (3.15) is a stochastic flow of diffeomorphismsgenerated by the jump SDE (3.16). It may be considered as a compound Poissonprocess with values in the group G of diffeomorphisms.

In the case V0(x, t) �= 0, let ϕs,t (x) be the deterministic flow generated by thevector field V0(x, t); i.e., ϕs,t (x) = x + ∫ t

sV0(ϕs,r (x), r) dr. We set

Φt0,t ={ϕt0,t , if t < τ1,

ϕτn,t ◦ φτn,Sn ◦ ϕτn−1,τn ◦ · · · ◦ φτ1,S1 ◦ ϕt0,τ1 , if τn ≤ t < τn+1.

Then Φt0,t ;Rd → Rd are diffeomorphisms a.s. for any t0 < t . Further, Xt :=

Φt0,t (X0), t ≥ t0 is a solution of equation (3.14). Hence the above {Φt0,t } is also astochastic flow of diffeomorphisms generated by SDE (3.14).

Now let us return to the SDE (3.10). We assume that the Lévy measure is finite.Then the equation is rewritten as

Xt = X0 +d ′∑

k=1

∫ t

t0


∫ t

t0

V0(Xr, r) dr

+∫ t

t0

∫

|z|>0g(Xr−, r, z)N(dr dz). (3.17)

Let {Φ0t0,t

} be the stochastic flow generated by the continuous SDE (3.1), replacing

the drift vector field V0 by V0. Set


Φt0,t (x) (3.18)

={Φ0

t0,t(x), if t < τ1,

Φ0τn,t

◦ φτn,Sn ◦Φ0τn−1,τn

◦ · · · ◦ φτ1,S1 ◦Φ0t0,τ1

(x), if τn ≤ t < τn+1.

Then Xt = Φt0,t (X0), t ≥ t0 satisfies

Xt −Xτm−1 =d ′∑

k=1

∫ t

τm−1


∫ t

τm−1

V0(Xr, r) dr,

if τm−1 ≤ t < τm. At t = τm, we have Xτm −Xτm− = g(Xτm−, τm, Sm). Therefore,if τn ≤ t < τn+1,

Xt −X0 = Xt −Xτn +n∑

m=1

(Xτm− −Xτm−1)+n∑

m=1

(Xτm −Xτm−)

=d ′∑

k=1

∫ t

t0


∫ t

t0

V0(Xr, r) dr

+n∑

m=1

g(Xτm−, τm, Sm). (3.19)

The last term of the right-hand side is equal to∫ t

t0g(Xr−, r, z)N(dr dz). Therefore

the above Xt, t ≥ t0 is a solution of equation (3.17). Consequently, {Φt0,t } definedby (3.18) is the stochastic flow generated by the SDE (3.17).

Remark In Proposition 1.4.4, we constructed a compound Poisson process on Rd ,

regarding it as an Abelian group. The expression (3.18) tells us that the stochasticflow {Φs,t } is a compound Poisson process with values in the non-Abelian group G

of diffeomorphisms on Rd . Since it has independent increments in the group G, the

stochastic flow may be considered as a Lévy process with values in the group G.Finally, consider the SDE (3.10) where the associated Lévy measure is an infinite

measure. The solution cannot be written as the above, since jumps may occurinfinitely many times in a finite interval. It should be the limit of a sequence ofabove solutions. For a given ε > 0, consider an SDE

Xt = X0 +d ′∑

k=1

∫ t

t0


∫ t

t0

V ε0 (Xr, r) dr

+∫ t

t0

∫

|z|≥ε

g(Xr−, r, z)N(dr dz), (3.20)


where V ε0 (r) = V0(r) − ∑

k bkεVk(r) and bkε = ∫

ε≤|z|≤1 zkν(dz). Let {Φε

s,t } bethe stochastic flow of diffeomorphisms generated by SDE (3.20). We will show inSect. 3.9 that Xε,x,t0

t := Φεt0,t

(x), t ≥ t0 converges to a stochastic process Xx,t0t as

ε → 0 and that the limit is a solution of the SDE (3.10). Further, it has a modificationΦt0,t (x) which are diffeomorphic maps on R

d for any t ≥ t0.A stochastic flow {Φs,t } is called right continuous, if Φs,t (x) is right continuous

with respect to t and s almost surely. Here a C∞-function with parameter t

denoted by φt (x) is called right continuous at t0 if sup|x|≤M |∂ iφt (x) − ∂ iφt0(x)|converges to 0 as t ↓ t0 for any M > 0 and multi-index i. It is expected that theSDE (3.10) satisfying (J.1) and (J.2) should define a right continuous stochastic flowof diffeomorphisms.

For the rigorous proof of these facts, we need long discussions. In Sect. 3.3we consider another SDE, which we will call the master equation. Coefficientsof the equation are parameterized by λ. We will discuss the continuity and thedifferentiability of the solution with respect to parameter λ. Then results of themaster equation will be applied to solutions of the SDE (3.10). Let X

x,t0t be the

solution of the SDE starting from x at time t0. By applying results to the masterequation, we show in Sect. 3.4 that the family of solutions {Xx,t0

t ; x ∈ Rd} has

a modification such that it is infinitely differentiable with respect to x for anyt0 ≤ t < ∞ a.s. Then, we may regard that for each t0 < t < ∞, X

x,t0t is a

C∞-map from Rd to itself a.s.

3.3 Master Equation

Let us consider another Itô SDE on Rd with parameter λ ∈ Λ given by

Xt = cλ +∫ t

t0

αλ(Xr−, r) dWr +∫ t

t0

βλ(Xr−, r) dr

+∫ t

t0

∫

Rd′0

χλ(Xr−, r, z)N(dr dz). (3.21)

Here, for the parameter set Λ we take a Euclidean space Re or its unit ball Be

1 ={λ ∈ R

e; |λ| < 1}. Let Xt be a cadlag {Ft }-adapted process with values in Rd . If

each component Xit of Xt = (X1

t , . . . , Xdt ) satisfies

Xit = ciλ +

d ′∑

k=1

∫ t

t0

αikλ (Xr−, r) dWk

r +∫ t

t0

βiλ(Xr−, r) dr

+∫ t

t0

∫

Rd′0

χiλ(Xr−, r, z)N(dr dz),

Xt , t ≥ t0 is called a solution of equation (3.21) starting from cλ at time t0.

3.3 Master Equation 87

For the initial condition cλ = (c1λ, . . . , c

dλ), we assume that it is non-random,

bounded and Lipschitz continuous with respect to λ. Coefficients

αλ(x, r) = (αikλ (x, r)), βλ(x, r) = (β1

λ(x, r), . . . , βdλ (x, r)),

χλ(x, r, z) = (χ1λ(x, r, z), . . . , χ

dλ (x, r, z)),

where x ∈ Rd , r ∈ T, z ∈ R

d ′ , are possibly random, jointly measurable andpredictable with respect to the filtration {Ft }. Further, χλ(x, r, 0) = 0 and χλ(x, r, z)

is continuously differentiable with respect to z ∈ S(ν), where S(ν) is the support ofthe Lévy measure ν.

Further, αλ, βλ, χλ and ∂zχλ satisfy the following three conditions:

1. At x = 0, coefficients are uniformly Lp-bounded with respect to λ and r;

supλ,r

{E[|αλ(0, r)|p] + E[|βλ(0, r)|p] + E

[ ∫

Rd′0

∣∣∣χλ(0, r, z)

γ (z)

∣∣∣p

μ(dz)]}

< ∞(3.22)

holds for any p ≥ 2. Here γ (z) = |z| ∧ 1 and μ(dz) = γ (z)2ν(dz).2. Coefficients are uniformly Lipschitz continuous with respect to x; there exist

positive constants c and cp such that

|αλ(x, r)−αλ(x′, r)| ≤ c|x − x′|, (3.23)

|βλ(x, r)−βλ(x′, r)| ≤ c|x − x′|,

( ∫

Rd′0

∣∣∣χλ(x, r, z)

γ (z)− χλ(x

′, r, z)γ (z)

∣∣∣p

μ(dz)) 1

p ≤ cp|x − x′|

hold for any x, x′, r, λ and p ≥ 2 a.s.3. With respect to parameter λ, coefficients are weakly Lipschitz continuous in the

following sense: For any T > 0 and p ≥ 2, there exists a positive random fieldKλ,λ′(r) such that

E[( ∫ T

0Kλ,λ′(r) dr

)p] 1p< cp|λ− λ′| (3.24)

holds with a positive constant cp, and coefficients satisfy

|αλ(x, r)−αλ′(x, r)| ≤ Kλ,λ′(r)(1 + |x|), (3.25)

|βλ(x, r)−βλ′(x, r)| ≤ Kλ,λ′(r)(1 + |x|),( ∫

Rd′0

∣∣∣χλ(x, r, z)

γ (z)− χλ′(x, r, z)

γ (z)

∣∣∣p

μ(dz)) 1

p ≤ Kλ,λ′(r)(1 + |x|)

a.s. for any x, r, λ, λ′.


Equation (3.21) is called a master equation (with parameter λ). If χλ(x, r) ≡ 0,the equation is called a continuous master equation.

For the analysis of the master equation, we will often use an inequality calledGronwall’s inequality:

Let f (t) and g(t) be nonnegative continuous functions on [t0,∞) satisfying theinequality

f (t) ≤ g(t)+ c

∫ t

t0

f (r) dr, ∀t ≥ t0,

where c is a positive constant. Then the inequality

f (t) ≤ g(t)+ c

∫ t

t0

g(r)ec(t−r) dr (3.26)

holds for any t ≥ t0. In particular, if f (t) ≤ c∫ t

t0f (r) dr holds for any t ≥ 0, then

f (t) ≡ 0 holds.The inequality (3.26) is called Gronwall’s inequality. The proof is straightfor-

ward and it is omitted.For an R

d -valued cadlag stochastic process Xt, t ∈ [t0,∞), we define norms‖ ‖t depending on parameter t0 ≤ t ≤ T by ‖X‖t = supt0≤s≤t |Xr−|.Lemma 3.3.1 Let p ≥ 2. For any λ, equation (3.21) has a solution belonging toLp.

Proof The existence of the solution can be verified by a standard argument using thesuccessive approximation. Set Xλ,0

t = cλ and define Xλ,nt for n ≥ 1 by induction as

follows:

Xλ,n+1t = cλ +

∫ t

t0

αλ(Xλ,nr− , r) dWr +

∫ t

t0

βλ(Xλ,nr− , r) dr

+∫ t

t0

∫χλ(X

λ,nr− , r, z)N(dr dz).

Then, using Doob’s inequality for martingales (Theorem 1.5.1), we have

E[‖Xλ,n+1 −Xλ,n‖pt

]

≤ C0

{E[∣∣∣

∫ t

t0

(αλ(Xλ,nr− , r)− αλ(X

λ,n−1r− , r)) dWr

∣∣∣p]

+ E[∣∣∣

∫ t

t0

|βλ(Xλ,nr− , r)− βλ(X

λ,n−1r− , r)| dr

∣∣∣p]

+ E[∣∣∣

∫ t

t0

∫ {χλ(X

λ,nr− , r, z)− χλ(X

λ,n−1r− , r, z)

}N(dr dz)

∣∣∣p]}

.


Using the Burkholder–Davis–Gundy inequality (Propositions 2.2.2 and 2.6.1) andthe uniform Lipschitz continuity for coefficients, we have for any t0 ≤ t ≤ T

E[( ∫ t

t0

(αλ(Xλ,nr− , r)− αλ(X

λ,n−1r− , r)) dWr

)p]

≤ CpE[ ∫ t

t0

|αλ(Xλ,nr− , r)− αλ(X

λ,n−1r− , r)|p dr

]

≤ C′p

∫ t

t0

E[‖Xλ,n −Xλ,n−1‖pr ] dr,

E[∣∣∣

∫ t

t0

|βλ(Xλ,nr− , r)− βλ(X

λ,n−1r− , r)| dr

∣∣∣p]

≤ C1

∫ t

t0

E[‖Xλ,n −Xλ,n−1‖pr ] dr,

E[( ∫ t

t0

∫ {χλ(X

λ,nr− , r, z)− χ(X

λ,n−1r− , r, z)

}N(dr dz)

)p]

≤ C′E[ ∫ t

t0

( ∫ ∣∣∣χλ(X

λ,nr− , r, z)

γ (z)− χλ(X

λ,n−1r− , r, z)

γ (z)

∣∣∣p

μ(dz))dr

]

≤ C′′∫ t

t0

E[‖Xλ,n −Xλ,n−1‖pr ] dr.

Therefore there exists a positive constant C such that

E[‖Xλ,n+1 −Xλ,n‖pt ] ≤ C

∫ t

t0

E[‖Xλ,n −Xλ,n−1‖pr ] dr.

Repeating this argument, we arrive at

E[‖Xλ,n+1 −Xλ,n‖pt ] ≤(C(t − t0))

n

n! E[‖Xλ,1 −Xλ,0‖pt ].

By the definition of Xλ,1t , it holds that supλ E[‖Xλ,1 − Xλ,0‖pt ] < ∞. Then

there exists a cadlag process Xλt such that supλ E[‖Xλ − Xλ,0‖pt ] < ∞ and

supλ E[‖Xλ,n − Xλ‖pt ] converges to 0 for any t0 ≤ t ≤ T . The process Xλt is a

solution of the above stochastic differential equation. ��


Lemma 3.3.2 Any solution Xλt of the master equation (3.21) belongs L∞− =⋂

p>1 Lp. Furthermore, for any T > t0 and p ≥ 2, there is a positive constant

c such that

E[‖Xλ‖pT ] < c(1 + |cλ|)p (3.27)

holds for all λ ∈ Λ.

Proof For n > 0, define a stopping time by

τn ={

inf{t ≥ t0; |Xλt | ≥ n}, if {· · · } �= ∅,

∞, if {· · · } = ∅.

Then setting t ′ = t ∧ τn and using the Burkholder–Davis–Gundy’s inequality, wehave the following Lp-estimate:

E[‖Xλ‖pt ′ ]

≤ cp

{|cλ|p + E

[ ∫ t ′

t0

|αλ(Xλr−, r)|p dr

]+ E

[ ∫ t ′

t0

|βλ(Xλr−, r)|p dr

]

+ E[ ∫ t ′

t0

( ∫ ∣∣∣χλ(X

λr−, r, z)γ (z)

∣∣∣p

μ(dz))dr

].

We have

|αλ(Xλr−, r)| ≤ |αλ(0, r)| + c|Xλ

r−|, |βλ(Xλr−, r)| ≤ |βλ(0, r)| + c|Xλ

r−|,( ∫ ∣∣∣

χλ(Xλr−, r, z)γ (z)

∣∣∣p

μ(dz)) 1

p ≤( ∫ ∣∣∣

χλ(0, r, z)

γ (z)

∣∣∣p

μ(dz)) 1

p + c|Xλr−|.

Therefore, there exist positive constants C1, C2 such that

E[‖Xλ‖p

t ′] ≤ C1(1 + |cλ|)p + C2E

[ ∫ t ′

t0

|Xλr−|p dr

].

We have |Xλr ′ | ≤ n a.s. for r ′ < t ′. Then the last integral is finite. Therefore,

E[‖Xλ‖pt ′ ] ≤ C1(1 + |cλ|)p + C2

∫ t

t0

E[‖Xλ‖pr ′ ] dr.

By Gronwall’s inequality, there exists a positive constant cp such that

E[‖Xλ‖pt∧τn] ≤ cp(1 + |cλ|)p


holds for all n and λ. Let n tend to ∞. The above inequality indicates that the set{τn ≤ t} shrinks to the empty set a.s., so that we get E[‖Xλ‖pt ] < cp(1 + |cλ|)p.The inequality is valid for all λ, t . Hence we get the inequality of the lemma. ��Lemma 3.3.3 Let {Xλ

t , λ ∈ Λ} be any family of solutions of master equa-tions (3.21). Then for any T > t0 and p ≥ 2, there exists a positive constant Csuch that

E[‖Xλ −Xλ′ ‖pT ] ≤ C|λ− λ′|p (3.28)

holds for all λ, λ′ ∈ Λ.

Proof Set Yt = Xλt −Xλ′

t . Then we have

Yt = cλ − cλ′ +∫ t

t0

(αλ(X

λr−, r)− αλ′(X

λ′r−, r)

)dWr

+∫ t

t0

(βλ(X

λr−, r)− βλ′(X

λ′r−, r)

)dr

+∫ t

t0

∫ {χλ(X

λr−, r, z)− χλ′(X

λ′r−, r, z)

}N(dr dz).

We shall consider terms of the right-hand side. We have by (3.25)

E[∣∣∣

∫ t

t0

(αλ(Xλr−, r)−αλ′(X

λ′r−, r)) dWr

∣∣∣p]

≤ c0E[ ∫ t

t0

|Xλr−−Xλ′

r−|p dr]+ c0E

[ ∫ t

t0

Kλ,λ′(r)p(1 + |Xλr−|)p dr

].

Since Kλ,λ′r satisfies (3.24) and since |Xλ

t | are Lp-bounded with respect to x, λ byLemma 3.3.2, the above is dominated by

c1

∫ t

t0

E[‖Xλ −Xλ′ ‖pr ] dr + c′1|λ− λ′|p.

Similarly, we can show that both

E[∣∣∣

∫ t

t0

(βλ(Xλr−, r)− βλ′(X

λ′r−, r)) dr

∣∣∣p]

,

E[∣∣∣

∫ t

t0

∫ {χλ(X

λr−, r, z)− χλ′(X

λ′r−, r, z)

}N(dr dz)

∣∣∣p]


satisfy the similar inequality. Further, note that cλ is Lipschitz continuous. Thenthere exist positive constants C,C′ such that

E[‖Y‖pt ] ≤ C|λ− λ′|p + C′∫ t

t0

E[‖Y‖pr ] dr

holds for all λ, λ′ and t . Then, Gronwall’s inequality implies the inequality of thelemma. ��Theorem 3.3.1 Assume Conditions 1–3 for the master equation. For any cλ, themaster equation (3.21) has a unique solution Xλ

t , which belongs to L∞−. The familyof solutions {Xλ

t , λ ∈ Λ} has a modification which is continuous with respect to λ

and cadlag with respect to t a.s. Further, for any T > t0 and p ≥ 2 there is apositive constant c such that {Xλ

t , λ ∈ Λ} satisfies

E[‖Xλ‖pT ] ≤ c(1 + |cλ|)p (3.29)

for all λ ∈ Λ.

Proof The existence of a solution belonging to L∞− is proved in Lemma 3.3.1.Suppose that Xλ

t and Xλt are solutions of the master equation with the same initial

value cλ. Then both belong to L∞− by Lemma 3.3.2. Then Lemma 3.3.3 tells usXλ

t = Xλt a.s., since λ = λ′.

Now the family of solutions {Xλt ; λ ∈ Λ} satisfies (3.28). Then Xλ

t has amodification which is continuous in λ by the Kolmogorov-Totoki theorem (The-orem 1.8.1). Further, it satisfies (3.29) for any p ≥ 2 by Lemma 3.3.1. ��

We shall next consider the differentiability of the solution with respect toparameter λ. We assume that coefficients αλ, βλ, χλ and the initial value cλ aredifferentiable with respect to λ. Let λ = (λ1, . . . , λe) and let ∂λi αλ be the partialderivative of αλ with respect to λi . We set ∂λαλ = (∂λ1αλ, . . . , ∂λeαλ).

Theorem 3.3.2 Assume that coefficients of the master equation are continuouslydifferentiable with respect to x and λ, and derivatives ∂xαλ(x, r), ∂xβλ(x, r),∂xχλ(x, r, z), ∂λαλ(x, r), ∂λβλ(x, r) and ∂λχλ(x, r, z) satisfy Conditions 1–3 of themaster equation. Assume further that ∂λcλ exists and is Hölder continuous.

1. The family of solutions {Xλt , λ ∈ Λ} has a modification which is continuously

differentiable with respect to λ for any t and the derivative is continuous in λ

and cadlag in t a.s. Further, for any T > t0 and p ≥ 2, it holds that

E[∥∥∥

1

ε(Xλ+εei −Xλ)− ∂λiX

λ∥∥∥p

T

]→ 0 (3.30)

as ε → 0 uniformly in λ. Here ei are unit vectors in Re.


2. The derivative ∂λXλt satisfies the following equation:

∂λXλt = ∂λcλ +

∫ t

t0

{∂λαλ(Xλr−, r)+ ∂xαλ(X

λr−, r)∂λXλ

r−} dWr

+∫ t

t0

{∂λβλ(Xλr−, r)+ ∂xβλ(X

λr−, r)∂λXλ

r−} dr (3.31)

+∫ t

t0

∫

Rd′0

{∂λχλ(Xλr−, r, z)+ ∂xχλ(X

λr−, r, z)∂λXλ

r−}N(dr dz).

3. For any T > t0 and p ≥ 2 there is a positive constant c such that

E[‖∂λXλ‖pT ] ≤ c(1 + |∂λcλ|)p (3.32)

holds for all λ.

Proof Let λ ∈ Λ and ε ∈ Θ = R \ {0}. For a positive integer i with 1 ≤ i ≤ e, weset εi = εei , where ei is an unit vector in R

e. We set

Yλ,εt = 1

ε

{X

λ+εit −Xλ

t

}.

We want to prove that limε→0 Yλ,εt exists uniformly in t for all λ almost surely and

it is continuous with respect to λ almost surely. For this purpose, we will find anSDE which governs Y

λ,εt . Since X

λ+εit and Xλ

t are solutions of (3.21), we find thatx = Y

λ,εt is a solution of a master equation on R

d with parameter λ, ε starting fromcλ,ε := (cλ+εi − cλ)/ε at time t0, whose coefficients are given by

αλ,ε(x, r) = 1

ε

{αλ+εi (X

λr− + εx, r)− αλ(X

λr−, r)

},

βλ,ε(x, r) = 1

ε

{βλ+εi (X

λr− + εx, r)− βλ(X

λr−, r)

},

χλ,ε(x, r, z) = 1

ε

{χλ+εi (X

λr− + εx, r, z)− χλ(X

λr−, r, z)

},

because Xλ+εir− = Xλ

r− + εYλ,εr− . We will show that these coefficients fulfill

three conditions of the master equation. We will consider αλ,ε only, since othercoefficients can be handled similarly. It holds that

|aλ,ε(x, r)−aλ,ε(x′, r)| ≤ 1

ε

∣∣∣aλ+εi (Xλr−+εx, r)−aλ+εi (X

λr−+εx′, r)

∣∣∣ ≤ c|x−x′|.

We will next consider |αλ,ε(x, r)− αλ′,ε′(x, r)|. Rewrite first the right-hand side ofaλ,ε(x, r) using the mean value theorem. Then it is written as


αλ,ε(x, r) = ∂xαλ+θεi (Xλr− + εθx, r)x + ∂λi αλ+θεi (X

λr− + εθx, r),

where 0 ≤ |θ | ≤ 1. Therefore, |αλ,ε(x, r)−αλ′,ε′(x, r)| is dominated by the sum ofthe following two;

|∂xαλ+θεi (Xλr− + εθx, r)− ∂xαλ′+θε′i (X

λ′r− + ε′θx, r)|, (3.33)

|∂λi αλ+θεi (Xλr− + εθx, r)− ∂λi αλ′+θε′i (X

λ′r− + ε′θx, r)|.

The first term of (3.33) is dominated by

|∂xαλ+θεi (Xλr− + εθx, r)− ∂xαλ′+θε′i (X

λr− + εθx, r)|

+ |∂xαλ′+θε′i (Xλr− + εθx, r)− ∂xαλ′+θε′i (X

λ′r− + ε′θx, r)|

≤ Kλ+θεi ,λ′+θε′i (1 + |x| + |Xλ

r−|)+ c(|Xλr− −Xλ′

r−| + |ε − ε′|(1 + |x|)),

where Kλ,λ′ is a positive functional satisfying E[(∫ T

0 Kλ,λ′(r) dr)p] ≤ cp|λ− λ′|pfor any p ≥ 2. A similar estimate is valid for the second term of (3.33).Consequently, we obtain the inequality

|αλ,ε(x, r)− αλ′,ε′(x, r)| ≤ 2K(λ,ε),(λ′,ε′)(r)(1 + |x|),

where

K(λ,ε),(λ′,ε′)(r) = Kλ+θεi ,λ′+θε′i (r)(1 + |Xλ′

r |)+ c(|Xλr− −Xλ′

r−| + |ε − ε′|).

It satisfies

E[( ∫ T

0K(λ,ε),(λ′,ε′)(r) dr

)p] ≤ c′p(|λ− λ′|p + |ε − ε′|p)

for any p ≥ 2. Therefore, αλ,ε(x, r) fulfills Conditions 1–3 of the master equation.In the same way, we can verify that βλ(x, r) and χλ(x, r, z) fulfill Conditions 1–3of the master equation.

Now Lemmas 3.3.2 and 3.3.3 tell us that for any K > 0, T > t0 and p ≥ 2, thereexists a positive constant C such that inequalities

E[‖Yλ,ε‖pT ] ≤ C(1 + |cλ,ε |)p, (3.34)

E[‖Yλ,ε − Yλ′,ε′ ‖pT ] ≤ C{|λ− λ′|p + |ε − ε′|p} (3.35)

hold for any |λ|, |λ′| ≤ K and 0 < |ε|, |ε′| < 1. Then, we apply the Kolmogorov–Totoki theorem to Yλ,ε , regarding it as a random field with values in


B = {Xt ; Rd -valued cadlag function of t ∈ T}.

Then the random field Yλ,ε has a modification such that it is uniformly continuouson the set {|λ| < K, 0 < |ε| < 1}. It means that Yλ,ε

t can be extended continuouslyat ε = 0, and for any t , Yλ,0

t = ∃ limε→0 Yλ,εt (uniformly in t) is continuous in λ

a.s. This means that Xλt is continuously differentiable with respect to λ for any t and

∂λiXλt = Y

λ,0t holds for all λ, t a.s. Further, set λ′ = λ and let ε′ tend to 0 for the

inequality (3.35). Then we get

E[‖Yλ,ε − ∂λiXλ‖pT ] ≤ Cpε

p.

Therefore (3.30) holds uniformly in λ.Next we will prove (3.31). The solution Xλ

t satisfies

Xλt = cλ +

∫ t

t0

αλ(Xλr−, r) dWr +

∫ t

t0

βλ(Xλr−, r) dr

+∫ t

t0

∫χλ(X

λr−, r, z)N(dr dz).

We want to show that each term of the right-hand side is continuously differentiablewith respect to λ and we can change the order of the derivative operator ∂λ andthe integral operator

∫ t

t0. We will check this for the Itô integral

∫ t

t0αλ(X

λr−, r) dWr .

Since αλ and Xλr are continuously differentiable with respect to λ, the composite

αλ(Xλr−, r) is also continuously differentiable with respect to λ. Further, the

derivative ∂λXλr− is Lipschitz continuous in Lp, i.e., it satisfies

supr

E[|∂λXλr− − ∂λX

λ′r−|p] ≤ cp|λ− λ′|p,

in view of (3.35). Therefore αλ(Xλr−, r) belongs to the class L1+Lip,p

T(Λ)

(Sect. 2.3). Consequently, the Itô integral is continuously differentiable in λ andwe get

∂λ

∫ t

t0

αλ(Xλr−, r) dWr =

∫ t

t0

∂λ(αλ(Xλr−, r)) dWr

=∫ t

t0

{∂λαλ(Xλr−, r)+ ∂xαλ(X

λr−, r)∂λXλ

r−} dWr,

in view of Proposition 2.3.1. We can show similar facts for∫ t

t0βλ(X

λr−, r) dr and

∫ t

t0

∫χλ(X

λr−, r)N(dr dz). Then we get the equality (3.31).

Finally, apply Lemma 3.3.2 to the master equation (3.31). Then we get (3.32).��


Remark Later, we will be concerned with a more general master equation. Letδ(x, r, z), (x, r, z) ∈ R

d × T× Rd ′ be a C

∞,0,0b function. Let (ri, zi), i = 1, . . . , n

be points in T× Rd ′0 . We consider an SDE:

Xt = cλ +∫ t

t0

αλ(Xr−, r) dWr +∫ t

t0

βλ(Xr−, r) dr (3.36)

+∫ t

t0

∫

Rd′0

χλ(Xr−, r, z)N(dr dz)+∑

i;t0≤ri≤t

δ(Xri−, ri , zi).

If coefficients αλ, βλ, χλ satisfy Conditions 1–3, the equation has a unique solutionXλ

t . Further, assertions of Theorems 3.3.1 and 3.3.2 are valid.

3.4 Lp-Estimates and Regularity of Solutions; C∞-Flows

Let us return to SDE (3.10). For its coefficients, we assume that V0(x, t), . . . ,

Vd ′(x, t) are C∞,1b -functions and g(x, t, z) satisfies Condition (J.1). Instead of the

above equation, consider the Itô SDE;

Xt = λ+∫ t

t0

α(Xr−, r) dWr +∫ t

t0

β(Xr−, r) dr

+∫ t

t0

∫

Rd′0

χ(Xr−, r, z)N(dr dz), (3.37)

where λ ∈ Rd , αik(x, r) = V i

k (x, r), χ(x, r, z) = g(x, r, z) and β(x, r) is defined

by (3.13). It is in fact a C∞,1b -function.

The functions α and β are uniformly Lipschitz continuous with respect to x.Further, since ∂xi χ(x, r, z) and ∂zk ∂xi χ(x, r, z) are bounded, for any p ≥ 2, thereexists a positive constant cp such that

( ∫ ∣∣∣χ(x, r, z)

γ (z)− χ(x′, r, z)

γ (z)

∣∣∣p

μ(dz)) 1

p ≤ cp|x − x′|

holds for any x, x′ and r . Therefore χ(x, r, z) satisfies the inequality (3.23).Then (3.37) is a master equation with parameter λ ∈ R

d and initial conditioncλ = λ. It has a unique solution Xλ

t , t ≥ t0 and the solution belongs to L∞− byTheorem 3.3.1.

Theorem 3.4.1 The family of solutions {Xλt ; λ ∈ R

d} of equation (3.37) has amodification, which is continuous with respect to λ and cadlag with respect to t a.s.

3.4 Lp-Estimates and Regularity of Solutions; C∞-Flows 97

The modification Xλt is continuously differentiable with respect to λ for all t and the

derivative is cadlag with respect to t a.s. Its derivative ∂λXλt satisfies

∂λXλt = I +

∫ t

t0

∂xα(Xλr−, r)∂λXλ

r− dWr +∫ t

t0

∂xβ(Xλr−, r)∂λXλ

r− dr

+∫ t

t0

∫

Rd′0

∂xχ(Xλr−, r, z)∂λXλ

r−N(dr dz). (3.38)

Further, for any T > t0 and p ≥ 2 there is a positive constant cp such that

E[‖Xλ‖pT ] ≤ cp(1 + |λ|)p, E[‖∂λXλ‖pT ] < cp. (3.39)

Proof Theorem 3.3.1 tells us that the solution Xλt has a modification such that it

is continuous in λ and cadlag with respect to t . For the differentiability, we willapply Theorem 3.3.2. We will again regard λ as a parameter. It satisfies propertiesof the master equation, by setting αλ(x, r) = α(x, r) etc. These coefficientssatisfy the condition of Theorem 3.3.2. Therefore Xλ

t has a modification which iscontinuously differentiable with respect to λ a.s. for any t . Further, the derivative∂λX

λt satisfies (3.38), since ∂λα = ∂λβ = ∂λχ = 0 holds in the formula (3.31).

Equation (3.39) follows from (3.29) and (3.32). ��We will next show the existence of higher derivatives of Xλ

t .

Theorem 3.4.2 For SDE (3.10), assume that diffusion and drift coefficients Vk, k =0, . . . , d ′ are C

∞,1b -functions of x, r and jump coefficient g satisfies Condition (J.1).

Then the solution starting from λ at time t0 has a modification Xλt , t ∈ T, λ ∈ Λ

which is infinitely differentiable with respect to λ and derivatives are cadlag withrespect to t a.s. Further, for any |i| ≥ 2, T > t0 and p ≥ 2 there is a positiveconstant c such that

supλ

E[‖∂ iλX

λ‖pT ] < c. (3.40)

Proof We saw in the previous lemma that Xλt , t ∈ T, λ ∈ Λ is continuously

differentiable with respect to λ and the derivative ∂Xλt := ∂λX

λt satisfies the

SDE (3.38). Further, it satisfies

E[‖∂Xλ − ∂Xλ′ ‖pT ] ≤ Cp|λ− λ′|p, (3.41)

because (3.35) holds with λ = λ, λ′ = λ′ and ε = ε′ = 0. We may regard (3.38)as an SDE for x = Xλ

t := ∂Xλt with initial data and random coefficients with

parameter λ given by

cλ = 1, αλ(x, r) = ∂xα(Xλr−, r)x, (3.42)

βλ(x, r) = ∂xβ(Xλr−, r)x, χλ(x, r, z) = ∂xχ(Xλ

r−, r, z)x.


We want to show the differentiability of Xλt with respect to λ. Note that ∂xα(Xλ

r−, r)etc. are bounded functional. We will consider the stopped process Xλ

t∧τ , where τ =τN is the stopping time defined by

τN = inf{t ≥ t0; sup|λ|≤K

|Xλt | > N} (= ∞ if {· · · } is empty),

where K is a positive constant. It holds that P(τN ≤ t) → 0 as N → ∞.Further, x = Xλ

t satisfies a master equation with coefficients α′λ(x, r), β

′λ(x, r) and

χ ′x(x, r, z), where

α′λ(x, r) =

{αλ(x, r), if r < τ,

0, if r ≥ τ,

etc. These coefficients are differentiable with respect to x and parameter λ andfurther derivatives ∂xα

′λ(x, r) and ∂λα

′λ(x, r) satisfy Conditions 1–3 of the master

equation. The first term is easily verified. We consider the second term. It holds that

∂λα′λ(x, r) = ∂x∂xα(X

λr∧τ , r)∂λX

λr∧τ x.

It is uniformly Lipschitz continuous with respect to x, since ∂x∂xα(Xλr∧τ )∂xX

λr∧τ is

bounded a.s. Further, it is weakly Lipschitz continuous with respect to the parameterλ. Hence they satisfy conditions of Theorem 3.3.2. Then Xλ

t∧τ is continuouslydifferentiable with respect to λ by Theorem 3.3.2. Let N tend to ∞. Then we findthat Xλ

t is continuously differentiable with respect to λ a.s.Now, the second derivative x = Xt = ∂ i

λXλt (|i| = 2) satisfies an another

linear SDE with random coefficients. Its initial data is 0 and diffusion coefficientis given by

αλ(x, r) := ∂ ixα(X

λr−, r)(∂λXλ

r−)2 + ∂xα(Xλr−, r)x. (3.43)

Drift coefficient and jump coefficient are given similarly. These coefficients satisfyConditions 1–3 of the master equation. Then we get (3.40) in the case |i| ≤ 2.

By a similar device, we can show that ∂ iλX

λt is again continuously differentiable

with respect to λ. Further, its derivatives satisfy (3.40) for |i| ≤ 3. Repeating thisargument inductively, we find that Xλ

t is infinitely continuously differentiable withrespect to λ for any t a.s. and (3.40) holds for any i. ��

For a continuous SDE, we get stronger assertions in Theorems 3.4.1 and 3.4.2.

Theorem 3.4.3 Let Xλ,st , t ∈ T, λ ∈ Λ be a solution of a continuous symmetric

SDE (3.1) starting from λ at time s. Then for any T > 0 and p ≥ 2, there is apositive constant c such that the inequality

E[|Xλ,st −X

λ′,s′t ′ |p] ≤ c

{|λ− λ′|p + |s − s′| p2 + |t − t ′| p2

}(3.44)

3.4 Lp-Estimates and Regularity of Solutions; C∞-Flows 99

holds for any s, s′, t, t ′ ∈ [0,∞) such that s ∨ s′ < t ∧ t ′ and λ, λ′ ∈ Rd . Further,

Xλ,st has a modification such that it is continuous in (λ, s, t), infinitely differentiable

with respect to λ and derivatives ∂ iλX

λ,st are continuous in (λ, s, t) a.s. for any i.

Proof We will show the inequality (3.44). Suppose s′ < s < t < t ′. Then Xλ′,s′t ′ =

XX

λ′,s′s ,s

t ′ holds a.s. Therefore,

E[|Xλ,st −X

λ′,s′t ′ |p]

≤ c1{E[E[|Xλ,st −X

z,st |p]|

z=Xλ′,s′s

] + E[|Xλ′,s′t −X

λ′,s′t ′ |p]}

≤ c2{E[|λ−Xλ′,s′s |p] + E[|Xλ′,s′

t −Xλ′,s′t ′ |p]}

≤ c3{|λ− λ′|p + E[|λ′ −Xλ′,s′s |p] + E[|Xλ′,s′

t −Xλ′,s′t ′ |p]}.

Here we used the inequality E[|Xλ,st −X

z,st |p] ≤ c2|λ− z|p. Since

Xλ′,s′s − λ′ =

∫ s

s′α(r) dWr +

∫ s

s′β(r) dr,

Xλ′,s′t −X

λ′,s′t ′ =

∫ t ′

t

α′(r) dWr +∫ t ′

t

β ′(r) dr

hold with bounded functionals α(r), β(r) etc., we get by the Burkholder–Davis–Gundy inequality,

E[|Xλ′,s′s − λ′|p] ≤ c4|s − s′| p2 , E[|Xλ′,s′

t −Xλ′,s′t ′ |p] ≤ c5|t − t ′| p2 .

Summing up these computations, we get (3.44). Then by the Kolmogorov–Totokitheorem, Xs,λ

t has a modification which is continuous in (λ, s, t).We can apply the similar argument for the processes ∂ i

λXλ,st and we find that

∂ iλX

λ,st is continuous in λ, s, t . ��

We saw in Theorem 3.4.2 that the solution Xλ,st of equation (3.37) starting from

λ at time s has a modification of a C∞-function of λ. We denote the modificationby Φs,t (λ). Then the maps λ ∈ R

d → Φs,t (λ) ∈ Rd are C∞-maps a.s. for any

s < t . Further, the equality Xλ,su = X

Xλ,st ,t

u holds a.s. for s < t < u by the path-wiseuniqueness of the solution. Therefore the family of maps {Φs,t } satisfies the equalityΦs,u(λ) = Φt,u(Φs,t (λ)) for any λ a.s. for any s < t < u.

In the following discussion, we write the spatial parameter λ ∈ Rd as x ∈ R

d .Then the family of maps {Φs,t ; 0 ≤ s < t < ∞} is a stochastic flow of C∞-mapsor simply C∞-flow.


Theorem 3.4.4 Consider a symmetric SDE with jumps given by (3.10). Solutionsof the SDE define a stochastic flow of C∞-maps {Φs,t }. Further, for any T > 0 andp ≥ 2 and multi-index i,

E[|Φs,t (x)− x|p], E[|∂ iΦs,t (x)|p] (3.45)

are bounded with respect to 0 < s < t < T and x.If the SDE is continuous, it defines a continuous stochastic flow of C∞-maps.

We want to prove that maps Φs,t ;Rd → Rd are diffeomorphic a.s. for any

s < t . However, discussions are not simple. The proof will be given in Sect. 3.6for continuous SDEs and in Sect. 3.8 for SDEs with jumps. Before we proceed tothe proof, we will develop the backward calculus. In the next section, we discussstochastic flows of C∞-maps generated by backward SDEs.

3.5 Backward SDE, Backward Stochastic Flow

We have so far discussed a continuous SDE (3.1) and an SDE with jumps (3.10);these equations describe the time evolution to the forward direction. However, weare also interested in solving these equations to the backward direction: For thispurpose, we need to define the backward SDEs. Solutions of backward SDEs willbe written as Xs , while solutions of forward ones have been written as Xt .

We will first consider a continuous backward symmetric SDE, which has thesame coefficients Vk(x, t), k = 0, . . . , d ′ of C∞,1-class as the forward SDE (3.1).Given t1 > 0 and x ∈ R

d , if a continuous backward Fs,t1 -semi-martingale Xs, s ≤t1 satisfies

Xs = x +d ′∑

k=0

∫ t1

s

Vk(Xr , r) ◦ dWkr (3.46)

for 0 ≤ s < t1, Xs, 0 ≤ s ≤ t1 is called a solution of the symmetricbackward equation with coefficients Vk(x, t) starting from x at time t1. Here◦dWk

r , k = 1, . . . , d ′ denote the backward symmetric integrals with respect toW k

r = Wkt −Wk

T , k = 1, . . . , d ′ and ◦dW 0r = dr . See Sect. 2.7.

The existence and the uniqueness of the solution of the backward equation canbe verified in the same way as those of the forward equation. Let Xx,t1

s , 0 ≤ s ≤ t1be the solution of the equation (3.46). Then it has a modification such that it is aC∞-function of x. We denote it by Φs,t1(x). Then x → Φs,t1(x);Rd → R

d areC∞-maps a.s. It satisfies the backward flow property Φs,u = Φs,t ◦ Φt,u a.s. for anyu > t > s ≥ 0. We call {Φs,t } a backward stochastic flow of C∞-maps defined bythe backward continuous symmetric SDE with coefficients Vk(x, t), k = 0, . . . , d ′.

3.6 Forward–Backward Calculus for Continuous C∞-Flows 101

We will next study the backward SDE with jumps. Let Vk(x, t), k = 0, . . . , d ′be time-dependent vector fields of C∞,1-class and let g(x, t, z) be a jump-map ofC∞,1,2-class, which are given in the definition of the forward SDE (3.10). Givent1 > 0 and x ∈ R

d , if a right continuous backward Fs,t1 semi-martingale Xs, s ≤ t1satisfies

Xs = x +d ′∑

k=0

∫ t1

s

Vk(Xr , r) ◦ dWkr (3.47)

+limε→0

{ ∫ t1

s

∫

|z|≥ε

g(Xr , r, z)N(dr dz)−d ′∑

k=1

bεk

∫ t1

s

Vk(Xr , r) dr},

for 0 ≤ s < t1, Xs, 0 ≤ s ≤ t1 is called a solution of the backward symmetric SDEwith coefficients (Vk(x, t), k = 0, . . . , d ′, g(x, t.z)) starting from x at time t1.

The existence and the uniqueness of the solution of the equation can be verified inthe same way as those of the forward equation. Let Xx,t1

s , 0 ≤ s ≤ t1 be the solutionof the equation. Then it has a modification such that it is a C∞-function of x. Wedenote it by Φs,t1(x). Then x → Φs,t1(x);Rd → R

d are C∞-maps a.s. It has thebackward flow property. {Φs,t } is called the backward stochastic flow of C∞-mapsdefined by the backward SDE with coefficients (Vk(x, t), k = 0, . . . , d ′, g(x, t, z)).

3.6 Forward–Backward Calculus for Continuous C∞-Flows

Associated with the continuous symmetric SDE (3.1), we define vector fields (first-order differential operators) with time parameter t by

Vk(t)f (x) =d∑

i=1

V ik (x, t)

∂f

∂xi(x), k = 0, . . . , d ′.

For a slowly increasing (or rapidly decreasing) C∞-function f , we define a second-order differential operator A(t) with time parameter 0 ≤ t < ∞ by

A(t)f (x) = 1

2

d ′∑

k=1

Vk(t)2f (x)+ V0(t)f (x). (3.48)

Then A(t)f (x) is a slowly increasing (or rapidly decreasing, respectively) C∞-function. We first discuss the forward differential calculus of the stochastic flowgenerated by a continuous SDE.


Proposition 3.6.1 Let {Φs,t } be a continuous stochastic flow of C∞-maps definedby a continuous symmetric SDE (3.1). It satisfies equation (3.6). Let f be a slowlyincreasing C∞-function. For each s, we have the forward differential rule withrespect to t:

f (Φs,t )=f+d ′∑

k=1

∫ t

s

Vk(r)f (Φs,r ) dWkr+

∫ t

s

A(r)f (Φs,r ) dr (3.49)

= f +d ′∑

k=0

∫ t

s

Vk(r)f (Φs,r ) ◦ dWkr . (3.50)

Proof Set φ(r) = (V1(Φs,r , r), . . . , Vd ′(Φs,r , r)) and υ(r) = V0(Φs,r , r). Then theSDE is written as Xt = x+∫ t

sφ(r)◦dWr+

∫ r

sυ(r) dr. We will apply Theorem 2.4.1

(Itô’s formula for symmetric integral). Since

∑

i

υi(r)∂f

∂xi(Φs,r ) = V0(r)f (Φs,r ),

∑

i

φik(r)∂f

∂xi(Φs,r ) = Vk(r)f (Φs,r ),

we get f (Φs,r ) = f +∑d ′k=0

∫ t

sVk(r)f (Φs,r ) ◦ dWk

r from (2.23).Next, since the martingale part of Vk(t)f (Φs,t ) is

∑

i,l

∫ t

s

∂Vk(r)f

∂xi(Φs,r )V

il (Φs,r , r) dW

lr ,

the symmetric integral is rewritten as

∫ t

s

Vk(r)f (Φs,r ) ◦ dWkr

=∫ t

s

Vk(r)f (Φs,r ) dWkr + 1

2

∑

i,l

⟨∫ t

s

∂Vk(r)f

∂xi(Φs,r )V

il (Φs,r , r) dW

lr ,W

kt

⟩

s,t

=∫ t

s

Vk(r)f (Φs,r ) dWkr + 1

2

∫ t

s

Vk(r)2f (Φs,r ) dr.

Therefore we get (3.49). ��Next, we will consider the stochastic flow {Φs,t } as a backward process of time

parameter s ∈ [0, t] starting from x at time t . Backward Itô integrals and backwardsymmetric integrals are defined in Sect. 2.7.

Let f be a C∞-function of x ∈ Rd . Then for any r < t , the composites f ◦Φr,t

are C∞-functions of x ∈ Rd , a.s. Then we may apply the differential operator Vk(r)

and A(r) to these functions. These are denoted by Vk(r)(f ◦Φs,t ) and A(r)(f ◦Φr,t ),respectively.

3.6 Forward–Backward Calculus for Continuous C∞-Flows 103

Proposition 3.6.2 Let {Φs,t } be the continuous stochastic flow of C∞-maps definedby the continuous symmetric SDE (3.1). For a slowly increasing C∞-function f , wehave the backward differential rule with respect to s:

f ◦Φs,t =f+d ′∑

k=1

∫ t

s

Vk(r)(f ◦Φr,t ) dWkr +

∫ t

s

A(r)(f ◦Φr,t ) dr (3.51)

= f +d ′∑

k=0

∫ t

s

Vk(r)(f ◦Φr,t ) ◦ dWkr . (3.52)

Proof Let Π = {s = rn < · · · < r0 = t} be a partition of the interval [s, t]. Wehave

f (Φs,t (x))− f (x) (3.53)

=n∑

m=1

{f (Φrm,t (x))− f (Φrm−1,t (x))}

=∑

i

{∑

m

∂

∂xi(f ◦Φrm−1,t )(x)(Φ

irm,rm−1

(x)− xi)}

+1

2

∑

i,j

{∑

m

∂2

∂xi∂xj(f ◦Φrm−1,t )(x)(Φ

irm,rm−1

(x)− xi)(Φjrm,rm−1(x)− xj )

}

+ O(Π).

Here we used the formula Φrm,t (x) = Φrm−1,t ◦ Φrm,rm−1(x). Let |Π | tend to 0.Then the term O(Π) converges to 0 in probability. Note that Φs,t (x) is a solutionof Itô’s SDE dXt = α(Xt , t) dWt + β(Xt , t) dt , where β(x, t) = V0(x, t) +12

∑k Vk(t)(Vk(t))(x). Then we have

Φirm,rm−1

(x)− xi =∑

k

αik(x, rm)(Wkrm−1

−Wkrm)+ βi(x, rm)(rm−1 − rm)+Om,

where∑

m Om → 0 in probability as |Π | → 0. In the above formula, we canreplace αik(x, rm) by αik(x, rm−1), since αik(x, r) is a C∞,1-function. Therefore,the above converges to the following backward integral:

∑

i,k

∫ t

s

∂

∂xi(f ◦Φr,t )(x)α

ik(x, r) dWkr +

∑

i

∫ t

s

∂

∂xi(f ◦Φr,t )(x)β

i(x, r) dr

+ 1

2

∑

i,j

∫ t

s

∂2

∂xi∂xj(f ◦Φr,t )(x)

(∑

k

αik(x, r)αjk(x, r))dr


=d ′∑

k=1

∫ t

s

Vk(r)(f ◦Φr,t )(x) dWkr +

∫ t

s

A(r)(f ◦Φr,t )(x) dr.

Therefore we get (3.51). Further, we have for 1 ≤ k ≤ d ′

∫ t

s

Vk(r)(f ◦Φr,t )(x) ◦ dWkr

=∫ t

s

Vk(r)(f ◦Φr,t )(x) dWkr + 1

2

∫ t

s

Vk(r)2(f ◦Φr,t )(x) dr,

in view of (2.48). From the above we get (3.52). ��Next, we consider the backward flow {Φs,t } defined by the backward continuous

SDE (3.46).

Proposition 3.6.3 Let {Φs,t } be the backward continuous stochastic flow of C∞-maps defined by the backward continuous symmetric SDE (3.46). Let f be a slowlyincreasing C∞-function on R

d . Then, we have the backward differential rule withrespect to s:

f (Φs,t ) = f +d ′∑

k=1

∫ t

s

Vk(r)f (Φr,t ) dWkr +

∫ t

s

A(r)f (Φr,t ) dr. (3.54)

Further, we have the forward differential rule with respect to t .

f ◦ Φs,t = f +d ′∑

k=1

∫ t

s

Vk(r)(f ◦ Φs,r ) dWkr +

∫ t

s

A(r)(f ◦ Φs,r ) dr. (3.55)

Equation (3.54) is shown similarly as the proof of (3.49), using Theorem 2.7.1.Equation (3.55) is shown similarly as the proof of (3.51).

3.7 Diffeomorphic Property and Inverse Flowfor Continuous SDE

Let us consider again the (forward) continuous stochastic flow {Φs,t } of C∞-mapsgenerated by the continuous SDE (3.1). In order to prove the diffeomorphic propertyof the flow Φs,t , we want to construct a stochastic process with parameters s, x ∈ R

d

denoted by Ψ ′s,t (x), which is continuous in x, t and satisfies

Φs,t (Ψ′s,t (x)) = Ψ ′

s,t (Φs,t (x)) = x, ∀x a.s.

3.7 Diffeomorphic Property and Inverse Flow for Continuous SDE 105

for any s < t . Then Φs,t (x);Rd → Rd should be diffeomorphisms a.s. Indeed, take

ω for which the equality Ψ ′s,t (Φs,t (x)) = Φs,t (Ψ

′s,t (x)) = x holds for all x ∈ R

d .Then for any x ∈ R

d , choose z = Ψ ′s,t (x). Then Φs,t (z) = x holds, so that the map

Φs,t is an onto map. Next suppose that Φs,t (x) = Φs,t (y) holds for some x, y. ThenΨ ′s,t (Φs,t (x)) = Ψ ′

s,t (Φs,t (y)) holds. Therefore we have x = y. This means that themap Φs,t ;Rd → R

d is one to one.Let φ(x) = (φ1(x), . . . , φd(x)) be a C1-map from R

d into itself. The d × d-matrix ∂φ(x) = ( ∂

∂xiφj (x))i=1,...,d,j=1,...,d is denoted by ∇φ(x) and is called the

Jacobian matrix. Further, the determinant of the Jacobian matrix is denoted bydet∇φ(x) and is called the Jacobian or Jacobian determinant of φ. Now, if theJacobian matrix ∇Φs,t (x) is invertible for all x, a.s., then the inverse map Φ−1

s,t

should be C∞ and hence Φs,t should be C∞-diffeomorphisms.In the next lemma, we will prove that ∇Φs,t (x) are invertible a.s and then in

Lemma 3.7.2, we will construct such a random field Ψ ′s,t (x).

Lemma 3.7.1 Let {Φs,t } be a continuous stochastic flow of C∞-maps generatedby a continuous symmetric SDE with coefficients Vk(x, t), k = 0, . . . , d ′. Then thefamily of Jacobian matrices {∇Φs,t (x)} satisfy the linear SDE

∇Φs,t (x) = I +d ′∑

k=0

∫ t

s

∇Vk(Φs,r (x), r)∇Φs,r (x) ◦ dWkr (3.56)

for any s < t and x, a.s. Further, ∇Φs,t (x) is invertible for any s < t, x almostsurely, and the inverse matrix Vs,t (x) satisfies

supx

E[|Vs,t (x)|p] < ∞ (3.57)

for any s < t and p ≥ 2.

Proof Differentiating each term of equation (3.6), we get (3.56). See Proposi-tion 2.4.3. Set Φt = Φs,t and ∇Φt = ∇Φs,t . Consider a stochastic differentialequation for matrix-valued process Vt with parameter x:

Vt = I −d ′∑

k=0

∫ t

s

Vr∇Vk(Φr(x), r) ◦ dWkr . (3.58)

It has a unique solution, which we denote by Vt (x). Then, using Itô’s formula forsymmetric integrals (Theorem 2.4.1), we have

◦d(Vt∇Φt) = ◦dVt∇Φt + Vt ◦ d∇Φt

= −d ′∑

k=0

Vt∇Vk∇Φt ◦ dWkt +

d ′∑

k=0

Vt∇Vk∇Φt ◦ dWkt = 0.


Therefore we have Vt (x)∇Φt(x) = I . This shows that the Jacobian matrix ∇Φt(x)

is invertible and the inverse Vt (x) = ∇Φt(x)−1 satisfies (3.58). The equation for Vt

satisfies the condition of the master equation with parameter λ = x. Then we getthe inequality (3.57) in view of Lemma 3.3.2. ��

Now, let us consider a forward SDE defined by

Xt = x −d ′∑

k=0

∫ t

s

∇Φs,r (Xr)−1Vk(Xr, r) ◦ dWk

r . (3.59)

It is an SDE with random coefficients ∇Φs,t (x)−1Vk(x, t), k = 0, . . . , d ′. We will

show that it has a unique solution Xx,st , t ≥ s. We fix s and M > 0. Let τn be the

first time t such that sup|x|≤M |∇Φs,t (x)−1| exceeds n. These are stopping times and

limn→∞ τn = ∞ holds a.s. We consider

dXnt = −

d ′∑

k=0

∇Φs,t∧τn(Xnt )

−1Vk(Xnt , t) ◦ dWk

t .

Then coefficients satisfy the Lipschitz condition. Therefore it is a master equation.Let Xn,x,s

t be the solution. It is a C∞-function of x. By the path-wise uniqueness ofthe solution, it holds that Xn,x,s

t = Xn+1,x,st for any |x| ≤ M if t < τn. Therefore

there exists Xx,st such that X

x,xt = X

n,x,st holds a.s. if t < τn. Then X

x,st is

the unique solution of equation (3.59). Further, the solution Xx,st of (3.59) has a

modification of C∞-maps. We denote it by Ψs,t (x).

Lemma 3.7.2 It holds that Φs,t (Ψs,t (x)) = Ψs,t (Φs,t (x)) = x for any x a.s. forany s < t .

Proof We denote Φs,t (x) and Ψs,t (x) by Φt(x) and Ψt (x), respectively. ThenF(x, t) ≡ Φt(x) is a stochastic process with parameter x belonging to the classLn+Lip,p

T(Rd) for any n, p. Further, it satisfies conditions of Theorem 2.4.2. Then,

setting F(x, t) = Φt(x) and Xt = Ψt (x), we have by Theorem 2.4.2,

◦dΦt(Ψt ) =d ′∑

k=0

Vk(Φt ◦ Ψt , t) ◦ dWkt +∇Φt(Ψt ) ◦ dΨt

=d ′∑

k=0

Vk(Φt ◦ Ψt , t) ◦ dWkt −

d ′∑

k=0

Vk(Φt ◦ Ψt , t) ◦ dWkt = 0.

Therefore we have Φt(Ψt (x)) = x for all x a.s. Apply Theorem 2.4.2 again forF(x, t) = Ψt (x) and Xt = Φt(x). Then we have

3.7 Diffeomorphic Property and Inverse Flow for Continuous SDE 107

◦dΨt (Φt ) = −d ′∑

k=0

(∇Φt)−1Vk(Ψt ◦Φt, t) ◦ dWk

t + ∇Ψt (Φt ) ◦ dΦt .

Since Φt ◦ Ψt (x) = x, we have ∇Φt(Ψt )∇Ψt = I . Therefore we have ∇Ψt =(∇Φt)(Ψt )

−1. Then we get ∇Ψt (Φt ) ◦ dΦt = ∑k(∇Φt)

−1Vk(Ψt ◦ Φt, t) ◦ dWkt .

Consequently, we get ◦dΨt (Φt ) = 0. This proves Ψt (Φt (x)) = x for all x a.s. ��By the above two lemmas, maps Φs,t ;Rd → R

d are diffeomorphic a.s. and Ψs,t

are inverse maps of Φs,t , i.e., Ψs,t (x) = Φ−1s,t (x) holds for any x a.s. We show that

Xs := Ψs,t (x) satisfies the backward SDE (3.46) with coefficients −Vk(x, t). Wehave Φs,t (Ψs,t (x)) = x for all x a.s. This implies ∇Φs,t (Ψs,t (x))∇Ψs,t (x) = I .Therefore we have ∇Φs,t (Ψs,t (x))

−1 = ∇Ψs,t (x). Then equation (3.59) can berewritten as

Ψs,t (x) = x −d ′∑

k=0

∫ t

s

∇Ψs,r (x)Vk(Ψs,r (x), r) ◦ dWkr . (3.60)

Consequently, we have by Theorem 2.4.1

f (Ψs,t (x)) = f (x)−d ′∑

k=0

∫ t

s

Vk(f ◦ Ψs,r )(x) ◦ dWkr ,

for any C∞-function f . Then Ψs,t (x) should satisfy the backward differential rule

f (Ψs,t (x)) = f (x)−d ′∑

k=0

∫ t

s

Vk(r)f (Ψr,t (x)) ◦ dWkr ,

in view of Proposition 3.6.3. The above equation shows that Ψs,t (x) satisfies thebackward symmetric SDE;

Ψs,t (x) = x −d ′∑

k=0

∫ t

s

Vk(Ψr,t (x), r) ◦ dWkr . (3.61)

It satisfies for any C∞-function f

f (Ψs,t ) = f (x)−d ′∑

k=0

∫ t

s

Vk(r)f (Ψr,t ) ◦ dWkr (3.62)

= f (x)−d ′∑

k=1

∫ t

s

Vk(r)f (Ψr,t ) dWkr +

∫ t

s

A(r)f (Ψr,t ) dr, (3.63)


where A(t) = 12

∑d ′k=1(−Vk(t))

2f + (−V0(t))f or simply

A(t)f = 1

2

d ′∑

k=1

Vk(t)2f − V0(t)f. (3.64)

Summing up these facts, we get the following theorem.

Theorem 3.7.1 Assume that coefficients Vk(x, t), k = 0, . . . , d ′ of a continuoussymmetric SDE (3.1) are C

∞,1b -functions. Then equation defines a continuous

stochastic flow of diffeomorphisms {Φs,t }. Set Ψs,t (x) = Φ−1s,t (x) for any x and

s < t . Then {Ψs,t } is a backward continuous stochastic flow and satisfies thebackward SDE (3.61). It satisfies (3.62)–(3.63).

Remark It is interesting to know that the equation for the forward flow givenby (3.6) and the backward equation for the inverse flow Ψs,t given by (3.7) aresymmetric. If we rewrite equations using Itô integrals, the symmetric propertydisappears. In fact, the symmetric equation (3.6) is equivalent to the Itô equation

Φs,t (x) = x +d ′∑

k=1

∫ t

s

Vk(Φs,r (x), r) dWkr +

∫ t

s

V ′0(Φs,r (x), r) dr, (3.65)

where V ′0(x, t) = V0(x, t) + 1

2

∑k≥1 Vk(t)Vk(x, t). Further, equation (3.61) is

equivalent to the backward Itô equation

Ψs,t (x) = x −d ′∑

k=1

∫ t

s

Vk(Ψr,t (x), r) dWkr −

∫ t

s

V ′′0 (Ψr,t (x), r) dr, (3.66)

where V ′′0 (x, t) = V0(x, t)− 1

2

∑k≥1 Vk(t)Vk(x, t).

Remark Consider a continuous forward SDE with coefficients −Vj (x, t), j =0, . . . , d ′: The equation is written as

Xt = X0 −d ′∑

k=0

∫ t

s

Vk(Xr, r) ◦ dWjr .

Theorem 3.7.1 tells us that the equation generates a stochastic flow of diffeomor-phisms, which we denote by {Ψs,t }. Set Ψ−1

s,t = Φs,t . The theorem tells us further

that Xt1,xs := Φs,t1(x) satisfies the backward SDE (3.46). Hence it coincides with

the backward stochastic flow {Φs,t } discussed in Sect. 3.5.

3.8 Forward–Backward Calculus for C∞-Flows of Jumps 109

Note The study of stochastic differential equations was initiated by Itô. In [50],stochastic differential equations driven by the Wiener process and Poisson randommeasure are discussed. Then continuous stochastic differential equations are studiedextensively.

The diffeomorphic property of maps x → Xxt received a lot of attention around

1980. We refer Elworthy [25], Malliavin [78], Bismut [8], Le Jan [75], Harris [36],Ikeda–Watanabe [41] and Kunita [59].

A method of proving the diffeomorphic property is to approximate SDEs by asequence of stochastic ordinary differential equations. It is described as follows.Instead of a Wiener process Wt , we consider a suitable sequence of piecewisesmooth stochastic processes Wn

t = (W1,nt , . . . ,W

d ′,nt ) which converges to Wt , and

consider a sequence of equations

d

dtXn

t =d ′∑

k=1

Vk(xnt , t)W

k,nt + V0(x

nt , t).

where Wk,nt = dW

k,nt

dt. Then its solution X

n,xt starting from x at time 0 defines a

diffeomorphic map for any n. The sequence {Xn,xt , n = 1, 2, . . .} should converge

to the solution Xxt of the SDE and it should be a diffeomorphic map. For detail, see

Ikeda–Watanabe [41], Bismut [8], Malliavin [78].Kunita [59] presented another method for proving the diffeomorphic property

through a skillful use of the Kolmogorov–Totoki theorem. In this monograph,we took another method by constructing a backward flow Ψs,t (x) which satisfiesLemma 3.7.2. An advantage of the new method is that it can be applied for provingthe diffeomorphic property of solutions of SDE on a manifold as in Sect. 7.1.

3.8 Forward–Backward Calculus for C∞-Flows of Jumps

We will consider stochastic flow of C∞-maps generated by a symmetric SDEwith jumps (3.10), where V0(x, t), . . . , Vd ′(x, t) are C

∞,1b -functions and g(x, t, z)

satisfies Condition (J.1). For a slowly increasing C∞-function f , we define anintegro-differential operator AJ (t) with time parameter t by

AJ (t)f (x) = A(t)f (3.67)

+∫

Rd′0

{f (φt,z(x))−f (x)− 1D(z)

d ′∑

k=1

zkVk(t)f (x)}ν(dz),

where A(t) is the differential operator defined by (3.48) and φt,z(x) = g(x, t, z)+x

and D = {z = (z1, . . . , zd′) ∈ R

d ′0 ; |z| ≤ 1}.


Proposition 3.8.1 Let {Φs,t } be a stochastic flow of C∞-maps defined by symmetricSDE (3.10) with jumps. Let f be a slowly increasing C∞-function on R

d . Then, thestochastic flow satisfies the forward differential rule with respect to t:

f (Φs,t ) = f +d ′∑

k=1

∫ t

s

Vk(r)f (Φs,r−) dWkr +

∫ t

s

AJ (r)f (Φs,r−) dr

+∫ t

s

∫

Rd′0

{f (φr,z ◦Φs,r−)− f (Φs,r−)

}N(dr dz) (3.68)

= f +d ′∑

k=0

∫ t

s

Vk(r)f (Φs,r ) ◦ dWkr

+ limε→0

{ ∫ t

s

∫

|z|≥ε


}N(dr dz)

−d ′∑

k=1

bkε

∫ t

s

Vk(r)f (Φs,r−) dr}. (3.69)

Proof The process Xt = Φs,t (x) is a solution of equation (3.10). Then it is an Itôprocess satisfying

Xt = x +∫ t

s

α(Xr, r) dWr +∫ t

s

β(Xr, r) dr +∫ t

s

∫χ(Xr−, r, z)ND(dr dz),

where

α(x, r) = (V1(x, r), . . . , Vd ′(x, r)), χ(x, r, z) = g(x, t, z),

β(x, r) = V0(x, r)+ 1

2

∑

k≥1

Vk(r)Vk(x, r)+∫

D

{g(x, r, z)−

∑

k≥1

zkVk(x, r)}ν(dz).

We apply Itô’s formula (Theorem 2.6.1) for the C3-function f . Then the diffusionpart of the stochastic process f (Xt ) is equal to

d∑

i=1

d ′∑

k=1

∫ t

s

αik(r)∂f

∂xi(Φs,r−) dWk

r =d ′∑

k=1

∫ t

s

Vk(r)f (Φs,r−) dWkr .


The jump part is equal to

∫ t

s

∫ {f (φr,z(Φs,r−))− f (Φs,r−)

}ND(dr dz)

=∫ t

s

∫ {f (φr,z(Φs,r−))− f (Φs,r−)

}N(dr dz)

+∫ t

s

∫

|z|>1

{f (φr,z(Φs,r−))− f (Φs,r−)

}n(dr dz).

The drift part is equal to

1

2

d∑

i,j=1

∫ t

s

( d ′∑

k=1

αik(r)αjk(r)) ∂2f

∂xi∂xj(Φs,r−) dr +

d∑

i=1

∫ t

s

βi(r)∂f

∂xi(Φs,r−) dr

+∫ t

s

∫

|z|<1

{f (φr,z(Φs,r−))− f (Φs,r−)−

∑

i

gi ∂f

∂xi(Φs,r−)

}n(dr dz)

=∫ t

s

(1

2

d ′∑

k=1

Vk(r)2f (Φs,r−)+ V0(r)f (Φs,r−)

)dr

+∫ t

s

∫

|z|<1

{f (φr,z(Φs,r−))− f (Φs,r−)−

d ′∑

k=1

zkVk(r)f (Φs,r−)}n(dr dz).

Summing these, we get the formula (3.68).Equation (3.69) follows from (3.68), rewriting Itô integrals by symmetric

integrals. ��Proposition 3.8.2 Let {Φs,t } be a stochastic flow of C∞-maps defined by thesymmetric SDE (3.10) with jumps. Let f be a slowly increasing C∞-function. Thenfor each t , the flow {Φs,t } has a modification which is right continuous with respectto s a.s. and the modification satisfies the backward differential rule with respectto s:

f ◦Φs,t = f +d ′∑

k=1

∫ t

s


∫ t

s

AJ (r)(f ◦Φr,t ) dr

+∫ t

s

∫ {f ◦Φr,t ◦ φr,z − f ◦Φr,t

}N(dr dz) (3.70)


= f +d ′∑

k=0

∫ t

s

Vk(r)(f ◦Φr,t ) ◦ dWkr

+ limε→0

{ ∫ t

s

∫

|z|≥ε

{f ◦Φr,t ◦ φr,z − f ◦Φr,t

}N(dr dz)

−∫ t

s

d ′∑

k=1

bkε Vk(r)(f ◦Φr,t ) dr}. (3.71)

In order to prove the proposition, we will approximate Φs,t (x) by a family ofstochastic flows {Φε

s,t (x), 0 < ε < 1} generated by SDEs:

Xt = x +d ′∑

k=1

∫ t

s


∫ t

s

V ε0 (Xr, r) dr

+∫ t

s

∫

|z|≥ε

g(Xr−, r, z)N(dr dz), (3.72)

where V ε0 = V0 −∑d ′

k=1 bkε Vk . The equation cuts off all jumps g(Xr−, r, z), |z| ≤ ε

from equation (3.10). Its Lévy measure νε(dz) = 1(ε,∞)(z)ν(dz) has finite mass.Let Φε

s,t (x) be stochastic flows generated by the above SDE. Let Φ0s,t (x) be the

stochastic flow generated by the continuous SDE

Xt = x +d ′∑

k=1

∫ t

s

Vk(Xr, r) ◦ dWir +

∫ t

s

V ε0 (Xr, r) dr. (3.73)

We saw in (3.18) that, taking s = t0, Φεs,t (x) is written as

Φεs,t (x) =

{Φ0

s,t (x), if t < τ1,

Φ0τn,t

◦ φτn,Sn ◦ · · · ◦ φτ1,S1 ◦Φ0s,τ1

(x), if τn ≤ t < τn+1,(3.74)

where τm, Sm are random variables defined in Sect. 3.2. It is a right continuousstochastic flow of diffeomorphisms. Further, it is a backward semi-martingale withrespect to s. Then we have

f (Φεs,t (x))− f (x)

=∑

m

{f (Φε

τm−,t (x))− f (Φετm−1,t

(x))}+

∑

m

{f (Φε

τm−,t (x))− f (Φετm,t (x)

}

= I ε1 + I ε2 .

Since Φεs,t (x) does not have jumps at s ∈ (τm−1, τm), we have by Proposition 3.6.2:


I ε1 =d ′∑

k=1

∫ t

s

Vk(r)(f ◦Φεr,t )(x) dW

kr +

∫ t

s

Aε(r)(f ◦Φεr,t )(x) dr,

where Aε(r) = 12

∑k≥1 Vk(r)

2 + V ε0 (r). Further, we have

I ε2 =∫ t

s

∫

|z|≥ε

{f ◦Φε

r,t ◦ φr,z(x)− f ◦Φεr,t (x)

}N(dr dz).

Therefore Φεs,t (x) satisfies the equality (3.70) replacing AJ (r) by

AεJ (r) = Aε(r)+

∫

|z|>ε

{f ◦ φr,z − f }ν(dz)

and N(dr dz) by Nε(dr dz) = 1(ε,∞)(|z|)N(dr dz).We will study the convergence of Φε

s,t (x) as ε → 0.

Lemma 3.8.1 For any p ≥ 2, 0 ≤ s < t ≤ T and i, we have

limε→0

supx′∈Rd

E[|∂ iΦεs,t (x

′)− ∂ iΦs,t (x′)|p] = 0. (3.75)

Proof We will consider the case where |i| = 0, 1. The case |i| ≥ 2 will be shownsimilarly. We shall consider pair processes Ys,t (x

′) = (Φs,t (x′), ∂Φs,t (x

′)) andY εs,t (x

′) = (Φεs,t (x

′), ∂Φεs,t (x

′)), regarding 0 < ε < 1 as a parameter. The latter can

be regarded as a solution of the master equation (3.21) on Rd×R

d2with coefficients

αε(x, y, r) = (α(x, r), ∂α(x, r)y), βε(x, y, r) = (βε(x, r), ∂βε(x, r)y),

χε(x, y, r, z) = (χ(x, r, z)1|z|≥ε, ∂χ(x, r, z)y1|z|≥ε),

and x = Φεs,t (x

′), y = ∂Φεs,t (x

′), where

βε(x, r) = V0(x, r)+ 1

2

d ′∑

k=1

Vk(r)Vk(x, r)

+∫

|z|≥ε

{g(x, r, z)− 1D(

d ′∑

k=1

zkVk(x, r))}ν(dz).

It holds that |βε(x, y, r)− β(x, y, r)| ≤ cϕ0(ε)(1 + |y|) and

∫

Rd′0

∣∣∣χ

γ (z)− χε

γ (z)

∣∣∣p

μ(dz) =∫

|z|<ε

∣∣∣g

|z|∣∣∣p

μ(dz) ≤ cϕ0(ε)(1 + |y|)p,

where ϕ0(ε) = μ({|z| < ε}) = ∫|z|<ε

|z|2ν(dz).


We shall estimate the integral E[|Y εs,t (x

′) − Ys,t (x′)|p] for p > 2. Set Y ε

t =Y εs,t (x

′) and Yt = Ys,t (x′). The drift term of Y ε

s,t (x′)− Ys,t (x

′) is estimated as

∣∣∣∫ t

s

(βε(Yεr , r)− β(Yr , r)) dr

∣∣∣

≤∫ t

s

|βε(Yεr , r)− β(Y ε

r , r)| dr +∫ t

s

|β(Y εr , r)− β(Yr , r)| dr

≤ cϕ0(ε)

∫ t

s

(1 + |Y εr |) dr + c

∫ t

s

|Y εr − Yr | dr.

Since E[|Y εr |p] are uniformly bounded with respect to 0 < s < t < T and x, ε, we

have

E[∣∣∣

∫ t

s

(βε(Yεr , r)− β(Yr , r)) dr

∣∣∣p] ≤ c1ϕ0(ε)

p + c2

∫ t

s

E[|Y εr − Yr |p] dr.

Similar estimates are valid for the diffusion term and jump term of Y εs,t (x

′) −Ys,t (x

′). We have

E[∣∣∣

∫ t

s

(αε(Y

εr , r)− α(Yr , r)

)dWr

∣∣∣p] ≤ c′1ϕ0(ε)

p + c′2∫ t

s

E[|Y εr − Yr |p] dr,

E[∣∣∣∫ t

s

∫{χε(Y

εr , r, z)−χ(Yr , r, z)}N(dr dz)

∣∣∣p]

≤ c′′1ϕ0(ε)+ c′′2∫ t

s

E[|Y εr −Yr |p] dr.

Consequently, for any p ≥ 2 there exist c1 > 0, c2 > 0 such that

E[|Y εt − Yt |p] ≤ c1(ϕ0(ε)

p + ϕ0(ε))+ c2

∫ t

s

E[|Y εr − Yr |p] dr.

Then Gronwall’s inequality implies

E[|Y εt − Yt |p] ≤ C1(ϕ0(ε)

p + ϕ0(ε)).

Since this is valid for any x′, s, we get (3.75) in the case |i| = 0, 1. ��We are interested in the uniform convergence of ∂ iΦε

s,t (x) with respect to x, s.For this problem, we need Sobolev’s inequality in Lp(D, dx)-space, where D isa bounded domain in R

d . Let f (x) be a real function on D. We denote the weakderivative of f by ∂ i by ∂ if . We denote by Dk,p the set of all f such that its weakderivatives ∂ if , |i| ≤ k belong to Lp(D, dx). We quote Morrey’s Sobolev inequalitywithout proof.


Lemma 3.8.2 (Brezis, [14]) Let D be a bounded subdomain in Rd with smooth

boundary. Let p > d. Then elements of Dk,p are Ck−1-functions. Further, thereexists a positive constant ck,p such that

∑

|i|≤k−1

supx∈D

|∂ if (x)| ≤ ck,p

( ∑

|i|≤k

∫

D

|∂ if (x′)|p dx′) 1

p(3.76)

holds for any f ∈ Dk,p.

Lemma 3.8.3 For any s < t , there exists a sequence {εn} converging to 0 suchthat solutions {Φεn

s,t (x)} and their derivatives {∂ iΦεns,t (x)} converge to Φs,t (x) and

∂ iΦs,t (x), uniformly on compact sets with respect to x a.s.

Proof By Lemma 3.8.2, we have

E[ sup|x|≤M

|Φs,t (x)−Φεs,t (x)|p] (3.77)

≤ CE[ ∫

|x′|≤M

|Φs,t (x′)−Φε

s,t (x′)|p dx′ +

∫

|x′|≤M

|∂Φs,t (x′)− ∂Φε

s,t (x′)|p dx′

].

It converges to 0 as ε → 0 in view of Lemma 3.8.1. Therefore a subsequence{Φεn

s,t (x)} converges to Φs,t (x) uniformly on compact sets with respect to x, t a.s.The convergence of {∂ iΦ

εns,t (x)} can be shown similarly. ��

Proof of Proposition 3.8.2 Let us consider the sequence of {Φεns,t (x)} obtained in

Lemma 3.8.3. We saw that each Φεns,t (x) is right continuous in s and satisfies (3.70)

replacing N(dr dz) by Nεn(dr dz). Therefore it is a right continuous backwardsemi-martingale with respect to s. Let n tend to ∞. Then terms

d ′∑

k=1

∫ t

s

Vk(r)(f ◦Φεnr,t )(x) dW

kr ,

∫ t

s

AεnJ (r)(f ◦Φ

εnr,t )(x) dr, etc.

should converge a.s. Therefore the limit Φs,t (x) is a backward semi-martingale withrespect to s for any t, x, and it has a right continuous modification. We denote it byΦs,t (x), again. Then the above integrals should converge to

d ′∑

k=1

∫ t

s

Vk(r)(f ◦Φr,t )(x) dWkr ,

∫ t

s

AJ (r)(f ◦Φr,t )(x) dr, etc.

Therefore {Φs,t } satisfies (3.70).Finally, the equality (3.71) follows from (3.70), if we rewrite backward symmet-

ric integrals using backward Itô integrals. ��


We shall next consider differential rules for the backward flow {Φs,t } definedby the backward symmetric SDE (3.47). The following is the counterpart ofProposition 3.8.1, whose proof is omitted.

Proposition 3.8.3 Let {Φs,t } be the backward stochastic flow of C∞-maps definedby the backward symmetric SDE (3.47). Let f be a slowly increasing C∞-functionon R

d . Then, we have the backward differential rule with respect to s:

f (Φs,t ) = f +d ′∑

k=1

∫ t

s

Vk(r)f (Φr,t ) dWkr +

∫ t

s

AJ (r)f (Φr,t ) dr (3.78)

+ limε→0

∫ t

s

∫

|z|≥ε

{f (φr,z ◦ Φr,t )− f (Φr,t )

}N(dr dz)

= f +d ′∑

k=0

∫ t

s

Vk(r)f (Φr,t ) ◦ dWkr (3.79)

+ limε→0

{ ∫ t

s

∫

|z|≥ε

{f (φr,z ◦ Φr,t )− f (Φr,t )}N(dr dz)

−d ′∑

k=1

bkε

∫ t

s

Vk(r)f (Φr,t ) dr}.

3.9 Diffeomorphic Property and Inverse Flow for SDEwith Jumps

In this section, we assume further that the jump-map g(x, t, z) of the SDE (3.10)satisfies Condition (J.2) and hence φt,z(x) = g(x, t, z) + x are diffeomorphic forany t, z. We will discuss the diffeomorphic property of the solution of the SDE. Letε > 0. Associated with equation (3.10), we consider a forward SDE (3.72). Thesolution defines a right continuous stochastic flow {Φε

s,t } of diffeomorphisms.

Lemma 3.9.1 Set Ψ εs,t (x) = (Φε

s,t )−1(x). Then Xε

s := Ψ εs,t (x), 0 ≤ s < t < ∞

satisfies the following backward equation with jumps:

Xεs = x −

d ′∑

k=0

∫ t

s

Vk(Xεr , r) ◦ dWk

r (3.80)

−{ ∫ t

s

∫

|z|≥ε

h(Xεr , r, z)N(dr dz)−

d ′∑

k=1

bkε

∫ t

s

Vk(Xεr , r) dr

},

where h(x, t, z) = x − φ−1t,z (x) and

Vk(x, t) = ∂zkg(x, t, z)

∣∣∣z=0

= ∂zkh(x, t, z)

∣∣∣z=0

.

3.9 Diffeomorphic Property and Inverse Flow for SDE with Jumps 117

Proof Since the flow Φεs,t is represented by (3.74), the inverse flow Ψ ε

s,t (x) isrepresented by

{Ψ 0s,t (x), if t < τ1

Ψ 0s,τ1

◦ φ−1τ1,S1

◦ · · · ◦ φ−1τn,Sn

◦ Ψ 0τn,t

(x), if τn ≤ t < τn+1.(3.81)

It is cadlag with respect to s and t , since Φεs,t (x) is cadlag with respect to s and t .

Note that Xs = Ψ 0s,t satisfies the continuous backward SDE

Xs = x −d ′∑

k=1

∫ t

s

Vk(Xr , r) ◦ dWkr −

∫ t

s

V ε0 (Xr , r) dr.

Then the composite process Xεs = Ψ ε

s,t (x) should satisfy the backward SDE

Xεs = x −

d ′∑

k=1

∫ t

s

Vk(Xεr , r) ◦ dWk

r −∫ t

s

V ε0 (X

εr , r) dr

−∫ t

s

∫

|z|≥ε

h(Xεr , r, z)N(dr dz). (3.82)

(We have shown a similar fact for a forward equation in Sect. 3.2.) The aboveequation can be rewritten as (3.80). ��Theorem 3.9.1 Assume that diffusion and drift coefficients of symmetricSDE (3.10) are C

∞,1b -functions and jump coefficients satisfy Conditions (J.1)

and (J.2). Then the equation defines a right continuous stochastic flow ofdiffeomorphisms {Φs,t }.

Let {Ψs,t } be the backward flow of C∞-maps generated by a backward symmetricSDE (3.47) with coefficients (−Vk(x, t), k = 0, . . . , d ′,−h(x, t, z)). Then it holdsthat Ψs,t (x) = Φ−1

s,t (x) for any x a.s. for any s < t .

Proof We saw in Lemma 3.8.3 that for any s < t , there is a sequence {εn}converging to 0 such that Φ

εns,t (x) converges to Φs,t (x) uniformly on compact

sets with respect to x almost surely. Then by the same reasoning, the sequenceΨ

εns,t (x) converges to Ψs,t (x) uniformly on compact sets almost surely. Since

Φεs,t (Ψ

εs,t (x)) = Ψ ε

s,t (Φεs,t (x)) = x holds for all x almost surely, we get

Φs,t (Ψs,t (x)) = Ψs,t (Φs,t (x)) = x for all x almost surely. This implies that mapsΦs,t ;Rd → R

d are one to one and onto a.s. for any s < t . Further, since maps Ψs,t

are smooth a.s., maps Φs,t are diffeomorphic a.s.Now we will fix t0. By the above argument, Φ−1

t0,t(x) exists for any t , a.s. Further,

Φ−1t0,t

(x) should be cadlag with respect to t for any x. Finally, we define for any s < t ,


Φs,t = Φt0,t ◦Φ−1t0,s

. Then it is right continuous in t and further, a diffeomorphic mapfor any s < t a.s. Therefore {Φs,t } is a right continuous stochastic flow generatedby the SDE (3.10).

Since Xεs = Ψ ε

s,t satisfies the backward SDE (3.82), the limit Ψs,t satisfies thebackward SDE with coefficients (−Vk(x, t), k = 0, . . . , d ′,−h(x, t, z)). ��

So far, we assumed that coefficients of the SDE are not big; we assumed thatcoefficients and their derivatives are bounded. Then we saw that the stochasticflow {Φs,t } belongs to L∞− (Theorem 3.4.1). The property is needed for provingthat Φs,t (x) is a smooth Wiener–Poisson functional in the Malliavin calculus. SeeChap. 6.

We will relax conditions for jump-maps φt,z in SDE (3.10) for a while. Insteadof Conditions (J.1) and (J.2), we introduce the following weaker conditions.

Condition (J.1)K . (i) The function g(x, t, z) := φt,z(x)−x is of C∞,1,2b -class on

Rd×T×{|z| ≤ c} for some c > 0. (ii) It is of C∞

b -class on Rd for any t ∈ T, |z| ≥ c

and is piecewise continuous in (t, z) ∈ T× {|z| > c}.Condition (J.2)K . For any t, z, the map φr,z;Rd → R

d is a diffeomorphism ofR

d . Further, h(x, t, z) := x − φ−1t,z (x) satisfy Condition (J.1)K .

Theorem 3.9.2 Assume Conditions (J.1)K and (J.2)K for jump -maps φt,z. Thenthe assertion of Theorem 3.9.1 is valid. Further, the integro-differential operatorAJ (t) of (3.67) is well defined for any f ∈ C∞

b , and further, the stochastic flowsatisfies equations (3.68), (3.69), (3.70) and (3.71) for any f ∈ C∞

b .

Proof We truncate big jumps and consider gc(x, t, z) = g(x, t, z)1|z|≤c andφct,z(x) = x + gc(x, t, z) instead of g(x, t, z), φt,z. Then the SDE with coefficients

(Vk(x, t), k = 0, . . . , d ′, gc(x, t, z)) defines a stochastic flow of diffeomorphisms{Φc

s,t (x)} in view of Theorem 3.9.1. We define a stochastic flow Φs,t by

Φs,t (x) ={Φc

s,t (x), if t < τ1,

Φcτn,t

◦ φτn,Sn ◦ · · · ◦ φτ1,S1 ◦Φcs,τ1

(x), if τn ≤ t < τn+1.

Here τn, n = 1, 2, . . . are jumping times of the Poisson process Nt with intensityλc := ν({|z| > c}) with initial time s, and Sn, n = 1, 2, . . . are independentrandom variables with the identical distribution νc = ν1|z|>c/λc. Then {Φs,t }is a right continuous stochastic flow of diffeomorphisms generated by SDE withcoefficients (Vk(x, t), g(x, t, z)). Further, the inverse Ψs,t = Φ−1

s,t satisfies thebackward equation with coefficients (−Vk(x, t), k = 0, . . . , d ′,−h(x, t, z)).

The operator AJ (t) is well defined if we restrict f in C∞b . Equations (3.68)–

(3.71) for f ∈ C∞b can be verified in the same way as in the proof of Proposi-

tions 3.8.1 and 3.8.2. ��The stochastic flow Φs,t (x) of the above theorem may not be Lp-bounded. (3.45)

may be infinite.We are also interested in the case where diffusion coefficients are unbounded. In

such a case, the solution of the SDE may explode in finite time. In Chap. 7, we will

3.10 Simple Expressions of Equations; Cases of Weak Drift and Strong Drift 119

discuss SDEs on manifolds, where the solution may explode in finite time. Resultsin Chap. 7 can be applied to SDEs on Euclidean spaces with unbounded coefficients.

Note The flow property of solutions for SDE with jumps was first studied byFujiwara–Kunita [30]. Then, it was developed further by Fujiwara–Kunita [31]and Kunita [60]. Some technical conditions posed for jump coefficient g(x, t, z)

in [30, 31] are removed in [60]. The latter is closed to the conditions in this book.However, concerning the proof of the regularity of the solution with respect to theinitial data, we took another method introducing the master equation. Further, for theproof of the diffeomorphic property, a method different from these works is taken,approximating SDE by a sequence of SDE with finite Lévy measure.

3.10 Simple Expressions of Equations; Cases of Weak Driftand Strong Drift

Let ν be a Lévy measure. If∫

0<|z|≤1 |z|ν(dz) is finite, then d ′-vector b =∫0<|z|≤1 zν(dz) is well defined. It is called a strong drift of the Lévy measure ν.

If ν is a finite measure, it has a strong drift. Suppose next that ν may not have thestrong drift. For a given 0 < ε < 1, we define d ′-vector bε = (b1

ε , . . . , bd ′ε ) by

bjε = ∫

ε<|z|≤1 zj ν(dz), where z = (z1, . . . , zd

′) ∈ R

d ′ . If b0 = limε→0 bε existsand is finite, the Lévy measure ν is said to have a weak drift b0.

So far we studied SDEs with jumps in the case where Lévy measures ν forPoisson random measures are most general. However, if the Lévy measure ν hasa weak drift, the equation can be written in a simple form.

To see this, we will reconsider the integral by Poisson random measure, assumingthat its Lévy measure ν has a weak drift. Let ψ(r, z) be a bounded measurablefunction on U. The improper integrals of ψ with respect to the measures n and N

are defined by

∫ t

s

∫

|z|>0+ψ(r, z)n(dr dz) := lim

ε→0

∫ t

s

∫

|z|>ε

ψ(r, z) drν(dz),

∫ t

s

∫

|z|>0+ψ(r, z)N(dr dz) := lim

ε→0

∫ t

s

∫

|z|>ε

ψ(r, z)N(dr dz),

if the right-hand sides exist and are finite almost surely, respectively.We introduce

L0U=

{ψ(r, z); predictable, C2-class in z, ψ(r, 0) = 0 for any r ∈ T,

∂kz ψ(r, z), k ≤ 2 are bounded in (r, z) a.s.}. (3.83)

Then L0U

is a subset of L∞−U

.


Proposition 3.10.1 Suppose that the Lévy measure ν has a weak drift.Let ψ(u) is an element of the functional space L0

Udefined by (3.83). Then the

improper integrals∫ t

s

∫|z|>0+ ψ(r, z)N(dr dz) and

∫ t

s

∫|z|>0+ ψ(r, z) drν(dz) exist.

Further, both are integrable with respect to P and satisfy

∫ t

s

∫

Rd′0

ψ(r, z)N(dr dz) (3.84)

=∫ t

s

∫

|z|>0+ψ(r, z)N(dr dz)−

∫ t

s

∫

|z|>0+ψ(r, z) drν(dz).

In particular, if ν has a strong drift, the above improper integrals are replaced byLebesgue integrals.

Proof We have ψ(r, z) = ∂zψ(r, 0) ·z+ 12 (∂

2z ψ(r, θz)z ·zT , by Taylor’s expansion.

Therefore,

∫ t

s

∫

1≥|z|>ε

ψ(r, z) dr dν =∫ t

s

∂zψ(r, 0) dr ·∫

1≥|z|>ε

zν(dz)

+1

2

∫ t

s

∫

1≥|z|>ε

∂2z ψ(r, θz)z · zT dr dν. (3.85)

The first term of the right-hand side converges as ε → 0, since ν has a weak drift.The second term of the right-hand side converges, since the function ∂2

z ψ(r, θz)z·zTis integrable on {|z| ≤ 1} with respect to the measure dr dν. Therefore (3.85)converges as ε → 0. The limit is the improper integral

∫ t

s

∫1≥|z|>0+ ψ(r, z) drν(dz).

Then improper integral∫ t

s

∫|z|>0+ ψ(r, z) drν(dz) is also well defined and it is

integrable with respect to P . Further, in the equality

∫ t

s

∫

|z|>ε

ψ(r, z)N(dr dz) =∫ t

s

∫

|z|>ε

ψ(r, z)N(dr dz)−∫ t

s

∫

|z|>ε

ψ(r, z) drν(dz),

the left-hand side converges, since ψ ∈ L∞−U

. Therefore∫ t

s

∫|z|>ε

ψ(r, z)N(dr dz)

should converge to the improper integral∫ t

s

∫|z|>0+ ψ(r, z)N(dr dz) and (3.84)

holds. Since∫ t

s

∫|z|>0+ ψ(r, z) drν(dz) is integrable,

∫ t

s

∫|z|>0+ ψ(r, z)N(dr dz) is

also integrable with respect to P . ��Now, for the Itô process Xt of (2.38), we assume that φ(t) is a semi-martingale

with bounded jumps and the symmetric integral by the Wiener process is welldefined and that ψ(r, z) ∈ L0

U. Then Itô process (2.38) is rewritten as

Xt = X0 +∫ t

t0

φ(r)◦dWr +∫ t

t0

υ(r) dr+∫ t

t0

∫

|z|>0+ψ(r, z)N(dr dz). (3.86)


For this Itô process, Itô’s formula is rewritten simply, which looks like a formula ofusual differential-difference calculus. Formula (2.39) is rewritten as follows.

Theorem 3.10.1 Assume that the Lévy measure has a weak drift. Let Xt, t ≥ t0 bean Itô process represented by (3.86). Let f (x, t) be a function of C3,1-class. Thenwe have for any t0 ≤ s < t < ∞,

f (Xt , t) = f (Xs, s)+∫ t

s

∂f

∂t(Xr, r) dr

+d∑

i=1

d ′∑

k=1

∫ t

s

∂f

∂xi(Xr, r)φ

ik(r) ◦ dWkr +

d∑

i=1

∫ t

s

∂f

∂xi(Xr, r)υ

i(r) dr

+∫ t

s

∫

|z|>0+{f (Xr− + ψ(r, z), r)− f (Xr−, r)}N(dr dz). (3.87)

Proof Rewrite the symmetric integral of the above using the Itô integral. Nextrewrite the last integral with Poisson random measure N , using the compensatedPoisson random measure N . Then we can show similarly to the proof of The-orem 2.6.1 that the right-hand side of the above is equal to the right-hand sideof (2.39). ��

Next, we consider the backward Itô process written as

Xt = X0+∫ t1

t

φ(r)◦dWr +∫ t1

t

υ(r) dr+∫ t1

t

∫

|z|>0+ψ(r, z)N(dr dz). (3.88)

Formula (2.50) is rewritten as follows.

Theorem 3.10.2 Assume that the Lévy measure has a weak drift. Let Xt be abackward Itô process represented by (3.88). Let f (x, t) be a function of C3,1-class.Then we have for any t0 ≤ s < t < ∞,

f (Xs, s) = f (Xt , t)−∫ t

s

∂f

∂t(Xr , r) dr

+d∑

i=1

d ′∑

k=1

∫ t

s

∂f

∂xi(Xr , r)φ

ik(r) ◦ dWkr +

d∑

i=1

∫ t

s

∂f

∂xi(Xr , r)υ

i(r) dr

+∫ t

s

∫

|z|>0+{f (Xr + ψ(r, z), r)− f (Xr , r)}N(dr dz). (3.89)

Let us reconsider SDE (3.10) on Rd . By Proposition 3.10.1, we have


limε→0

{ ∫ t

t0

∫

|z|≥ε


k=1

bkε

∫ t

t0

Vk(Xr−, r) dr}

=∫ t

t0

∫

|z|>0+g(Xr−, r, z)N(dr dz)−

d ′∑

k=1

bk0

∫ t

t0

Vk(Xr−, r) dr.

Therefore, rewriting the vector V0(t) − ∑k b

k0Vk(t) by V0(t) and setting φt,z(x) =

g(x, t, z)+ x, we can rewrite equation (3.10) simply as

Xt = X0 +d ′∑

k=0

∫ t

t0

Vk(Xr, r) ◦ dWkr

+∫ t

t0

∫

|z|>0+{φr,z(Xr−)−Xr−}N(dr dz). (3.90)

If the Lévy measure has a strong drift, the last improper integral coincides with theLebesgue integral. This expression of the SDE interprets the geometric meaning ofthe solution more than expression (3.10). Indeed, if the Lévy measure is of finitemass, the solution is given by the stochastic flow Φs,t represented by (3.18), i.e., theflow is written as composites of stochastic flow Φ0

s,t generated by the continuousSDE dXt = ∑

k Vk(Xt , t)◦dWkt and diffeomorphic maps φt,z. If the Lévy measure

has infinite mass, the solution should be the limit of flows given by (3.18).Further, for the operator AJ (t) the last term of (3.67) is equal to

∫

|z|>0+{f ◦ φt,z−f

}ν(dz)−

d ′∑

k=1

bk0Vk(t)f.

Then, rewriting V0(t) − ∑k b

k0Vk(t) as V0(t) again, the operator AJ (t) given

by (3.67) is rewritten in a simple form:

AJ (t)f = A(t)f +∫

|z|>0+{f ◦ φt,z−f }ν(dz). (3.91)

If the Lévy measure has a strong drift, the last improper integral coincideswith the Lebesgue integral. We call the triple (Vk(t), k = 0, . . . , d ′, φt,z, ν) thecharacteristics of the SDE (3.90).

Let {Φs,t } be the stochastic flow generated by the SDE (3.90). Then, rules offorward and backward calculus given in (3.69) and (3.71) are rewritten simply as


f (Φs,t ) = f +d ′∑

k=0

∫ t

s

Vk(r)f (Φs,r ) ◦ dWkr (3.92)

+∫ t

s

∫

|z|>0+{f (φr,z ◦Φs,r−)− f (Φs,r−)

}N(dr dz),

= f +d ′∑

k=1

∫ t

s

Vk(r)f (Φs,r−) dWkr +

∫ t

s

AJ (r)f (Φs,r ) dr

+∫ t

s

∫

|z|>0


}N(dr dz), (3.93)

f ◦Φs,t = f +d ′∑

k=0

∫ t

s

Vk(r)(f ◦Φr,t ) ◦ dWkr

+∫ t

s

∫

|z|>0+{(f ◦Φr,t ) ◦ φr,z − f ◦Φr,t

}N(dr dz) (3.94)

= f +d ′∑

k=1

∫ t

s


∫ t

s

AJ (r)(f ◦Φr,t ) dr

+∫ t

s

∫

|z|>0

{(f ◦Φr,t ) ◦ φr,z − f ◦Φr,t

}N(dr dz). (3.95)

Next, we will consider a backward SDE, which governs the inverse flow Ψs,t =Φ−1

s,t . Rewriting V0(t) − ∑k b

k0Vk(t) as V0(t) again, the backward SDE for Xs =

Ψs,t1(x) can be rewritten as

Xs = x−d ′∑

k=0

∫ t1

s

Vk(Xr , r)◦dWkr −

∫ t1

s

∫

|z|>0+{Xr−φ−1

r,z (Xr )}N(dr dz). (3.96)

We define an integro-differential operator AJ (t) by

AJ (t)f = A(t)f +∫

|z|>0+{f ◦ φ−1

t,z −f }ν(dz), (3.97)

where A(t) is the differential operator defined by (3.64). Then the backwarddifferential rule (3.78) for the inverse flow Ψs,t is rewritten as


f ◦ Ψs,t = f −d ′∑

k=0

∫ t

s

Vk(r)f (Ψr,t ) ◦ dWkr

+∫ t

s

∫

|z|>0+{f (φ−1

r,z ◦ Ψr,t )− f (Ψr,t )}N(dr dz) (3.98)

= f +∫ t

s

AJ (r)f (Ψr,t ) dr −d ′∑

k=1

∫ t

s

Vk(r)f (Ψr,t ) dWkr

+∫ t

s

∫

|z|>0

{f (φ−1

r,z ◦ Ψr,t )− f (Ψr,t )}N(dr dz). (3.99)

Chapter 4Diffusions, Jump-Diffusions and HeatEquations

Abstract We study diffusions and jump-diffusions on a Euclidean space deter-mined by SDE studied in Chap. 3. We select topics which are related to thestochastic flow generated by the SDE; topics are concerned with heat equationsand backward heat equations.

In Sects. 4.1, 4.2, 4.3, and 4.4, we consider diffusion processes on a Euclideanspace. In Sect. 4.1, we show that a stochastic flow generated by a continuoussymmetric SDE defines a diffusion process. Its generator A(t) is representedexplicitly as a second order linear partial differential operator with time parametert , using coefficients of the SDE. Kolmogorov’s forward and backward equationassociated with the operator A(t) will be derived. In Sect. 4.2, we discuss expo-nential transformation of the diffusion process by potentials. It will be shownthat solutions of various types of backward heat equations will be obtained byexponential transformations by potentials. In Sect. 4.3, we study backward SDE andbackward diffusions. We will see that Kolmogorov equations for backward diffusionwill give the solution of heat equations. In Sect. 4.4, we present a new method ofconstructing the dual (adjoint) semigroup, making use of the geometric propertyof diffeomorphic maps Φs,t of stochastic flows. The method will present a cleargeometric explanation of the adjoint operator A(t)∗ and the dual semigroup. We willsee that the dual semigroup will be obtained by a certain exponential transformation(Feynman–Kac–Girsanov transformation) of a backward diffusion.

In Sects. 4.5 and 4.6, we consider jump-diffusions on a Euclidean space. We willextend results for diffusion studied in Sects. 4.2, 4.3, and 4.4 to those for jump-diffusions. We show that the dual semigroup of jump-diffusion is well defined ifthe jump coefficients are diffeomorphic. In Sect. 4.7, we return to a problem ofthe stochastic flow. We discuss the volume-preserving property of stochastic flowsby applying properties of dual jump-diffusions. In Sect. 4.8, we consider processeskilled outside of a subdomain of an Euclidean space and its dual processes.


125


https://doi.org/10.1007/978-981-13-3801-4_4

126 4 Diffusions, Jump-Diffusions and Heat Equations

4.1 Continuous Stochastic Flows, Diffusion Processesand Kolmogorov Equations

We will study diffusion processes on a Euclidean space determined by a continuousstochastic differential equations. Let us consider a continuous symmetric SDE onR

d :

dXt =d ′∑

k=0

Vk(Xt , t) ◦ dWkt , (4.1)

where Vk(x, t), k = 0, . . . , d ′ are C∞,1b -functions of (x, t) ∈ R

d × T and Wt =(W 1

t , . . . ,Wd ′t ), t ∈ T is a d ′-dimensional Wiener process. Here, T = [0,∞),

W 0t ≡ t and V0(Xt , t) ◦ dW 0

t ≡ V0(Xt ) dt , conventionally. Let {Φs,t } be thestochastic flow of diffeomorphisms generated by the above continuous SDE. ThenXt = X

x,st = Φs,t (x), t ≥ s is a solution of equation (4.1) starting from x at time s.

Proposition 4.1.1 Laws of Φs,t (x):

Ps,t (x, E) = P(Φs,t (x) ∈ E), 0 ≤ s < t < ∞, E ∈ B(Rd) (4.2)

are determined uniquely from SDE (4.1). Further, {Ps,t (x, E)} is a transitionprobability.

Proof We consider the equation (4.1) with the initial condition t0 = s and X0 = x.Using the Itô integral it is written as Xt = x+∫ t

sα(Xr, r) dWr+

∫ t

sβ(Xr, r) dr . We

saw in Sect. 3.3 that the equation has a pathwise unique solution (Theorem 3.3.1).Further, the solution Xt is approximated successively; setting X0

s = x and

Xnt = Xn−1

s +∫ t

s

α(Xn−1r , r) dWr +

∫ t

s

β(Xn−1r , r) dr, n = 1, 2, . . . ,

Xnt converges to the solution Xt in Lp-sense, in view of Lemma 3.3.1. Since the

law of Xnt is uniquely determined by coefficients α, β and x, the law of Xt is also

determined uniquely from these coefficients and initial condition.For a bounded continuous function f , we set

Ps,tf (x) :=∫

Rd

Ps,t (x, dy)f (y) = E[f (Φs,t (x))]. (4.3)

Let {Fs,t , 0 ≤ s < t < ∞} be the two-sided filtration generated by the Wienerprocess Wt . Then Φt,u(y) is independent of Fs,t for any s < t < u. Therefore wehave the semigroup property

4.1 Continuous Stochastic Flows, Diffusion Processes and Kolmogorov. . . 127

Ps,uf (x) = E[E[f (Φt,u(Φs,t (x)))

∣∣Fs,t

]] = E[E[f (Φt,u(y))

]∣∣y=Φs,t (x)

]

= E[Pt,uf (Φs,t (x))

] = Ps,tPt,uf (x),

for s < t < u. We have further, limt↓s Ps,tf (x) = f (x), since Φs,t (x) → x ast ↓ s. Therefore, {Ps,t (x, E)} is a transition probability. ��

Now, for a given (x, t0) ∈ Rd × T, let us consider the stochastic process with

parameter x, t0 given by Xx,t0t = Φt0,t (x). In view of the flow property Φs,t ◦Φt0,s =

Φt0,t , the equality Xx,t0t = Φs.t ◦ X

x,t0s holds a.s. for any t0 < s < t . Since Φs,t (x)

and Ft0,s are independent, we have

P(Xx,t0t ∈ E|Ft0,s) = Ps,t (X

x,t0s , E) a.s.

Therefore the stochastic process Xx,t0t , t ∈ [t0,∞) is a continuous conservative

Markov process on Rd with semigroup {Ps,t }. Further, if f is a bounded continuous

function, Ps,tf (x) is also a bounded continuous function of x, since the flowΦs,t (x) is continuous in x a.s. Therefore, X

x,t0t has the strong Markov property

(Proposition 1.7.1). Then Xx,t0t is a diffusion process of the initial state (x, t0)

with semigroup {Ps,t }. Consequently, {Xx,st } is a system of diffusion processes with

semigroup {Ps,t }.A continuous function f on R

d is called slowly increasing if |f (x)|/(1 + |x|)mis a bounded function for some positive integer m. A Ck-function f is called slowlyincreasing if for any |i| ≤ k, |∂ if (x)|/(1 + |x|)m are bounded for some positiveinteger m. A C∞-function f is called slowly increasing if it is a slowly increasingCk-function for any k. We denote by O(Rd) the set of all real slowly increasingC∞-functions on R

d .Let f (x) be a slowly increasing continuous function. Let T > 0. Since

E[|Φs,t (x)|p] ≤ cp(1 + |x|)p holds for any 0 ≤ s < t ≤ T , x ∈ Rd and

p ≥ 2 (Theorem 3.4.1), the integral Ps,tf (x) = E[f (Φs,t (x))] is well definedfor any s < t and x. Further, it is a slowly increasing function of x. Suppose furtherthat f (x) is a slowly increasing C∞-function (or C∞

b function, respectively). SinceE[|∂ iΦs,t (x)|p] is bounded with respect to x for any p ≥ 2 (Theorem 3.4.1), wecan change the order of the derivative and expectation. In fact, E[f (Φs,t (x))] is aC∞-function of x and satisfies

∂ iE[f (Φs,t (x))] = E[∂ i(f ◦Φs,t (x))]

for any i. Therefore Ps,tf (x) is a slowly increasing C∞-function of x (or C∞b -

function of x, respectively) for each s < t . Hence {Ps,t } is the family of lineartransformations on O(Rd) (or on C∞

b (Rd)), satisfying the semigroup propertyPs,tPt,uf = Ps,uf for any s < t < u.

Associated with SDE (4.1), we define differential operators (vector fields)Vk(t), k = 0, . . . , d ′ by Vk(t)f (x) = ∑

k Vik (x, t)

∂f∂xi

and we define a second-orderpartial differential operator A(t) with time parameter t by


A(t)f (x) = 1

2

d ′∑

k=1

Vk(t)2f (x)+ V0(t)f (x). (4.4)

If f is a slowly increasing C∞-function (or C∞b -function), then both Vk(t)f and

A(t)f (x) are slowly increasing C∞-functions (or C∞b -functions, respectively).

Let ft,u(x), f (x, t) be C∞-functions of x with parameters t, u. Denote themby fλ(x), regarding {t, u}, t as parameter λ. Apply the differential operatorsVk(t), A(t) to the function fλ(x). The functions Vk(t)fλ(x), A(t)fλ(x) are denotedby Vk(t)ft,u(x), A(t)ft,u(x), Vk(t)f (x, t), A(t)f (x, t), respectively.

Theorem 4.1.1 Solutions of the continuous symmetric SDE (4.1) define a systemof conservative diffusion process with semigroup {Ps,t }. If f is a slowly increasingC∞-function, Ps,tf (x) is also a slowly increasing C∞-function for any s < t .Further, it is continuously differentiable with respect to s, t for s < t and it satisfiesthe following forward and backward partial differential equations, respectively:

∂

∂tPs,tf (x) = Ps,tA(t)f (x), if t > s, (4.5)

∂

∂sPs,tf (x) = −A(s)Ps,tf (x), if t > s. (4.6)

Equation (4.5) is called Kolmogorov’s forward equation and (4.6) is calledKolmogorov’s backward equation. Equation (4.6) shows that the function v(x, s) :=Ps,tf (x), s < t is a solution of the backward heat equation

∂

∂sv(x, s) = −A(s)v(x.s)

with the final condition lims↑t v(x, s) = f (x).

Proof Let f be a slowly increasing C∞-function. Then for any s < t and x, wehave the forward differential calculus (Proposition 3.6.1):

f (Φs,t (x)) = f (x)+d ′∑

k=1

∫ t

s

Vk(r)f (Φs,r (x)) dWkr +

∫ t

s

A(r)f (Φs,r (x)) dr.

(4.7)In equation (4.7), terms written in the stochastic integrals as dWk

r are martingaleswith mean 0. Therefore, taking the expectation of each term of (4.7), we have

Ps,tf (x) = f (x)+∫ t

s

E[A(r)f (Φs,r (x))] dr

= f (x)+∫ t

s

Ps,rA(r)f (x) dr. (4.8)

4.2 Exponential Transformation and Backward Heat Equation 129

Therefore, the function Ps,tf (x) is continuously differentiable with respect to t .Differentiating both sides, we get the equality (4.5).

We saw in Proposition 3.6.2 the rule of the backward differential calculus for theflow {Φs,t };

f (Φs,t (x)) = f (x)+d ′∑

k=1

∫ t

s

Vk(r)(f ◦Φr,t )(x) dWkr +

∫ t

s

A(r)(f ◦Φr,t )(x) dr.

(4.9)Take the expectation of each term of (4.9). Since the second term of the right-handside is a backward martingale, its expectation is 0. We shall consider the last term.We have

E[ ∫ t

s

A(r)(f ◦Φr,t )(x) dr]=

∫ t

s

E[A(r)(f ◦Φr,t )(x)

]dr

=∫ t

s

A(r)E[f (Φr,t (x))

]dr.

In the last equality, we used ∂ iE[f (Φs,t (x))] = E[∂ i(f ◦Φs,t (x))]. Then we get

Ps,tf (x) = f (x)+∫ t

s

A(r)Pr,tf (x) dr. (4.10)

Differentiate both sides with respect to s (< t) we get (4.6). ��

4.2 Exponential Transformation and Backward HeatEquation

We are interested in partial differential equations (4.5) and (4.6) for more generaltypes of operator. Let A(t) be the differential operator defined by (4.4). Letck(x, t), k = 0, . . . , d ′ be C

∞,1b -functions on R

d×T. We will transform the operatorA(t) by the functions c ≡ {c0, . . . , cd ′ }. We define another differential operatorAc(t) by

Ac(t)f = 1

2

d ′∑

k=1

(Vk(t)+ ck(t))2f + (V0(t)+ c0(t))f. (4.11)

Here, (Vk(t)+ ck(t)), k = 0, . . . , d ′ are linear operators defined by

(Vk(t)+ ck(t))f (x) =∑

i

V ik (x, t)

∂f

∂xi(x)+ ck(x, t)f (x)


and (Vk(t)+ ck(t))2f (x) = (Vk(t)+ ck(t)){(Vk(t)+ ck(t))f }(x). Then Ac(t)f (x)

is rewritten as

Ac(t)f (x) = A(t)f (x)+d ′∑

k=1

ck(x, t)Vk(t)f (x)+ c(x, t)f (x),

where

c(x, t) = c0(x, t)+ 1

2

d ′∑

k=1

(Vk(t)ck(x, t)+ ck(x, t)

2). (4.12)

We consider an exponential functional with coefficients c = {c0, . . . , cd ′ }:

Gs,t (x) = Gcs,t (x) = exp

{ d ′∑

k=0

∫ t

s

ck(Φs,r (x), r) ◦ dWkr

}. (4.13)

Lemma 4.2.1 For any i, p ≥ 2 and T > 0, E[|∂ iGs,t (x)|p] is bounded with respectto 0 ≤ s < t ≤ T and x ∈ R

d .

Proof Apply Itô’s formula (Theorem 2.4.1) for the function f (x) = ex and Xt =∑k

∫ t

sc′k(r) ◦ dWk

r , where c′k(r) := ck(Φs,r (x), r). Then we have

Gs,t = 1 +d ′∑

k=0

∫ t

s

Gs,rc′k(r) ◦ dWk

r

= 1 +d ′∑

k=0

∫ t

s

Gs,rc′k(r) dW

kr + 1

2

d ′∑

k=1

〈Gs,t c′k(t),W

kt −Wk

s 〉s,t .

For the computation of the last term, note that the martingale part of the continuoussemi-martingale Gs,t c

′k(t) is

∑

k≥1

∫ t

s

Gs,rVk(r)ck(Φs,r , r) dWkr +

∑

k≥1

∫ t

s

Gs,rc′k(r)c

′k(r) dW

kr

in view of Itô’s formula. Then we have for k = 1, . . . , d ′

〈Gs,t c′k(t),W

kt −Wk

s 〉 =∫ t

s

Gs,r {Vk(r)ck(Φs,r , r)+ ck(Φs,r , r)2} dr.


Therefore,

Gs,t = 1 +d ′∑

k=1

∫ t

s

Gs,rck(Φs,r , r) dWkr + 1

2

∫ t

s

c(Φs,r , r) dr.

Define

αx(y, r) = yc(Φs,r (x), r), βx(y, r) = yc(Φs,r (x), r), γx(x, r, z) = 0,

where c(x, r) = (c1(x, r), . . . , cd ′(x, r)) and c(x, r) is given by (4.12). Thesefunctionals satisfy Conditions 1–3 of the master equation with parameter x. Further,Yt = Y

x,st := Gs,t (x) satisfies the equation

Yx,st = 1 +

∫ t

s

αx(Yx,sr , r) dWr +

∫ t

s

βx(Yx,sr , r) dr.

Then in view of Theorem 3.3.1, E[|Yx,st |p] is bounded with respect to x, s for any

p ≥ 2.Since Y

x,st = Gs,t (x) satisfies the above equation, the derivative ∂Gs,t (x) =

(∂x1Gs,t (x), . . . , ∂xdGs,t (x)) satisfies

∂Gs,t=∫ t

s

(∂Gs,rc(Φs,r , r)+Gs,r∂c(Φs,r , r)) dWr

+∫ t

s

(∂Gs,r c(Φs,r , r)+Gs,r∂c(Φs,r , r)) dr.

Since c(Φs,r (x), r), c(Φs,r (x), r) and ∂c(Φs,r (x), r), ∂c(Φs,r (x), r) are boundedwith respect to r and x (x is a parameter), coefficients of the equation satisfy Condi-tions 1–3 of the master equation. Therefore E[|∂Gs,t (x)|p] should be bounded withrespect to 0 ≤ s < t ≤ T and x ∈ R

d by Theorem 3.3.1. Repeating this argumentinductively, we arrive at the assertion of the lemma. ��

Now, Yt = Gs,t satisfies the linear SDE:

dYt =d ′∑

k=0

ck(Xt , t)Yt ◦ dWkt . (4.14)

Let R+ be the open half line (0,∞). We shall consider an SDE on the product spaceR

d ×R+ for the process (Xt , Yt ) ∈ R

d ×R+ defined by the pair of equations (4.1)

and (4.14). Then the solution starting from (x, y) ∈ Rd × R

+ at time s is given by(Φs,t (x),Gs,t (x)y). For any p ≥ 2, there exists cp > 0 such that

E[|Φs,t (x)|p + |Gs,t (x)y|p] ≤ cp(1 + |x| + |y|)p, ∀0 ≤ s < t ≤ T .


For a slowly increasing C∞-function f on Rd × R

+, we can define the integral

Ps,t f (x, y) = E[f (Φs,t (x),Gs,t (x)y)]

for any s, t, x and y. It is a slowly increasing C∞-function. We can apply thearguments of Sect. 4.1 to the diffusion process (Xt , Yt ) = (Φs,t (x),Gs,t (x)y). Thenits generator A(t) is written by

A(t)f = 1

2

d ′∑

k=1

(Vk(t)+ ck(t)y

∂

∂y

)2f +

(V0(t)+ c0(t)y

∂

∂y

)f .

Now take f (x, y) = f (x)y, where f (x) is a slowly increasing C∞ function on Rd .

Apply it to the operator A(t). Then Ac(t)f (x) = A(t)f (x, 1) holds.We will define another transition function on R

d and its transformation by

P cs,t (x, E) := E[1E(Φs,t (x))G

cs,t (x)],

P cs,t f (x) :=

∫P cs,t (x, dx

′)f (x′) = E[f (Φs,t (x))Gcs,t (x)]. (4.15)

It is the transformation of Ps,tf (x) by the exponential functional Gcs,t (x). Since

{Ps,t } satisfies the semigroup property Ps,t Pt,u = Ps,u for any s < t < u,{P c

s,t } satisfies the semigroup property P cs,tP

ct,u = P c

s,u. Further, for any boundedcontinuous function f , P c

s,t f converges to f as t → s. Hence {P cs,t (x, E)} is a

transition function. It is called the transition function of Φs,t (x) weighted by c orsimply a weighted transition function.

Since Kolmogorov’s forward and backward equations are valid for the diffusionprocess (Xt , Yt ) = (Φs,t (x),Gs,t (x)y) on R

d×R+, we have the following assertion

in view of Theorem 4.1.1.

Theorem 4.2.1 If f is a slowly increasing C∞-function of x, P cs,t f (x) is also a

slowly increasing C∞-function of x. Further, it is continuously differentiable withrespect to s, t for s < t and satisfies the following forward and backward partialdifferential equations, respectively:

∂

∂tP cs,t f (x) = P c

s,tAc(t)f (x), (4.16)

∂

∂sP cs,t f (x) = −Ac(s)P c

s,t f (x). (4.17)

Equation (4.16) is called Kolmogorov’s forward equation and (4.17), Kol-mogorov’s backward equation.

As an application of the above theorem, we will consider the final value problemfor a backward heat equation. Let 0 < t1 < ∞ be a fixed time, called the finaltime. Let f1(x) be a slowly increasing continuous function on R

d . We want to find


a function v(x, s) of C2,1-class defined on Rd × (0, t1) satisfying the following

backward equation:

⎧⎨

⎩

∂

∂sv(x, s) = −Ac(s)v(x, s), ∀0 < s < t1,

lims↑t1 v(x, s) = f1(x), ∀x ∈ Rd .

(4.18)

If the above function v(x, s) exists, it is called a solution of the final value problemfor the backward heat equation associated with the operator Ac(t).

Theorem 4.2.2 If f1 is a slowly increasing C∞-function, the final value problemfor the backward heat equation (4.18) has a unique slowly increasing solutionv(x, s). The solution is of C∞,1-class. It is represented by

v(x, s) = E[f1(Φs,t1(x)) exp

{ d ′∑

k=0

∫ t1

s


}]. (4.19)

Proof The function v(x, s) given by (4.19) coincides with P cs,t1

f1(x). SinceP cs,t1

f1(x) satisfies (4.17), v should be a solution of equation (4.18). It is a C∞-function of x, since v(x, s) = E[f1(Φs,t1(x))G

cs,t1

(x)] is infinitely differentiable

with respect to x. Further, we have ∂ ixv(x, s) = ∂ i

x

(f1(x)+

∫ t1s

Ac(r)P cr,t1

f1(x) dr).

Therefore ∂ ixv(x, s) is continuously differentiable with respect to s.

We give the uniqueness of the solution. Let v′(x, s) be any slowly increasingsolution of equations (4.18) of C2,1-class. Set v0(x, s) = v(x, s) − v′(x, s). Thenwe have by Itô’s formula,

v0(Φs,t , t)Gcs,t = v0(x, s)+

d ′∑

k=1

∫ t

s

(Vk(r)+ ck(r))v0(Φs,r , r)Gcs,r dW

kr

+∫ t

s

{Ac(r)v0(Φs,r , r)+ ∂v0

∂r(Φs,r , r)

}Gc

s,r dr.

Since v0(x, s) is a solution of (4.18), Ac(s)v0(x, s) + ∂∂sv0(x, s) = 0 holds.

Therefore Mt = v0(Φs,t (x), t)Gcs,t (x) is a local martingale. Further, since v0 is

of polynomial growth, E[|Mt |p] < ∞. Therefore Mt is actually a martingale.Then we have Mt = E[MT |Ft ] = 0, because MT = 0 holds a.s. Thereforev0(Φs,t (x), t) = 0 holds for all x a.s. Let Ψs,t (x) be the inverse map ofΦs,t (x). Then we have v0(Φs,t ◦ Ψs,t (x), t) = 0 a.s. for any x. Therefore weget v0(x, t) = 0 for any x. Since this is valid for any t , v0(x, t) should beidentically 0. ��

A semigroup {Ps,t } is said to be of C2,1-class if for any C2b -function f , Ps,tf (x)

is a C2,1 function of x, s for any t . The above theorem shows that for a given partialdifferential operator Ac(t), the semigroup of C2,1-class satisfying Kolmogorov’s


backward equation exists uniquely. The semigroup is said to be generated by theoperator Ac(t) and Ac(t) is called the generator of the semigroup. In particular,when c = 0, a system of diffusion processes {Xx,s

t = Φs,t (x); (x, s) ∈ Rd×T} with

the semigroup {Ps,t } is called a diffusion process with the generator A(t)(= A0(t))

or a system of diffusion processes generated by the operator A(t). Further, sincethe law of Φs,t (x) is uniquely determined by the operator A(t), {Φs,t } is called thestochastic flow associated with the operator A(t).

Remark It is not obvious whether the semigroup satisfying Kolmogorov’s forwardequation is unique or not. However, if the semigroup is defined by a diffusionprocess, is of C2,1-class and satisfies Kolmogorov’s forward equation, then it isunique. We will prove this fact in the case c = 0. Let Xt is the diffusion processwith the semigroup Ps,t mentioned above. Then Ns = Ps,tf (Xs) is a martingale,because for s < u < t ,

E[Nu|Fs] = E[Pu,tf (Xu)|Fs] = Ps,uPu,tf (Xs) = Ps,tf (Xs) = Ns.

Note that the operator A(t) is written as

A(t)f (x) = 1

2

∑

i,j

αij (x, t)∂2f

∂xi∂xj(x)+

∑

i

βi(x, t)∂f

∂xi(x), (4.20)

where

αij (x, t) =d ′∑

k=1

V ik (x, t)V

jk (x, t), β(x, t) = V0(x, t)+ 1

2

d ′∑

k=1

Vk(t)Vk(x, t).

Set βir = βi(Xr, r) and βr = (β1

r , . . . , βdr ), Since f (Xt ) −

∫ t

t0A(r)f (Xr) dr is a

martingale, both

Xt −∫ t

t0

br dr and XitX

jt −

∫ t

t0

Xirβ

jr dr −

∫ t

t0

Xjr β

ir dr −

∫ t

t0

αij (Xr, r) dr

are martingales. On the other hand, by Itô’s formula (Theorem 2.2.1),

XitX

jt = Xi

t0X

jt0+

∫ t

t0

Xir dX

jr +

∫ t

t0

Xjr dXi

r + 〈Xi,Xj 〉t0,t .

Comparing these equations, we find that

〈Xi,Xj 〉t0,t =∫ t

t0

αij (Xr, r) dr.

Now, apply Itô’s formula to the function f (x, s) = Ps,tf (x) and x = Xs . Then wehave


Nt = f (Xt0 , t0)+∫ t

t0

{ ∂

∂rf (Xr, r)+ A(r)f (Xr, r)

}dr + a local martingale.

Therefore the integral∫ t

t0{· · · } dr is a continuous martingale, so that it should be 0

a.s. Therefore the integrand {· · · } is 0 a.s. Since this is valid for any Xt , we have{ ∂∂r

+ A(r)}f (x, r) = 0. This shows that f (x, s) = Ps,tf satisfies Kolmogorov’sbackward equation. Then the semigroup is unique by Theorem 4.2.1.

Now, we will rewrite the exponential functional Gs,t (x) using the Itô integral.Setting c′k(r) = ck(Φs,r (x), r), k = 0, . . . , d ′ and c′(r) = c(Φs,r (x), r), we have

d ′∑

k=0

∫c′k(r) ◦ dWk

r =d ′∑

k=1

∫ t

s

c′k(r) dWkr + 1

2

d ′∑

k=1

〈c′k(t),Wkt 〉s,t +

∫ t

s

c′0(r) dr

=d ′∑

k=1

∫ t

s

c′k(r) dWkr − 1

2

∫ t

s

d ′∑

k=1

c′k(r)2 dr +∫ t

s

c′(r) dr.

Therefore, Gcs,t (x) can be decomposed as the product of G(0)

s,t and G(1)s,t , where

G(0)s,t = exp

{ ∫ t

s

c(Φs,r , r) dr},

G(1)s,t = exp

{ d ′∑

k=1

∫ t

s

ck(Φs,r , r) dWkr − 1

2

d ′∑

k=1

∫ t

s

ck(Φs,r , r)2 dr

}.

Set P (0)s,t f (x) = E[f (Φs,t (x))G

(0)s,t (x)]. Then {P (0)

s,t } is a semigroup and satisfies

P(0)s,t f (x) = f (x)+

∫ t

s

P (0)s,r (A(r)+ c(r))f (x).

Hence by the transformation Ps,t → P(0)s,t , the potential term c(t) is added to

the generator. The generator is changed from A(t) to the operator A(0)(t) =A(t) + c(t). It is called the Feynman–Kac transformation. Next, G(1)

s,t is a positive

martingale with mean 1. In fact, apply Proposition 2.2.3 to Zt ≡ G(1)s,t . Since

∑d ′k=1

∫ t

sck(Φs,r (x), r)

2 dr belongs to L∞−, Zt is a positive L∞−-martingale with

mean 1. Then we can define another probability measure P (1) by dP (1) = G(1)s,t dP .

With respect to P (1), W kt = Wk

t − ∫ t

sck(Φs,r , r) dr, k = 1, . . . , d ′ is a Wiener

process by Girsanov’s theorem (Theorem 2.2.2). Then Xt = Φs,t (x) satisfies

Xit = xi +

d ′∑

k=0

∫ t

s

V ik (Xr, r) ◦ dW k

r +d ′∑

k=1

∫ t

s

ck(Xr, r)Vik (Xr, r) dr, a.s. P (1).


Therefore, with respect to P (1), Xt = Φs,t (x) is a diffusion process with the

generator A(1)(t)f = A(t)f + ∑d ′k=1 ck(t)Vk(t)f. Hence by the transformation

Ps,t → P(1)s,t , the drift term (first-order differential operator)

∑k ck(t)Vk(t) is added

to the generator. The generator is changed from A(t) to the operator A(1)(t). It iscalled the Girsanov transformation. Consequently, our exponential transformationcoincides with the product of the Feynman–Kac transformation and the Girsanovtransformation. We will call it the Feynman–Kac–Girsanov transformation. Equa-tion (4.19) is called the Feynman–Kac–Girsanov formula.

We are also interested in the initial value problem for a heat equation associatedwith the operator Ac(t). Let 0 ≤ t0 < ∞ be a fixed time, called the initial time. Letf0(x) be a slowly increasing continuous function on R

d . We want to find a functionu(x, t) of C2,1-class defined on R

d × (t0,∞) satisfying the equation

⎧⎨

⎩

∂

∂tu(x, t) = Ac(t)u(x, t), ∀t0 < t < ∞,

limt↓t0 u(x, t) = f0(x), ∀x ∈ Rd .

(4.21)

If such a function exists, it is called a solution of the initial value problem for theheat equation associated with the operator Ac(t).

The problem can be solved in time homogeneous case, using the Feynman–Kac–Girsanov formula.

Theorem 4.2.3 For the operator Ac(t), we assume that Vk(t), k = 1, . . . , d ′and ck(t), k = 0, . . . , d ′ are time homogeneous (not depending on t). Then forany slowly increasing C∞-function f0, the initial value problem for the heatequation (4.21) has a unique slowly increasing C∞-solution. It is given by theFeynman–Kac–Girsanov formula

u(x, t) = E[f0(Φt0,t (x)) exp

{ d ′∑

k=0

∫ t

t0

ck(Φt0,r (x)) ◦ dWkr

}]. (4.22)

Proof In the time homogeneous case, the semigroup P cs,t depends on t − s only,

which we denote by P ct−s . Then we have ∂

∂sP cs,t f = ∂

∂sP ct−sf = − ∂

∂tP ct−sf.

Therefore equation (4.17) is written as ∂∂tP ct f = AcP c

t f. Consequently, u(x, t) =P ct−t0

f0(x) is the solution of the heat equation (4.21). Further, u(x, t) is representedby (4.22). ��

The above argument can be applied to complex-valued potentials ck, k =0, . . . , d ′. Suppose that these are pure imaginary, written as ck(x) = √−1Θk(x).We define a complex-valued exponential functional by

Gt0,t (x) = exp{√−1

( d ′∑

k=0

∫ t

t0

Θk(Φt0,r (x)) ◦ dWkr

)}(4.23)

4.3 Backward Diffusions and Heat Equations 137

and define a semigroup by P ct0,t

f (x) = E[f (Φt0,t (x))Gt0,t (x)]. Then u(x, t) =P ct0,t

φ0(x) is a solution of the heat equation associated with the operator AΘ

defined by

AΘf = 1

2

d ′∑

k=1

(Vk +√−1Θk)

2f + (V0 +√−1Θ0)f.

If A is a Laplacian, i.e., Vk = ∂xk for k = 1, . . . , d ′ where d ′ = d, the operatorAΘ is called a Schrodinger operator with magnetic field Θ = (Θ0, . . . , Θd ′).(Matsumoto–Taniguchi [83]). The solution of the heat equation associated withthe operator AΘ is represented by Feynman–Kac–Girsanov formula (4.22) withck =

√−1Θk, k = 0, . . . , d.In the time-dependent case, discussions are complicated. It will be studied in the

next section.

Remark Backward heat equations are used in mathematical finance. Consider theequation on (0,∞)× [0, t1] defined by

∂

∂sv(x, s)+

(σ(s)2

2x2 ∂2

∂x2 + rx∂

∂x− r

)v(x, s) = 0, 0 < s < t1,

(final condition) lims→t1

v(x, s) = (x −K) ∨ 0.

where σ(t) > 0 and r,K are positive constants, called the volatility, the interest rateand the exercise price, respectively. The equation is called the Black–Scholes partialdifferential equation. The solution v(x, s) is the pricing function of the call optionof exercise price K . For details, see Karatzas–Shreve [56] and Lamberton–Lapeyre[72].

The function f1(x) = (x −K) ∨ 0 is a continuous function but it is not smooth.Then it is not clear whether the possible solution v(x, s) given by (4.19) is smoothor not with respect x. This problem will be solved by constructing the fundamentalsolution for the Black–Scholes partial differential equation, which will be done inSect. 6.3, using the Malliavin calculus.

4.3 Backward Diffusions and Heat Equations

Let us consider a continuous backward symmetric SDE on Rd with coefficients

Vk(x, t), k = 0, . . . , d ′, which are C∞,1-functions. For a given x, t , suppose thatthere exists a d-dimensional continuous backward semi-martingale Xs, s < t whichsatisfies


Xs = x +d ′∑

k=0

∫ t

s

Vk(Xr , r) ◦ dWkr , (4.24)

where the stochastic integrals ◦dWkr , k = 1, 2, . . . , d ′ are backward symmetric

integrals and ◦dW 0r = dr . Then Xs is called a solution of the equation starting

from x at time t . The equation has a unique solution, which we denote by Xx,ts .

Then, there exists a continuous backward stochastic flow of diffeomorphisms {Φs,t }such that Xx,t

s = Φs,t (x) holds for any x, t , as we studied in Chap. 3. We set

Ps,t (x, E) = P(Φs,t (x) ∈ E), 0 ≤ s < t < ∞, E ∈ B(Rd).

Let {Fs,t , 0 ≤ s < t < ∞} be the two-sided filtration generated by the Wienerprocess Wt . Then Φt,u(x) is independent of Fs,t for any s < t < u. Hencetransformations Ps,t f (x) := ∫

f (y)Ps,t (x, dy) have the backward semigroupproperty Ps,uf (x) = Pt,uPs,t f (x) for any s < t < u. The stochastic processXs = X

x,ts , s ∈ [0, t] has the backward Markov property P(Xs ∈ E|Ft,T ) =

Ps,t (Xt , E). It has the (backward) strong Markov property. Hence Xx,ts is a

backward diffusion process with the backward transition probabilities Ps,t (x, ·). Thesystem of processes {Xx,t

s , (x, t) ∈ Rd×T} is called a system of backward diffusion

process with transition probabilities Ps,t (x, E). By Proposition 3.6.3, the backwardflow {Φs,t } satisfies the rules of the following backward and forward differentialcalculus, respectively:

f ◦ Φs,t = f +∫ t

s

A(r)f ◦ Φr,t dr +d ′∑

k=1

∫ t

s

Vk(r)f ◦ Φr,t dWkr (4.25)

= f +∫ t

s

A(r)(f ◦ Φs,r ) dr +d ′∑

k=1

∫ t

s

Vk(r)(f ◦ Φs,r ) dWkr .

Expectations of the last terms of the above two equations are both equal to 0.Therefore, taking expectations for each terms of the above, we have

Ps,t f = f +∫ t

s

Pr,tA(r)f dr, Ps,t f = f +∫ t

s

A(r)Ps,rf dr. (4.26)

Consequently, the backward semigroup {Ps,t } is generated by the same differentialoperator A(t) given by (4.4).

Consider a backward exponential functional with coefficients c = {c0, . . . , cd ′ }:

4.3 Backward Diffusions and Heat Equations 139


{ d ′∑

k=0

∫ t

s

ck(Φr,t (x), r) ◦ dWkr

}. (4.27)

For any slowly increasing continuous function f on Rd , we can define the integral

P cs,t f (x) := E

[f (Φs,t (x))G

cs,t (x)

](4.28)

for any s, t and x. It is a slowly increasing C∞-function. Further, {P cs,t } satisfies the

backward semigroup property P ct,uP

cs,t f = P c

s,uf for any u > t > s. The followingtheorem corresponds to Theorem 4.2.1 for forward diffusion.

Theorem 4.3.1 If f is a slowly increasing C∞-function of x, P cs,t f (x) is also a

slowly increasing C∞-function of x. Further, it is continuously differentiable withrespect to s, t for s < t and satisfies the following backward and forward partialdifferential equations, respectively:

∂

∂sP cs,t f (x) = −P c

s,tAc(s)f (x), (4.29)

∂

∂tP cs,t f (x) = Ac(t)P c

s,t f (x). (4.30)

Proof We can show similarly to equations (4.26) that P cs,t f satisfies

P cs,t f = f +

∫ t

s

P cr,tA

c(r)f dr, P cs,t f = f +

∫ t

s

Ac(r)P cs,rf dr.

Therefore P cs,t f is differentiable with respect to t (> s) and s(< t). Differentiating

the first equation by s, we get (4.29) and differentiating the second equation by t ,we get (4.30). ��

Consequently, the backward semigroup {P cs,t } is generated by the differential

operator Ac(t) given by (4.11). Equation (4.30) tells us that u(x, t) = P ct0,t

f0(x)

is a solution of the heat equation (4.21). Therefore we have the followings:

Theorem 4.3.2 If f0 is a slowly increasing C∞-function, the heat equation (4.21)has a unique slowly increasing solution u(x, t). The solution is a C∞,1-function ofx, t . It is represented by the backward Feynman–Kac–Girsanov formula

u(x, t) = E[f0(Φt0,t (x)) exp

{ d ′∑

k=0

∫ t

t0

ck(Φr,t (x), r) ◦ dWkr

}], (4.31)

where {Φs,t } is the backward flow defined by a backward symmetric SDE withcoefficients Vk(x, t), k = 0, . . . , d ′.


So far, we have discussed the existence of solutions for a heat equation with asmooth initial function f0 and for a backward heat equation with a smooth finalfunction f1. If the function f0 is not smooth, it is not easy to show that u(x, t)

defined by (4.31) is a solution of the heat equation, since it is not clear whetherthe function u(x, t) may be smooth or not with respect to x. We will show inChap. 6 that if the operator Ac(t) is ‘nondegenerate’, the function u(x, t) givenby (4.31) is smooth and satisfies the heat equation. An approach to the problemis to construct the fundamental solution for the equation (4.21). Another method isto extend smooth functions f0 in formulas (4.31) to a generalized function f0. Itwill be completed by using the Malliavin calculus, which will be studied in Chap. 5.

4.4 Dual Semigroup, Inverse Flow and Backward Diffusion

We will denote by C∞0 or C∞

0 (Rd) the set of all real-valued C∞-functions ofcompact supports. Let A(t) be the differential operator defined by (4.4). It is a linearmap from C∞

0 (Rd) into itself. We shall study its dual (adjoint) operator A(t)∗ withrespect to the Lebesgue measure dx. A linear map A(t)∗;C∞

0 (Rd) → C∞0 (Rd) is

called a dual of A(t) if it satisfies

∫

Rd

A(t)f (x) · g(x) dx =∫

Rd

f (x) · A(t)∗g(x) dx (4.32)

for any f, g ∈ C∞0 (Rd).

Proposition 4.4.1 The dual A(t)∗ is well defined and is given by

A(t)∗g = 1

2

d ′∑

k=1

(Vk(t)+ divVk(t))2g − (V0(t)+ divV0(t))g, (4.33)

where divV (x, t) := ∑i∂V i

∂xi(x, t) is a C

∞,1b -function.

Proof For a vector field V = ∑V i ∂

∂xi, we have a formula of the integration by

parts:

∫Vf (x) · g(x) dx = −

∫f (x) · Vg(x) dx −

∫f (x) · divV (x)g(x) dx,

for f, g of C∞0 (Rd). Therefore we have V ∗g = −Vg− divVg. Then the adjoint of

A(t) is given by A(t)∗g = ∑d ′k=1

12 (Vk(t)

∗)2g + V0(t)∗g. ��

The geometric meaning of terms divVk(x, t) in the expression (4.33) is not clear.We want to define directly the dual semigroup of {Ps,t }, making use of the inverse

4.4 Dual Semigroup, Inverse Flow and Backward Diffusion 141

of the stochastic flow {Φs,t }. It is interesting to know how the expression of the dualoperator A(t)∗ is related to the geometric property of diffeomorphic maps Φs,t ofthe stochastic flow.

Let {Φs,t } be the stochastic flow of diffeomorphisms generated by SDE (4.1) andlet {Ps,t } be the semigroup defined by (4.3). Given 0 ≤ s < t < ∞, a linear operatorP ∗s,t ;C∞

b (Rd) → C∞b (Rd) is called the dual of Ps,t , if the equality

∫

Rd

Ps,tf (x) · g(x) dx =∫

Rd

f (x) · P ∗s,t g(x) dx

holds for all f, g ∈ C∞0 (Rd). We will show the dual P ∗

s,t exists uniquely for any s, t

and the family {P ∗s,t } is a backward semigroup with the generator A(t)∗.

Let Ψs,t ;Rd → Rd be the inverse map of the diffeomorphism Φs,t . Let ∇Ψs,t

be the Jacobian matrix of the diffeomorphism Ψs,t and let det∇Ψs,t be the Jacobian(Jacobian determinant). We will first remark that it is positive a.s. for any s < t .If t is fixed, Ψs,t is continuous in s a.s., since it satisfies the backward SDE withcoefficients −Vk(x, t), k = 0, . . . , d ′ by Theorem 3.7.1. Then its Jacobian det∇Ψs,t

is also continuous with respect to s. Note that it does not take the value 0, since Ψs,t

is diffeomorphic for any s. Further, it holds that det∇Ψt,t = 1, since Ψt,t is theidentity map. Consequently, det∇Ψs,t should be positive for all s < t a.s.

Now we will define the dual semi-group {P ∗s,t }. We need the formula of the

change of variables on Rd . Let f be an element of C∞

0 and let φ be a diffeomorphicmap from R

d onto itself. Then we have the formula of the change of variables;

∫

Rd

f (x) dx =∫

Rd

f (φ(x))| det∇φ(x)| dx, (4.34)

where ∇φ is the Jacobian matrix of the map φ;Rd → Rd and det∇φ is the Jacobian

of the map φ. Now take another g ∈ C∞0 and replace f by f ◦ φ · g and φ by φ−1

in the above formula. Then we obtain the formula∫

Rd

f (φ(x))g(x) dx =∫

Rd

f (x)g(φ−1(x))| det∇φ−1(x)| dx.

Setting φ(x) = Φs,t (x) in the above formula, we have

∫

Rd

f (Φs,t (x)) · g(x) dx =∫

Rd

f (x) · g(Ψs,t (x)) det∇Ψs,t (x) dx, (4.35)

almost surely. Now applying Theorem 3.4.2 to the backward SDE, we find that forany p ≥ 2, E[|∇Ψs,t (x)|p] are bounded with respect to x and 0 ≤ s < t ≤ T .Therefore we can define a bounded kernel by

P ∗s,t (x, E) := E[1E(Ψs,t (x)) det∇Ψs,t (x)],


and a linear transformation C∞0 (Rd) → C∞

b (Rd) by

P ∗s,t g(x) :=

∫

Rd

P ∗s,t (x, dy)g(y) = E[g(Ψs,t (x)) det∇Ψs,t (x)]. (4.36)

Taking the expectation of each side of (4.35), we have the formula of the duality:

∫

Rd


Rd

f (x) · P ∗s,t g(x) dx. (4.37)

Consequently, P ∗s,t of (4.36) is the dual of Ps,t with respect to the Lebesgue measure

dx.Since {Ps,t } satisfy Ps,u = Ps,tPt,u, dual operators {P ∗

s,t } satisfy the backwardsemigroup property P ∗

s,ug = P ∗t,uP

∗s,t g for u > t > s (Sect. 1.7). Further, the dual

semigroup and the dual operator A(t)∗ are related by

P ∗s,t g(x) = g(x)+

∫ t

s

P ∗r,tA(r)∗g(x) dr

= g(x)+∫ t

s

A(r)∗P ∗s,rg(x) dr

for any g ∈ C∞0 (Rd). Therefore, A(t)∗ is the generator of the dual semigroup {P ∗

s,t }.We will study further the relation between the dual semigroup {P ∗

s,t } and the

inverse flow {Ψs,t }. Recall that the generator of the diffusion defined by thestochastic flow {Φs,t } is given by (4.4). Recall further that Xs = Ψs,t (x) isthe solution of the continuous backward SDE with coefficients −Vk(x, t), k =0, . . . , d ′ starting from x at time t . Hence Xs = Ψs,t is a backward diffusion. Wedefine its backward transition probabilities by

Ps,t (x, E) := E[1E(Ψs,t (x))].

It defines the backward semigroup {Ps,t }.We saw in Theorem 3.7.1 that the backward flow {Ψs,t } satisfies

f (Ψs,t (x)) = f (x)+∫ t

s

A(r)f (Ψr,t (x)) dr −d ′∑

k=1

∫ t

s

Vk(r)f (Ψr,t (x)) dWkr

=f (x)+∫ t

s

A(r)(f ◦ Ψs,r )(x) dr−d ′∑

k=1

∫ t

s

Vk(r)(f ◦ Ψs,r )(x) dWkr ,

where


A(t) = 1

2

∑

k≥1

Vk(t)2f − V0(t)f. (4.38)

Take expectations for each term of the above. Then the backward semigroup {Ps,t }satisfies equations

Ps,t f = f (x)+∫ t

s

Pr,tA(r)f dr, Ps,t f = f (x)+∫ t

s

A(r)Ps,rf dr. (4.39)

Therefore A(t) is the generator of the backward semigroup {Ps,t }. We define abackward exponential functional with coefficients c associated with the inverse flow{Ψs,t } by

Gcs,t (x) = exp

{ d ′∑

k=0

∫ t

s

ck(Ψr,t (x), r) ◦ dWkr

}(4.40)

Lemma 4.4.1 Jacobian det∇Ψs,t (x) is represented as G−div Vs,t (x), where

−div V = −(divV0(x, t), . . . , divVd ′(x, t)).

Proof We will prove that det∇Ψs,t (x) satisfies the following backward linearequation for all x a.s.:

det∇Ψs,t (x) = 1 −d ′∑

k=0

∫ t

s

divVk(Ψr,t (x), r) det∇Ψr,t (x) ◦ dWkr . (4.41)

Let us recall the rule of the backward differential calculus (3.52). It is written as

f (Φs,t (x))− f (x) =d ′∑

k=0

∫ t

s

Vk(r)(f ◦Φr,t )(x) ◦ dWkr ,

using the backward symmetric integrals, where f is a function of C∞0 (Rd). Since∫

f (Φs,t (x)) dx = ∫det∇Ψs,t (x)f (x)dx holds by the formula of the change of

variables, we have

∫

Rd

f (x)(det∇Ψs,t (x)− 1) dx =d ′∑

k=0

∫

Rd

( ∫ t

s


)dx.

We shall compute the right-hand side. Use the Fubini theorem (Proposition 2.4.2)for integrals ◦dWk

r and dx. Then apply the formula of the integration by parts andthe formula of the change of variables. Then we have for any k = 0, . . . , d ′,


∫

Rd

( ∫ t

s


)dx

=∫ t

s

( ∫

Rd

Vk(r)(f ◦Φr,t )(x) dx)◦ dWk

r

= −∫ t

s

( ∫

Rd

divVk(x, r)f (Φr,t (x)) dx)◦ dWk

r

= −∫ t

s

( ∫

Rd

f (x)divVk(Ψr,t (x), r) det∇Ψr,t (x) dx)◦ dWk

r

= −∫

Rd

f (x)( ∫ t

s

divVk(Ψr,t (x), r) det∇Ψr,t (x) ◦ dWkr

)dx.

Since this is valid for any f ∈ C∞0 (Rd), we get the equality

det∇Ψs,t (x)− 1 = −d ′∑

k=0

∫ t

s

divVk(Ψr,t (x), r) det∇Ψr,t (x) ◦ dWkr ,

a.e. x, almost surely. We know that each symmetric integral of the right-hand sideis continuous in x (Proposition 2.4.3). Therefore the above equality holds for all xalmost surely. This proves (4.41).

Apply Theorem 3.10.2 for f (x) = ex and Xt = −∑k

∫ t

sdivVk(Ψr,t , r) ◦ dWk

r .

Then we find that G−div Vs,t (x) satisfies the following backward linear SDE:

G−div Vs,t (x) = 1 −

d ′∑

k=0

∫ t

s

divVk(Ψr,t (x), r)G−div Vr,t ◦ dWk

r . (4.42)

The equation is of the same form as (4.41) replacing det∇Ψs,t by G−div Vs,t . Then we

have G−div Vs,t (x) = det∇Ψs,t (x), by the uniqueness of the solution of the SDE. ��

We transform the operator A(t) by c = −div V. Then we get the relation

A(t)∗g = A−div V(t)g (4.43)

Theorem 4.4.1 Let {Ps,t } be the semigroup defined by the stochastic flow {Φs,t }generated by a continuous SDE (3.1) and let {P ∗

s,t } be its dual semigroup.Then if g is a slowly increasing C∞-function, P ∗

s,t g(x) is a slowly increasingC∞-function for any s < t . Further, the dual semigroup is obtained from thebackward semigroup {Ps,t } defined by the inverse flow {Ψs,t }, through the backwardexponential transformation with coefficients −div V, i.e., for any s < t and g, itholds that

P ∗s,t g(x) = E[g(Ψs,t (x)) det∇Ψs,t (x)] = P−div V

s,t g(x). (4.44)

Its generator is given by (4.43).


So far, we have studied the dual of the operator A(t) and the dual of the semi-group {Ps.t }. We will extend these to the dual of Ac(t) and {P c

s,t } which were definedin Sect. 4.2. Let Ac(t) be the differential operator defined by (4.11). Then its dual(adjoint) Ac(t)∗ is given by

Ac(t)∗g=1

2

d ′∑

k=1

(Vk(t)+ ck(t)+ divVk(t))2g − (V0(t)+ c0(t)+ divV0(t))g

= Ac−div V(t)g. (4.45)

Next we will define the dual of the semigroup {P cs,t }. For any f, g ∈ C∞

0 (Rd), wehave the formula of the change of variables:

∫

Rd

f (Φs,t (x))Gs,t (x)g(x) dx

=∫

Rd

f (x)Gs,t (Ψs,t (x))g(Ψs,t (x)) det∇Ψs,t (x) dx, (4.46)

almost surely. Since Gs,t (Ψs,t (x)) = exp{∑d ′

k=0

∫ t

sck(Ψr,t (x)) ◦ dWk

r

}, we have

Gs,t (Ψs,t (x)) det∇Ψs,t (x) = exp{ d ′∑

k=0

∫ t

s

(ck − divVk)(Ψr,t (x), r) ◦ dWkr

}.

It is Lp-bounded for any p ≥ 2. Then

Pc−div Vs,t g = E

[g(Ψs,t ) exp

{ d ′∑

k=0

∫ t

s

(ck − divVk)(Ψr,t , r) ◦ dWkr

}](4.47)

is well defined. We can take the expectation for both sides of (4.46). Then we get

∫P cs,t f (x)g(x) dx =

∫f (x)Pc−div V

s,t g(x) dx. (4.48)

Therefore Pc−div Vs,t is the dual of P c

s,t .

Theorem 4.4.2 Let {P cs,t } be the semigroup defined by (4.15), which is an exponen-

tial transformation of the semigroup {Ps,t }. Then the dual of {P cs,t } exists and it is

given by (4.47), where {Ψs,t } is the inverse flow of {Φs,t }. Further, its generator isgiven by (4.45).

Finally, we will discuss briefly the dual of time homogeneous diffusions. Weassume that coefficients Vk, k = 0, . . . , d ′ of the SDE do not depend on t . Then itsgenerator A is given by Af = 1

2

∑k≥1 V

2k +V0. Let P ∗

t be the dual of Pt . Then these


satisfy P ∗s P

∗t = P ∗

t+s for any s, t > 0. Hence the dual semigroup is again stationary.We can construct the dual diffusion as a forward process. Indeed, consider an SDEon R

d given by

dXt = −d ′∑

k=0

Vk(Xt ) ◦ dWkt . (4.49)

Let {Φs,t } be the stochastic flow generated by the above SDE. It is a diffusionprocess with the generator A = 1

2

∑k≥1 V

2k − V0. In particular, if div Vk(x) = 0

holds for k = 0, . . . , d ′, A∗ coincides with A. Hence the dual P ∗t is a conservative

semigroup defined by the forward diffusion Φs,t (x). Then Φ0,t (x) coincides withthe dual process of Φ0,t (x) in Hunt’s potential theory for Markov processes.

Note There are many studies of diffusion processes and their generators. Fordiffusion determined by SDE, we refer to Itô [50], Stroock–Varadhan [109], Dynkin[24], Ikeda–Watanabe [41] etc. Backward equations for diffusion determined bySDEs are studied in Freidlin–Wentzell [27] and Oksendal [90]. Our method usingthe rule of the backward differential calculus might be new.

Feynman–Kac transformation of the diffusion process is widely known. Girsanovtransformation (Theorem 2.2.1) is also widely used. See Ikeda–Watanabe [41] forthe application to the problem of solving SDE. See Karatzas–Shreve [56] and Kunita[62] for the application to mathematical finance. Our exponential transformation isthe product of the above two transformations. Its application to the Schrodingeroperator is discussed in Matsumoto–Taniguchi [83].

Feynman–Kac formula for the solution of time-homogeneous heat equationsis well known. However, we should remark that the formula should not hold forsolutions of non-homogeneous (non-stationary) heat equation. In such a case wehave to consider a similar formula for a backward diffusion. Extensions to theFeynman–Kac–Girsanov formula might be new.

The dual semigroup is defined usually as P ∗s,t g(y) = ∫

Rd ps,t (x, y)g(x) dx, ifthe transition function Ps,t (x, dy) has a density function ps,t (x, y) with respectto the Lebesgue measure dy. However, in our definition, we do not assume theexistence of the density ps,t (x, y).

4.5 Jump-Diffusion and Heat Equation; Case of SmoothJumps

Let us consider a stochastic differential equation with jumps on a Euclidean space.To avoid complicated notations, we assume that the Lévy measure ν has a weakdrift. We shall consider the equation studied in Sect. 3.10. It is given by

Xt=X0+d ′∑

k=0

∫ t

t0

Vk(Xr, r)◦dWkr+

∫ t

t0

∫

|z|>0+{φr,z(Xr−)−Xr−}N(dr dz), (4.50)

4.5 Jump-Diffusion and Heat Equation; Case of Smooth Jumps 147

where the last integral by the Poisson random measure is the improper integral. Fordiffusion and drift coefficients, we assume that these are C

∞,1b -functions and for

jump coefficients, we assume that g(x, t, z) ≡ φt,z(x) − x satisfies Condition (J.1)in Sect. 3.2.

Let {Φs,t } be the stochastic flow of C∞-maps generated by equation (4.50). Then,similarly to the proof of Proposition 4.1.1, we can show that its law

Ps,t (x, E) = P(Φs,t (x) ∈ E), 0 ≤ s < t < ∞, E ∈ B(Rd).

is determined uniquely from SDE (4.50) and {Ps,t (x, E)} is a transition probability.Further, the solution Xt, t ≥ t0 is a conservative strong Markov process on R

d

with the transition probability. The fact can be verified as shown in Sect. 4.1 forthe diffusion process. It is called a jump-diffusion process. In particular, if there isno diffusion part and no drift part, i.e., Vk(x, t) ≡ 0, k = 0, . . . , d ′, the solutionXt, t ≥ t0 is called a jump process.

For a slowly increasing continuous function f (x) on Rd , we define Ps,tf (x) as in

Sect. 4.1. Ps,tf (x) is again a slowly increasing continuous function for any s < t ,since E[|Φs,t (x)|p]1/p ≤ cp(1 + |x|) holds for any x. It satisfies the semigroupproperty Ps,tPt,uf = Ps,uf for any s < t < u and slowly increasing continuousfunction f . Next, if f (x) is a slowly increasing C∞-function (or C∞

b -function), wecan change the order of the derivative and expectation and we have the equality∂ iE[f (Φs,t (x))] = E[∂ i(f ◦ Φs,t (x))] for any i. Therefore Ps,tf (x) is a slowlyincreasing C∞-function of x (or C∞

b -function) for each s < t .Now, consider the rule of the forward differential calculus (3.93) for the flow

{Φs,t }. Take expectations for each term of (3.93). Since expectations of the secondand fourth terms of the right-hand side are 0, we get

Ps,tf (x) = f (x)+∫ t

s

Ps,rAJ (r)f (x) dr, (4.51)

for any slowly increasing C∞-function f . Here AJ (t) is an integro-differentialoperator defined by

AJ (t)f = A(t)f +∫

|z|>0+{f ◦ φt,z−f }ν(dz), (4.52)

where A(t) is the differential operator defined by (4.4) and the last integral by theLévy measure ν is the improper integral. If the Lévy measure has a strong drift, itcoincides with the Lebesgue integral. Furthermore, using the rule of the backwarddifferential calculus (3.95), we can show that Ps,tf satisfies

Ps,tf (x) = f (x)+∫ t

s

AJ (r)Pr,tf (x) dr, (4.53)


for any slowly increasing C∞-function f . Therefore the semigroup {Ps,t } satisfiesKolmogorov’s forward and backward equations associated with the operator AJ (t)

of (4.52). We will summarize the result.

Theorem 4.5.1 The solution Xt, t ≥ t0 of (4.50) is a conservative jump-diffusionprocess. If f (x) is a slowly increasing C∞-function, Ps,tf (x) is a slowly increasingC∞-function of x. Further, it is continuously differentiable with respect to s, t fors < t and satisfies the Kolmogorov forward equation (4.5) and the Kolmogorovbackward equation (4.6) associated with the operator AJ (t) given by (4.52).

The integro-differential operator AJ (t) of (4.52) is called the generator of thejump-diffusion process Xt . Further the flow {Φs,t } is called the stochastic flowassociated with the operator AJ (t).

Let us next consider an exponential transformation of the jump-diffusion. We willdefine an exponential Wiener–Poisson functional Gs,t (x), which is an extension ofthe exponential Wiener functional defined by (4.13). Let ck(x, t), k = 0, . . . , d ′be C


d × T and let dt,z(x) be a positive C∞,1,2b -function of

(x, t, z) such that dt,0(x) = 1 holds for any x, t . Then both dt,z(Φs,r (x)) − 1and log dr,z(Φs,r (x)) belong to the class L0

Udefined by (3.83). Then in view of

Proposition 3.10.1, improper integrals

∫ t

s

∫

|z|>0+{dr,z(Φs,r (x))− 1} drν(dz),

∫ t

s

∫

|z|>0+log dr,z(Φs,r−(x))N(dr dz)

etc. are well defined for any s < t and x, since the Lévy measure has a weak drift.For a given x and s < t , we define an exponential functional with coefficientsd = (dt,z(x)) by

Gds,t (x) = exp

{ ∫ t

s

∫

|z|>0+log dr,z(Φs,r−(x))N(dr dz)

}. (4.54)

We set Gs,t (x) = Gc,ds,t (x) = Gc

s,t (x)Gds,t (x), where Gc

s,t (x) is defined by (4.13) forthe stochastic flow {Φs,t }.Lemma 4.5.1 For any i, T > 0 and p ≥ 2, E[|∂ iGs,t (x)|p] is bounded with respectto 0 ≤ s < t ≤ T and x ∈ R

d .

Proof We first show that Gs,t satisfies a linear SDE

Gs,t = 1 +d ′∑

k=1

∫ t

s

Gs,r−c′k(r) dWkr +

∫ t

s

Gs,r−c′(r) dr

+∫ t

s

∫

Rd′0

Gs,r−{d ′z(r)− 1}N(dr dz), (4.55)


where c′k(r) = ck(Φs,r−(x), r), c′(r) = c(Φs,r−(x), r), d ′z(r) = dr,z(Φs,r−(x)).Here,

c(x, r) : = c0(x, r)+ 1

2

d ′∑

k=1

{Vk(r)ck(x, r)+ ck(x, r)2}

+∫

|z|>0+{dr,z(x)− 1}ν(dz). (4.56)

For the proof, apply the rule of the forward differential calculus (3.87) (Theo-rem 3.10.1) to the function f (x) = ex and the Itô process Xt := logGs,t , t > s.Then eXt − 1 is equal to

d ′∑

k=0

∫ t

s

eXr c′k(r) ◦ dWkr +

∫ t

s

∫

|z|>0+{eXr−+log d ′z(r) − eXr−}N(dr dz)

=d ′∑

k=1

∫ t

s

eXr−c′k(r) dWkr+

∫ t

s

eXr− c′(r) dr+∫ t

s

∫

|z|>0eXr−{d ′z(r)− 1}N(dr dz).

This proves (4.55).We may regard Gs,t as a solution of the master equation (3.21). In fact, note that

coefficients ci(x, r), c(x, r) and dr,z(x)− 1 are bounded functions. Then, regardingx as a parameter, we set c = (c1, . . . , cd ′), αx(y, r) = yc(Φs,r−(x), r) and

βx(y, r) = yc(Φs,r−(x), r), χx(y, r, z) = y(dr,z(Φs,r−(x), r)− 1).

These functionals satisfy Conditions 1–3 of the master equation and y = Gs,r (x)

is the solution of the master equation with the above coefficients. Then we find thatE[|Gs,t (x)|p] is bounded with respect to x ∈ R

d , 0 ≤ s < t ≤ T for any p ≥ 2.Next, the derivative ∂Gs,t (x) satisfies

∂Gs,t=∫ t

s

(∂Gs,r−c′(r)+Gs,r−∂c′(r)) dWr

+∫ t

s

(∂Gs,r−c′(r)+Gs,r−∂c′(r)) dr

+∫ t

s

∫ {∂Gs,r−(d ′z(r)− 1)+Gs,r−∂(d ′z(r)− 1)

}N(dr dz),

since Gs,t satisfies (4.55). We will apply Theorem 3.3.1 again. Since

E[|Gs,r−∂c′(r)|p], E[|Gs,r−|p

∫|∂(d ′z(r)− 1)|pν(dz)

]


are bounded with respect to s < r and x (x is a parameter), E[|∂Gs,t (x)|p] shouldbe bounded with respect to s < t, x. Repeating this argument inductively, we arriveat the assertion of the lemma. ��

We can define

Pc,ds,t f (x) = E

[f (Φs,t (x))G

c,ds,t (x)

](4.57)

for any slowly increasing continuous function f . Then {P c,ds,t } has the semigroup

property. We define further an integro-differential operator with parameter c,d by

Ac,dJ (t)f (x) = Ac(t)f (x)+

∫

|z|>0+{dt,z(x)f (φt,z(x))− f (x)}ν(dz) (4.58)

for slowly increasing C∞-function f , where Ac(t) is the differential operatordefined by (4.11). If the Lévy measure has a strong drift, the last integral coincideswith the Lebesgue integral. It is rewritten by a direct calculation as

Ac,dJ (t)f = AJ (t)f +

d ′∑

k=1

ck(t)Vk(t)f +∫

|z|>0+(dt,z−1){f ◦φt,z−f }ν(dz)+ c(t)f,

where c(x, t) is defined by (4.56).

Theorem 4.5.2 If f is a slowly increasing C∞-function of x, P c,ds,t f (x) is also a

slowly increasing C∞-function of x. Further, it is continuously differentiable withrespect to s, t for s < t and satisfies the following forward and backward integro-differential equations, respectively:

∂

∂tP

c,ds,t f (x) = P

c,ds,t A

c,dJ (t)f (x), (4.59)

∂

∂sP

c,ds,t f (x) = −A

c,dJ (s)P

c,ds,t f (x). (4.60)

Proof The first assertion follows from Lemma 4.5.1, immediately. We prove thesecond assertion. Using a symmetric integral, we can show that equation (4.55) isrewritten as

Gs,t = 1+d ′∑

k=0

∫ t

s

Gs,rc′k(r)◦dWk

r +∫ t

s

∫

|z|>0+Gs,r−{d ′z(r)−1}N(dr dz), (4.61)

similarly to the diffusion case (Sect. 4.2). The pair (Φs,t (x),Gs,t (x)y) may beregarded as a (d+ 1)-dimensional jump-diffusion process determined by the pair ofSDEs (4.50) and (4.61). Let f (x, y) be a slowly increasing C∞-function on R

d+1.Then, using the rule of the forward differential calculus (3.93) for the pair process,


we find that

f (Φs,t (x),Gs,t (x)y)− f (x, y)

=∫ t

a

AJ (r)f (Φs,r (x),Gs,r (x)y) dr + a martingale with mean 0.

Here AJ (t) is the operator given by

AJ (t)f (x, y) =d ′∑

k=1

(Vk(t)+ ck(t)y

∂

∂y

)2f (x, y)+

(V0(t)+ c0(t)y

∂

∂y

)f (x, y)

+∫

|z|>0+{f (φt,z(x), ψt,z(x, y))− f (x, y)

}ν(dz),

where ψt,z(x, y) = dt,z(x)y. Hence Ps,t f (x, y) := E[f (Φs,t (x),Gs,t (x)y)]satisfies Ps,t f (x, y) = f (x, y) + ∫ t

sPs,r AJ (r)f (x, y) dr . Further, from the rule

of the backward differential calculus (3.95), we have Ps,t f (x, y) = f (x, y) +∫ t

sAJ (r)Pr,t f (x, y) dr . Therefore AJ (t) is the generator of the semigroup {Ps,t }.Now take f (x, y) = f (x)y, where f (x) is a slowly increasing smooth function

on Rd . Apply it to the operator AJ (t). Then we have Ac,d(t)f (x) = AJ (t)f (x, 1).

Consequently, we can show that Pc,ds,t f (x) satisfies the forward and backward

equations (4.59) and (4.60), similarly to the case of diffusion discussed in Sect. 4.2.��

Set Xt = ∑d ′k=0

∫ t

sc′k(r)◦dWk

r +∫ t

s

∫|z|>0+ d ′z(r)N(dr dz). Using the Itô integral,

it is rewritten as Xt = X(0)t +X

(1)t +X

(2)t , where X

(0)t = ∫ t

sc′(r) dr and

X(1)t =

d ′∑

k=1

∫ t

s

c′k(r) dWkr − 1

2

d ′∑

k=1

∫ t

s

c′k(r)2 dr,

X(2)t =

∫ t

s

∫

|z|>0+log d ′z(r)N(dr dz)−

∫ t

s

∫

|z|>0+{d ′z(r)− 1} drν(dz).

Hence exponential functional Gs,t (x) = Gc,ds,t (x) is decomposed as the product of

three exponential functionals G(0)s,t (x) = eX

(0)t , G

(1)s,t (x) = eX

(1)t and G

(2)s,t (x) =

eX(2)t . We saw in Sect. 4.2 that the transformation by G

(0)s,t (x) is the Feynman–

Kac transformation. By this transformation, the potential term c(t)f is addedto the operator AJ (t). Further, the transformation by G

(1)s,t (x) is the Girsanov

transformation. By this transformation, the drift term∑

ckVk(t)f is added toAJ (t). We shall consider the transformation by G

(2)s,t (x). We denote it by Zt . Apply

Theorem 3.10.1 by setting F(x) = ex . Then we get


Zt = 1 −∫ t

s

∫Zr−(d ′z(r)− 1) drν(dz)+

∫ t

s

∫Zr−(d ′z(r)− 1)N(dr dz)

= 1 +∫ t

s

∫Zr−(d ′z(r)− 1)N(dr dz). (4.62)

Hence Zt is a positive local martingale. Further, E[Zpt ] < ∞ holds for any

p ≥ 2. Then Zt is a positive martingale with mean 1. Then we can define anotherprobability measure P (2) by dP (2) = G

(2)s,t (x) dP .

Lemma 4.5.2 Set Nd(dr dz) = N(dr dz)− d ′z(r) drν(dz) and

Yt =∫ t

s

∫

Rd′0

ψ(r, z)Nd(dr dz), (4.63)

where ψ(r, z) is a bounded predictable random field. Then Yt is a martingale withrespect to P (2).

Proof Let Zt = eX(2)t . If the product ZtYt is a martingale with respect to the measure

P , Yt is a martingale with respect to P (2). In fact, if ZtYt is a martingale with respectto P , we have E[ZtYt ;B] = E[ZsYs;B] for any t > s and B ∈ Fs . Therefore,

E(2)[Yt ;B] = E[ZtYt ;B] = E[ZsYs;B] = E(2)[Ys;B], ∀B ∈ Fs ,

showing that Yt is a martingale with respect to the measure P (2).Apply Theorem 3.10.1 to the product of Zt and Yt . Then we have

ZtYt =∫ t

s

∫(Zr− + d ′z(r)− 1)(Yr− + ψ(r, z)− Zr−Yr−)N(dr dz)

−∫ t

s

∫{Yr−(d ′z(r)− 1)+ Zr−ψ(r, z)} drν(dz)

=∫ t

s

∫(Yr−(d ′z(r)− 1)+ Zr−ψ(r, z))N(dr dz).

Therefore ZtYt is a martingale. ��The solution Xt = Φs,t (x) of equation (4.50) starting from x at time s

satisfies (3.93). We can rewrite it as

f (Φs,t ) = f +d ′∑

k=1

∫ t

s

Vk(r)f (Φs,r ) dWkr +

∫ t

s

AdJ (r)f (Φs,r ) dr

+∫ t

s

∫

|z|>0+{f ◦ φr,z ◦Φs,r− − f ◦Φs,r−}Nd(dr dz),


where

AdJ (t)f = AJ (t)f +

∫

|z|>0+(dt,z − 1)(f ◦ φt,z − f )ν(dz). (4.64)

Define the semigroup {P (2)s,t } by

P(2)s,t f (x) =

∫f (Φs,t (x)) dP

(2) = E[f (Φs,t (x))G(2)s,t (x)].

Then it satisfies

P(2)s,t f (x) = f (x)+

∫ t

s

P (2)s,r A

dJ (r)f (x) dr.

Therefore the generator of the semigroup {P (2)s,t } is Ad

J (t)f . Consequently, by the

transformation Ps,t → P(2)s,t , the jump term

∫(dt,z − 1){f ◦φt,z − f }ν(dz) is added

to AJ (t)f . It may be considered as a Girsanov transformation for a jump process.

The final value problem for the backward heat equation associated with theintegro-differential operator A

c,dJ (t) is defined in the same way as the problem

associated with the differential operator Ac(t) defined by (4.18).

Theorem 4.5.3 The final value problem for the backward heat equation associatedwith operator A

c,dJ (t) has a unique slowly increasing solution for any slowly

increasing C∞-function f1. Further, the solution v is represented by v(x, s) =E[f1(Φs,t1(x))G

c,ds,t1

(x)].The proof, which is carried out similarly to the proof of Theorem 4.2.2 will

be omitted here. Consequently, for a given integro-differential operator Ac,dJ (t)

of (4.58), there exists a unique semigroup of C2,1-class satisfying (4.59) and (4.60).The operator Ac,d

J (t) is called the generator of the semigroup {P c,ds,t }. In particular, if

c = 0 and d = 1, the operator Ac,dJ (t) coincides with the operator AJ (t). Then the

semigroup {Ps,t } satisfying Kolmogorov’s backward equation associated with AJ (t)

is unique. The semigroup is said to be generated by the operator AJ (t). The jump-diffusion process with the semigroup {Ps,t } is called the jump-diffusion process withthe generator AJ (t).

The initial value problem for the heat equation associated with the integro-differential operator A

c,dJ (t) is defined in the same way as the problem associated

with the differential operator Ac(t) defined by (4.11). In order to get the solutionof the initial value problem, we consider the backward SDE with characteristics(Vk(x, t), k = 0, . . . , d ′, g(x, t, z), ν). It is written as


Xs = x +d ′∑

k=0

∫ t

s

Vk(Xr , r) ◦ dWkr +

∫ t

s

∫g(Xr , r, z)N(dr dz). (4.65)

The solution exists uniquely, which we denote by Xx,ts . There exists a backward

stochastic flow {Φs,t } of C∞-maps such that Xx,ts = Φs,t (x) holds a.s. for any

t, s, x. Set Ps,t (x, E) = P(Φs,t (x) ∈ E). It is the transition function of thebackward Markov process Xs . Further, similarly to the case of diffusion in Sect. 4.3,we can show that the backward semigroup {Ps,t } satisfies

Ps,t f = f +∫ t

s

Pr,tAJ (r)f dr, Ps,t f = f +∫ t

s

AJ (r)Ps,rf dr,

where AJ (t) is the integro-differential operator given by (4.52). Therefore, thegenerator of the backward semigroup {Ps,t } is the same as that of the forwardsemigroup {Ps.t }.

Let c = (c0(x, t), . . . , cd ′(x, t)) and d = (dt,z) be functions satisfying the sameconditions as those for Gs,t of (4.54). We define backward exponential functionalsby

Gcs,t = exp

{ d ′∑

k=0

∫ t

s

ck(Φr,t , r) ◦ dWkr ,

Gds,t = exp

{ ∫ t

s

∫

|z|>0+log dr,z(Φr,t )N(dr dz)

}. (4.66)

and Gc,ds,t (x) = Gc

s,t (x)Gds,t (x). We set P c,d

s,t f (x) = E[f (Φs,t (x))Gc,ds,t (x)]. Then

{P c,ds,t } is a backward semigroup. Its generator is equal to A

c,dJ (t). Hence we have

the following theorem.

Theorem 4.5.4 If f0 is a slowly increasing C∞-function, the initial value problemfor the heat equation associated with the operator A

c,dJ (t) has a unique slowly

increasing solution u(x, t), (x, t) ∈ Rd × (t0,∞). It is of C∞,1-class and is

represented by

u(x, t) = E[f0(Φt0,t (x))G

c,dt0,t

(x)], (4.67)

where {Φs,t } is the right continuous backward stochastic flow generated by the sym-metric backward SDE with characteristics (Vk(x, t), k = 0, . . . , d ′, g(x, t, z), ν).

In particular, if Vk(t), k = 1, . . . , d ′ and ck(t), k = 0, . . . , d ′, dz(t) are timehomogeneous (not depending on t), the solution is given by the Feynman–Kac–Girsanov formula u(x, t) = E[f0(Φt0,t (x))G

c,dt0,t

(x)], making use of the forwardstochastic flow {Φs,t } with characteristics (Vk, k = 0, . . . , d ′, g(·, z), ν).

4.6 Dual Semigroup, Inverse Flow and Backward Jump-Diffusion; Case of. . . 155

4.6 Dual Semigroup, Inverse Flow and BackwardJump-Diffusion; Case of Diffeomorphic Jumps

In this section, we assume that jump-maps φt,z;Rd → Rd defined by φt,z(x) =

g(x, t, z) + x are diffeomorphic i.e., they satisfy Condition (J.2) in Chap. 3. Thenthe stochastic flow {Φs,t } is a flow of diffeomorphisms by Theorem 3.9.1.

Proposition 4.6.1 Let AJ (t) be an integro-differential operator given by (4.52).We assume that coefficients Vk(x, t), k = 0, . . . , d ′ are C

∞,1b -functions and that

jump coefficient g(x, t, z) = φt,z(x) − x satisfies Conditions (J.1) and (J.2) inChap. 3. Then the dual operator AJ (t)

∗;C∞0 (Rd) → C∞

b (Rd) is well defined. Itis represented by

AJ (t)∗g = 1

2

d ′∑

k=1

(Vk(t)+ divVk(t))2g − (V0(t)+ divV0(t))g

+∫

|z|>0+

{(det∇φ−1

t,z )g ◦ φ−1t,z − g

}ν(dz), (4.68)

where φ−1t,z is the inverse map of the diffeomrphic map φt,z and det∇φ−1

t,z is itsJacobian determinant.

Before we proceed to the proof of the proposition, let us check that the lastimproper integral of (4.68) is well defined and is a C∞

b -function of x, if g ∈C∞

0 (Rd). Note that φ−1t,z (x) is written as x−h(x, t, z), where h(x, t, z) has the same

property as that of g(x, t, z). Indeed, we have ∇xh(x, t, z) = O(|z|) uniformly inx. Therefore,

det∇φ−1t,z (x) = det(I − ∇h(x.t, z)) = 1 +O(|z|)

uniformly in x. Then det∇φ−1t,z > 0 holds for sufficiently small |z|. Since it is

continuous in z and does not take the value 0 (because of diffeomorphic maps), itshould be positive for all z. Further, we get

det∇φ−1t,z g ◦ φ−1

t,z − g = (det∇φ−1t,z − 1)g ◦ φ−1

t,z + (g ◦ φ−1t,z − g) = O(|z|).

uniformly in x. Then the improper integral of the above function with respect to theLévy measure with weak drift is well defined, and in fact, a C∞

b -function.

Proof of Proposition 4.6.1 The operator AJ (t) is written as A(t) + A2(t), whereA2(t)f = ∫

|z|>0+{f ◦ φt,z−f }ν(dz). We have computed the dual of the operatorA(t) in Sect. 4.4. So we will consider the operator A2(t). We want to show that thedual of the operator A2(t) is represented as


A2(t)∗g =

∫

|z|>0+{

det∇φ−1t,z · g ◦ φ−1

t,z − g}ν(dz). (4.69)

Let φ be a diffeomorphic map on Rd such that det∇φ(x) is a bounded function.

Then from formula of change of variables (4.34), we have

∫f (φ−1(x)) dx =

∫f (x)| det∇φ(x)| dx,

∫f (φ−1(x))g(x) dx =

∫f (x)g(φ(x))| det∇φ(x)| dx,

for any functions f, g of C∞0 (Rd). Therefore,

∫ ( ∫

|z|>0+{f (φt,z(x))− f (x)}ν(dz)

)· g(x) dx

=∫

|z|>0+

( ∫ {f (x)g(φ−1

t,z (x))| det∇φ−1t,z (x)| − f (x)g(x)

}dx

)ν(dz)

=∫

f (x) ·( ∫

|z|>0+

{| det∇φ−1

t,z (x)|g(φ−1t,z (x))− g(x)

}ν(dz)

)dx.

Since det∇φ−1t,z (x) is positive,

∫A2(t)f (x) · g(x) dx = ∫

f (x) · A2(t)∗g(x) dx

holds. These computations yield the dual formula. ��Now we will define the dual semigroup {P ∗

s,t }, making use of the inverse flow.Discussions are similar to the diffusion case in Sect. 4.4. Let f, g ∈ C∞

0 (Rd)

and Ψs,t = Φ−1s,t . Then we have the formula (4.35). We saw in Sect. 3.9 that

the inverse flow {Ψs,t } is a solution of a backward SDE with characteristics(−Vk(t), k = 0, . . . , d ′, φ−1

t,z (x) − x, ν). Applying Theorem 3.38 to the backwardSDE, we find that for any p > 1, E[|∂Ψs,t (x)|p] are bounded with respect to x and0 < s < t < T . Then E[| det∇Ψs,t (x)|p] is also bounded for any p ≥ 2. Thereforefor g ∈ C∞

0 (Rd), we can define P ∗s,t g(x) by (4.36). It is a function of C∞

b (Rd)

and satisfies (4.37). Therefore P ∗s,t defined by (4.36) is the dual of Ps,t . The dual

semigroup {P ∗s,t } and the dual operator AJ (t)

∗ are related by

P ∗s,t g(x) = g(x)+

∫ t

s

P ∗r,tAJ (r)

∗g(x) dr (4.70)

= g(x)+∫ t

s

AJ (r)∗P ∗

s,rg(x) dr

for any g ∈ C∞0 (Rd). Therefore, AJ (t)

∗ is the generator of the dual semi-group{P ∗

s,t }.


For a fixed x, t1, Xs = Xx,t1s = Ψs,t1(x), s ∈ [0, t1] is a right continuous

backward stochastic process satisfying the backward SDE (3.96). It is a backwardMarkov process associated with the backward transition probability Ps,t (x, E) =P(Ψs,t (x) ∈ E). Since Ψs,t satisfies (3.99), the backward semigroup Ps,t satisfies

Ps,t f = f +∫ t

s

Pr,tAJ (r)f dr,

where

AJ (t)f = A(t)f +∫

|z|>0+{f ◦ φ−1

t,z − f }ν(dz), (4.71)

where A(t) is the differential operator defined by (4.38).The dual operator AJ (t)

∗ can be regarded as a transformation of the operatorAJ (t). It is rewritten as

AJ (t)∗g = AJ (t)g +

d ′∑

k=1

divVk(t)Vk(t)g

+∫

|z|>0+(det∇φ−1

t,z − 1){g ◦ φ−1t,z − g}ν(dz)+ c∗(t)g, (4.72)

where

c∗(t) = −divV0(t)+ 1

2

d ′∑

k=1

{divVk(t)Vk(t)(divVk(t))+ divVk(t)

2}

+∫

|z|>0(det∇φ−1

t,z − 1)ν(dz). (4.73)

We will next consider | det∇Ψs,t (x)|. We set Js,t (x) = det∇Ψs,t (x).

Lemma 4.6.1 |Js,t | satisfies the following backward equation:

|Js,t | = 1 −d ′∑

k=0

∫ t

s

divVk(Ψr,t , r)|Jr,t | ◦ dWkr

+∫ t

s

∫

|z|>0+

{det∇φ−1

r,z (Ψr,t )− 1}|Jr,t |N(dr dz). (4.74)

Proof For f ∈ C∞0 (Rd), we have a rule of the backward differential calculus with

respect to r:


f (Φs,t (x))− f (x) =d ′∑

k=0

∫ t

s


+∫ t

s

∫

|z|>0+

{f (Φr,t (φr,z(x)))− f (Φr,t (x))

}N(dr dz).

See (3.94). In the proof of Lemma 4.4.1, we showed the equality

∫ (f (Φs,t (x))− f (x)−

d ′∑

k=0

∫ t

s


)dx

=∫

f (x)(|Js,t (x)| − 1 +

d ′∑

k=0

∫ t

s

divVk(Ψr,t (x), r)|Jr,t (x)| ◦ dWkr

)dx.

Further, we have by the Fubini theorem and formula of change of variables,

∫

Rd

( ∫ t

s

∫

|z|>0+

{f (Φr,t ◦ φr,z(x))− f (Φr,t (x))

}N(drdz)

)dx

=∫ t

s

∫

|z|>0+

( ∫

Rd

{f (Φr,t (x))| det∇(φ−1

r,z (x))| − f (Φr,t (x))}dx

)N(dr dz)

=∫ t

s

∫

|z|>0+

( ∫

Rd

f (x)(| det∇φ−1r,z (Ψr,t (x))| − 1)|Jr,t (x)| dx

)N(dr dz)

=∫

Rd

f (x)( ∫ t

s

∫

|z|>0+(| det∇φ−1

r,z (Ψr,t (x))| − 1)|Jr,t (x)|N(dr dz))dx.

Since these two equalities are valid for any f ∈ C∞0 (Rd), we get the equality

|Js,t | − 1 =d ′∑

k=0

∫ t

s

−divVk(Ψr,t , r)|Jr,t | ◦ dWkr

+∫ t

s

∫

|z|>0+

{| det∇φ−1

r,z (Ψr,t )| − 1}|Jr,t |N(dr dz).

This proves the equality of the lemma. ��Associated with the inverse flow {Ψs,t }, we set

Gds,t (x) = exp

{ ∫ t

s

∫

|z|>0+log dr,z(Ψr,t (x))N(dr dz)

}. (4.75)

and Gc,ds,t (x) = Gc

s,t (x)Gds,t (x), where Gc

s,t (x) is given by (4.40).


Proposition 4.6.2 Assume the same condition as in Proposition 4.6.1. Then

det∇Ψs,t is positive and is represented by det∇Ψs,t = G−divV,det ∇φ−1

s,t , where−divV = (−divV0(x, r), . . . ,−divVd ′(x, r)) and det ∇φ−1 = det∇φ−1

r,z (x).

Proof Set

Ys = −d ′∑

k=0

∫ t

s

divVk(Ψr,t , r) ◦ dWkr +

∫ t

s

∫

|z|>0+log det∇φ−1

r,z (Ψr,t )N(dr dz)

and apply the rule of the backward differential calculus (3.89) (Theorem 3.10.2) forthe exponential function f (x) = ex . Then we get

eYs = 1 −d ′∑

k=0

∫ t

s

eYr divVk(Ψr,t ) ◦ dWkr

+∫ t

s

∫

|z|>0+eYr {det∇φ−1

r,z (Ψr,t )− 1}N(dr dz),

similarly to the proof of Lemma 4.5.1. Hence both eYs and |Js,t | satisfy the same

backward SDE (4.72). Then we have eYs = |Js,t |, by the uniqueness of the solutionof the SDE.

It remains to show that Js,t (y) is positive a.s. Instead of {Ψs,t }, we will considera family of flows {Ψ ε

s,t }, ε > 0, which are obtained from {Ψs,t } by cutting off all

jumps φ−1t,z , |z| < ε, as in the proof of Lemma 3.8.1. It is written as Ψ ε

s,t = Ψ 0s,t if

t < τ1 and

Ψ εs,t = Ψ 0

s,τ1◦ φ−1

τ1,S1◦ · · · ◦ φ−1

τn,Sn◦ Ψ 0

τn,t, if τn ≤ t < τn+1,

where {Ψ 0s,t } is a continuous flow determined by a continuous backward SDE,

and {τi, Si; i = 1, 2, . . .} is a sequence of stopping times and Rd ′0 -valued random

variables. Then Jacobian of Ψ εs,t satisfies det Ψ ε

s,t = det Ψ 0s,t > 0 if t < τ1 and

det∇Ψ εs,t = det∇Ψ 0

s,τ1det∇φ−1

τ1,S1· · · det∇φ−1

τn,Sndet∇Ψ 0

τn,t> 0,

if τn ≤ t < τn+1, since det∇Ψ 0s,τi

, det∇φ−1τi ,Si

etc. are all positive. Now a

subsequence Ψεns,t (x) and ∇Ψ

εns,t (x) converges to Ψs,t (x) and ∇Ψs,t (x) in probability.

Therefore det∇Ψs,t (x) is nonnegative. Then it is strictly positive a.s. ��By the above proposition, the dual semigroup P ∗

s,t g(x) is obtained from the

backward semigroup Ps,t g(x) = E[g(Ψs,t (x))] by the exponential transformationby (4.75). Then the operator AJ (t) is transformed to the operator AJ (t)

∗ of (4.68).Consequently, we have the following theorem.


Theorem 4.6.1 Assume the same condition as in Proposition 4.6.1.

1. Let {Ψs,t } be the inverse flow of {Φs,t }. Then Xs = Xx,ts = Ψs,t (x) is a backward

Markov process with the backward semigroup {Ps,t } having the generator AJ (t)

defined by (4.71).2. The Jacobian of Ψs,t (x) is represented as G−divV,det ∇φ−1

3. The dual semigroup {P ∗s,t } is obtained by a backward exponential transformation

of the Ψs,t ;

P ∗s,t g(x) = E

[g(Ψs,t (x)) det∇Ψs,t (x)

] = P−div V,det∇φ−1

s,t g(x). (4.76)

Further, if g is a slowly increasing C∞-function of x, then P ∗s,t g(x) is also a

slowly increasing C∞-function of x.4. The generator of the dual semi-group is given by (4.68).

So far, we have studied the dual of the operator A(t) and the dual of the semi-group {Ps.t }. We will extend these to duals of Ac,d

J (t) and Pc,ds,t which were defined

in Sect. 4.5. Let Ac,dJ (t) be the integro-differential operator defined by (4.58). Then,

by a direct computation, we find that its dual Ac,dJ (t)∗ is expressed by

Ac,dJ (t)∗g = Ac−div V,d·det∇φ−1

J (t)g. (4.77)

We will define the dual semigroup {P c,d,∗s,t }. Let Gs,t (x) = G

c,ds,t (x) be an

exponential functional with coefficients c,d. For any C∞-functions of compactsupports f, g, we have the formula of the change of variables:

∫

Rd

f (Φs,t (x))Gs,t (x)g(x) dx

=∫

Rd

f (x)Gs,t (Ψs,t (x))g(Ψs,t (x)) det∇Ψs,t (x) dx, (4.78)

almost surely. Note Gc,ds,t (Ψs,t ) = Gc,d

s,t . Then E[|Gs,t (x)|p] is also bounded for anyp. Therefore,

Pc,d,∗s,t g(x) = E

[g(Ψs,t (x))Gc,d

s,t (x) det∇Ψs,t (x)]

is well defined for any g ∈ C∞0 (Rd). Taking expectations for both sides of (4.78),

we find that P c,d,∗s,t is the dual of P c,d

s,t . Further, it is written as

Pc,d,∗s,t g(x) = E

[g(Ψs,t (x))G

∗s,t (x)

], (4.79)

where G∗s,t (x) = Gc,d

s,t (x) det∇Ψs,t (x).

4.7 Volume-Preserving Flows 161

Theorem 4.6.2 Assume the same condition as in Proposition 4.6.1. Let {P c,ds,t }

be the simigroup defined by (4.57), which is an exponential transformation of thesemigroup {Ps,t }. Then the dual of P c,d

s,t exists and it is given by (4.79), where {Ψs,t }is the inverse flow of {Φs,t } and G∗

s,t = Gc,ds,t det∇Ψs,t . Further, its generator is

given by (4.77).

Remark The backward process Xs = Ψs,t (x) = Φ−1s,t (x) has the left limit

Xs− = limε↓s Xs−ε . It is a modification of Xs and has the backward strong Markovproperty. See Sect. 1.7.

Remark For the existence of the dual operator AJ (t)∗ and the dual semigroup

P ∗s,t , we assumed that jump-maps φt,z are diffeomorphic. If jump-maps are not

diffeomorphic, it should be difficult to consider dual processes.

4.7 Volume-Preserving Flows

In this section, we will discuss a different topic. We are interested how the volumeof a set is changed through the transformation by stochastic flows.

Let B be a Borel subset of Rd . We denote its volume (=Lebesgue measure)

by |B|. Suppose that a map φ : Rd → Rd is diffeomorphic. It is called volume-

preserving if the volume of the set φ−1(B) = {φ−1(x); x ∈ B} is equal to |B| forany Borel set B. Note that φ is volume-preserving if and only if φ−1 is volume-preserving. Next, φ is called volume-gaining (or volume-losing) if |φ−1(B)| ≤ |B|(or |φ−1(B)| ≥ |B|) holds for any Borel set B. Note that φ is volume-losing if andonly if φ−1 is volume-gaining.

By the formula of the change of the variable (4.34) we have the equality

∫

Rd

f (φ(x)) dx =∫

Rd

f (x)| det∇φ−1(x)| dx.

For a bounded Borel subset B of Rd , we set f = 1B . It holds that 1B ◦φ = 1φ−1(B).Therefore we have

|φ−1(B)| =∫

B

| det∇φ−1(x)| dx, ∀B (Borel subset of Rd). (4.80)

Hence φ is volume-preserving if and only if | det∇φ−1(x)| = 1 holds for any x.Further, φ is volume-gaining if and only if | det∇φ−1(x)| ≤ 1 holds for any x.

Given a vector field V (x) on Rd of C∞

b -class, let φt (x), t ≥ 0, x ∈ Rd be the

deterministic flow such that φ0(x) = x and satisfies dφt

dt(x) = V (φt (x)) for any

t ∈ R. Then maps φt , t ≥ 0;Rd → Rd are diffeomorphic for all t . If maps φt are

volume-preserving for all t ≥ 0, the flow φt is called volume-preserving. Volume-losing and volume-gaining flows are defined similarly. It is known that a flow φt is


volume-preserving if and only if the divergence of the associated vector field V isidentically 0, and further, the flow φt is volume-gaining if and only if divV (x) ≥ 0for any x.

We shall study the similar problem for stochastic flows. Let {Φs,t } be thestochastic flow generated by SDE (4.50) with jumps. It is called volume-preservingor volume-gaining if maps Φs,t ;Rd → R

d are volume-preserving a.s. for any s < t

or maps Φs,t are volume-gaining a.s. for any s < t .


1. The flow {Φs,t } is volume-preserving if and only if either one of the followingholds:

(a) det∇Ψs,t (x) = 1 holds a.s. for any s < t and x, where Ψs,t = Φ−1s,t .

(b) divV0(x, t) = 0, divVk(x, t) = 0, k = 1, . . . , d ′ and det∇φt,z(x) = 1hold for all x, t, z.

(c) AJ (t)∗ = AJ (t), namely, the dual semigroup P ∗

s,t coincides with the

semigroup of the inverse flow {Ψs,t }.2. The flow {Φs,t } is volume-gaining if and only if either one of the following

holds:

(a’) For any s < t and x det∇Ψs,t (x) is less than or equal to 1 a.s.(b’) divV0(x, t) ≥ 0, divVk(x, t) = 0, k = 1, . . . , d ′ and det∇φt,z(x) ≥ 1

holds for all x, t, z.

3. The flow {Φs,t } is volume-losing if and only if either one of the following holds:

(a”) For any s < t and x, det∇Ψs,t (x) is greater than or equal to 1 a.s.(b”) divV0(x, t) ≤ 0, divVk(x, t) = 0, k = 1, . . . , d ′ and det∇φt,z(x) ≤ 1

holds for all x, t, z.

Proof We give the proof of assertions 1 and 2. Assertion 3 can be verified similarly.Since maps Φs,t ;Rd → R

d are diffeomorphic, we have (4.80). Therefore the flowΦs,t is volume preserving (or gaining) if and only if | det∇Ψs,t (x)| = 1 a.s. (or ≤ 1a.s., respectively). Now, | det∇Ψs,t (x)| satisfies the following backward equation:

| det∇Ψs,t | − 1 =−d ′∑

k=0

∫ t

s

divVk(Ψr,t , r)| det∇Ψr,t | ◦ dWkr

+∫ t

s

∫

|z|>0+{| det∇φ−1

r,z (Ψr,t )| − 1}| det∇Ψr,t |N(dr dz).

See Lemma 4.6.1. If | det∇Ψs,t (x)| = 1 (or ≤ 1) holds, the right-hand side (semi-martingale) is identically 0 (or nonpositive, respectively). This implies that themartingale part is 0 and the process of bounded variation part is 0 (or nonpositive,respectively). Note that det∇φr,z(x) = 1/ det∇φ−1

r,z (φr,z(x)). Then (a) implies (b)

and (a’) implies (b’). Conversely if (b) (or (b’)) holds, then | det∇Ψs,t (x)| = 1 (or

4.7 Volume-Preserving Flows 163

≤ 1) a.s. for any s, t, x. Therefore we have proved (b) → (a) (or (b′) → (b′)). Next,

since AJ (t)∗g = A−div V,det∇φ−1

J (t)g, the equivalence of (b) and (c) is obvious. ��Corollary 4.7.1 If the flow is volume-preserving, we have P ∗

s,t = Ps,t for anys < t . Further, consider time homogeneous continuous flow. If the flow is volume-preserving and V0 = 0, then Ps,t is self adjoint, i.e., P ∗

s,t = Ps,t holds for anys < t .

We are interested in the case where the dual semigroup {P ∗s,t } is conservative or

Markovian. The dual semigroup is conservative if and only if the Lebesgue measuredx is an invariant measure of the transition probabilities, i.e.,

∫Rd Ps,t (x, E) dx =

|E| holds for any s < t . The dual semigroup is Markovian if and only if dx is anexcessive measure of the transition probabilities, i.e.,

∫Rd Ps,t (x, E) dx ≤ |E| holds

for any s < t . We will give their geometric characterization.The stochastic flow {Φs,t } is said to be volume-preserving in the mean or volume-

gaining in the mean, if E[|Φ−1s,t (B)|] = |B| or E[|Φ−1

s,t (B)|] ≤ |B| holds for anyt > s and Borel sets B, respectively.


1. The following assertions are equivalent:

(a) The flow {Φs,t } is volume-preserving in the mean.(b) E[| det∇Ψs,t (x)|] = 1 for any s < t and x.(c) Dual semigroup {P ∗

s,t } is conservative, i.e., P ∗s,t1(x) = 1 holds for any x, s <

t .(d) The function c∗(t) defined by (4.73) satisfies c∗(x, t) = 0 for any x, t .

2. The following assertions are equivalent:

(a’) The flow {Φs,t } is volume-gaining in the mean.(b’) E[| det∇Ψs,t (x)|] ≤ 1 for any s < t and x.(c’) Dual semigroup {P ∗

s,t } is Markovian, i.e., P ∗s,t1(x) ≤ 1 holds for any x, s < t .

(d’) The function c∗(t) defined by (4.73) satisfies c∗(x, t) ≤ 0 for any x, t .

Proof We have from (4.80) the equality |Ψs,t (B)| = ∫B| det∇Ψs,t (y)| dy. Taking

expectations for both sides, we have

E[|Ψs,t (B)|] =∫

B

E[| det∇Ψs,t (x)|] dx, ∀B. (4.81)

Therefore, the flow is volume-preserving in the mean (or volume-gaining in themean), if and only if E[| det∇Ψs,t (x)|] = 1 (or ≤ 1, respectively) holds for anys < t and x. The equivalence of assertions (b), (c) and (d) (or (b’), (c’) and (d’),respectively) will be obvious. ��

Finally, we will consider the case where the dual semigroup {P ∗s,t } is noncon-

servative. Then the flow is not volume-preserving. We will study the speed of thevolume-gaining (or volume-losing). It should be related to the potential functionc∗(x, t) of the dual semigroup defined by (4.73).


Theorem 4.7.3 For any Borel subset B of Rd with finite volume, we have

∃ limt→s

1

t − s(E[|Ψs,t (B)|] − |B|) =

∫

B

c∗(x, s) dx. (4.82)

Proof We have from (4.81)

E[|Ψs,t (B)|] − |B|t − s

=∫

B

E[| det∇Ψs,t (x)|] − 1

t − sdx =

∫

B

P ∗s,t1(x)− 1

t − sdx

=∫

B

∫ t

sP ∗r,tA(r)∗1(x)dr

t − sdx →

∫

B

c∗(x, s) dx,

as t → s. ��For a given s ∈ T, define a subset of Rd by C+(s) = {x ∈ R

d; c∗(x, s) > 0},C−(s) = {x ∈ R

d ; c∗(x, s) < 0} and C0(s) = {x ∈ Rd; c∗(x, s) = 0}. Then the

above theorem tells us that if t − s is small, Ψs,t (B) is volume-gaining (or volume-losing) in the mean if B ⊂ C−(s) (or B ⊂ C+(s)). Further, it is volume-preservingin the mean if B ⊂ C0(s).

Note We discussed the volume-preserving problem with respect to the Lebesguemeasure. It may be interesting to consider the problem with respect to othermeasures such as the Gaussian measure. We refer to Kunita–Oh [65].

4.8 Jump-Diffusion on Subdomain of Euclidean Space

Let {Φs,t (x)} be the right continuous stochastic flow defined by SDE (4.50) and letG

c,ds,t (x) be an exponential functional defined by (4.54). Let D be a bounded open

subset of Rd . Let τ(x, s) be the first leaving time of the process Xt = X

x,st =

Φs,t (x), t ∈ [s,∞) from the set D. We shall consider the killed process at timeτ(x, s). It is defined by X

′x,st = Φs,t (x) if t < τ(x, s) and by X

′x,st = ∞ if

t ≥ τ(x, s). We define the weighted law of the killed process by

Qc,ds,t (x, E) = E

[1E(X

′x,st )G

c,ds,t (x)] (4.83)

= E[1E(Φs,t (x))G

c,ds,t (x)

∏1D(Φs,r (x))

],

where the product∏

is taken for all rationals r such that s < r < t . We setQ

c,ds,t f (x) = ∫

DQ

c,ds,t (x, dy)f (y). Then it is written as

Qc,ds,t f (x) = E

[f (Φs,t (x))G

c,ds,t (x)

∏1D(Φs,r (x))

]. (4.84)

4.8 Jump-Diffusion on Subdomain of Euclidean Space 165

Proposition 4.8.1 {Qc,ds,t } has the semigroup property: Qc,d

s,t Qc,dt,u = Q

c,ds,u holds for

any s < t < u. Further, for any f ∈ C∞0 (D), Qc,d

s,t f is differentiable with respectto t and satisfies

∂

∂tQ

c,ds,t f (x) = Q

c,ds,t (A

c,dJ (t)f )(x) (4.85)

for any s < t and x ∈ D.

Proof For any s < t < u, we have

{τ(x, s) > u} = {τ(x, s) > t} ∩ {τ(Xt , t) > u}.

Then Qc,ds,uf (x) is equal to

E[f (Φs,u(x))Gc,ds,u(x)1τ(x,s)>u]

=E[E[f (Φt,u(Φs,t (x)))G

c,dt,u (Φs,t (x))G

c,ds,t (x)1τ(Φs,t (x),t)>u

∣∣∣Fs,t

]1τ(x,s)>t

]

= E[Q

c,dt,uf (Φs,t (x))G

c,ds,t (x)1τ(x,s)>t

] = Qc,ds,t Q

c,dt,uf (x).

Therefore Qc,ds,t has the semigroup property. We have by Itô’s formula

f (Φs,t (x))Gc,ds,t (x) = f (x)+

∫ t

s

Ac,dJ (r)f (Φs,r (x))G

c,ds,r (x) dr +Mt,

where Mt is a martingale with mean 0. Consider the stopped process of each termof the above by the stopping time τ = τ(x, s). Then we have

f (Φs,t∧τ (x))Gc,ds,t∧τ (x) = f (x)+

∫ t∧τ

s


c,ds,r (x) dr +Mt∧τ

=f (x)+∫ t

s


c,ds,r (x)1τ>r dr +Mt∧τ .

It holds that f (Φs,t∧τ ) = f (Φs,t ) if t < τ and f (Φs,t∧τ ) = 0 if t ≥ τ . Therefore,taking the expectation of each term of the above equality, we get

E[f (Φs,t (x))Gc,ds,t (x)1τ>t ] = f (x)+ E

[ ∫ t

s


c,ds,r (x)1τ>r dr

],

since Mt∧τ is a martingale with mean 0 by Doob’s optional sampling theorem(Theorem 1.5.1). Therefore we get the equality


Qc,ds,t f (x) = f (x)+

∫ t

s

Qc,ds,r (A

c,dJ (r)f )(x) dr.

This proves (4.85). ��In the case c = 0 and d = 1, we denote Q

c,ds,t (x, ·) by Qs,t (x, ·). Then the killed

process X′x,st is a strong Markov process with transition function Qs,t (x, ·).

We are interested in the dual of the above semigroup Qc,ds,t f with respect to dx.

We define

Qc,d,∗s,t g(x) = E

[g(Ψs,t (x))Gc,d

s,t (x)∏

1D(Ψr,t (x))| det∇Ψs,t (x)|], (4.86)

where Ψr,t (x) = Φ−1r,t (x) and det∇Ψs,t (x) is its Jacobian determinant. (We set

dr,z = 1 if Xt is a diffusion.) It satisfies the backward semigroup propertyQ

c,d,∗t,u Q

c,d,∗s,t g = Q

c,d,∗s,u g for any s < t < u.

Proposition 4.8.2 Qc,d,∗s,t is the dual of Qc,d

s,t ; It holds that

∫

D

Qc,ds,t f (x) · g(x) dx =

∫

D

f (x) ·Qc,d,∗s,t g(x) dx (4.87)

for any continuous function f, g supported by compact subsets of D. Further, itsatisfies

Qc,d,∗s,t g(x) = g(x)+

∫ t

s

Qc,d,∗r,t (A

c,dJ (r)∗g)(x) dr, (4.88)

where Ac,dJ (r)∗ is the dual operator of Ac,d(r).

Proof Since Φs,r ◦Φ−1s,t = Ψr,t holds for 0 < r < t , we have

∫

D

f (Φs,t (x))g(x)Gc,ds,t (x)

(∏1D(Φs,r (x))

)dx

=∫

D

f (x)g(Ψs,t (x))Gc,ds,t (x)

(∏1D(Ψr,t (x))

)| det∇Ψs,t (x)| dx.

Taking the expectation for each term of the above, we get the formula (4.87).Equation (4.88) is proved similarly to the proof of (4.85). ��

The semigroup {Qc,ds,t } satisfies ∂

∂tQ

c,ds,t f (x) = Q

c,ds,t A

c,dJ (t)f (Kolmogorov’s

forward equation) for any C∞-function f satisfying f (x) = 0 on Dc, in viewof (4.85). However, Kolmogorov’s backward equation for {Qc,d

s,t } is not evident,

since it is not clear whether Qc,ds,t f (x) is a smooth function of x. In Sect. 6.10,

we show that it is true if the jump-diffusion is ‘pseudo-elliptic’, making use of theMalliavin calculus.

Chapter 5Malliavin Calculus

Abstract We will discuss the Malliavin calculus on the Wiener space and on thespace of Poisson random measures, called the Poisson space. There are extensiveworks on the Malliavin calculus on the Wiener space. Also, there are someinteresting works on the Malliavin calculus on the Poisson space. These two typesof calculus have often been discussed separately, since the Wiener space and thatof the Poisson space are quite different. Even so, we are interested in the Malliavincalculus on the product of these two spaces; we want to develop these two theoriesin a compatible way.

In Sects. 5.1, 5.2, and 5.3, we study the Malliavin calculus on the Wiener space.We define ‘H -derivative’ operator Dt and its adjoint δ (Skorohod integral by Wienerprocess) on the Wiener space. Then, after introducing Sobolev norms for Wienerfunctionals, we get a useful estimate of the adjoint operator with respect to Sobolevnorms (Theorem 5.2.1). In Sect. 5.3, we give Malliavin’s criterion for which the lawof a given Wiener functional has a smooth density. It will be stated in terms of theMalliavin covariance.

In Sects. 5.4, 5.5, 5.6, and 5.7, we study the Malliavin calculus on the spaceof Poisson random measure. Difference operator Du and its adjoint δ (Skorohodintegral by Poisson random measure) are defined for Poisson functionals followingPicard [92]. We will define a family of Sobolev norms conditioned to a family ofstar-shaped neighborhoods {A(ρ), 0 < ρ < 1}. Then a criterion for the smoothdensity of the law of a Poisson functional will be given using this family of Sobolevnorms.

In Sects. 5.8, 5.9, and 5.10, we study the Malliavin calculus on the product of theWiener space and Poisson space. Sobolev norms for Wiener–Poisson functionals arestudied in Sect. 5.9. In Sect. 5.10, we study criteria for the smooth density. Finallyin Sect. 5.11, we discuss the composition of a ‘nondegenerate’ Wiener–Poissonfunctional and Schwartz’s distribution.

Results of this chapter will be applied in the next chapter for getting thefundamental solution of heat equations discussed in Chap. 4.


167


https://doi.org/10.1007/978-981-13-3801-4_5

168 5 Malliavin Calculus

5.1 Derivative and Its Adjoint on Wiener Space

The Malliavin calculus on the Wiener space has been studied extensively since the1980s. It is not our purpose to present the theory in full. We will restrict our attentionto the problem of finding smooth density of the law of a given random variable.

Let us define a Wiener space. In this chapter, it is convenient to take a finiteinterval T = [0, T ] for the time parameter. Let d ′ be a positive integer. Let W = Wd ′

0

be the set of all continuous maps w;T → Rd ′ such that w(0) = 0. We denote by

B(W) the smallest σ -field with respect to which w(t) are measurable for any t ∈ T.We denote w(t) by Wt(w) = (W 1

t (w), . . . ,Wd ′t (w)). A probability measure P on

(W,B(W)) is called a Wiener measure if Wt is a d ′-dimensional Wiener process.The triple (W,B(W), P ) is called a (classical) Wiener space. Let S be a completemetric space. An S-valued B(W)-measurable function F(w) (random variable) iscalled an S-valued Wiener functional. If S is the space of real numbers, it is calledsimply a Wiener functional.

In our discussion of the Malliavin calculus, we will often study Lp-estimates ofWiener functionals for large p instead of L2-estimates. Let p > 1. We denote byLp, the set of all real Wiener functionals X such that E[|X|p] < ∞. The norm of

X is defined by ‖X‖Lp = E[|X|p] 1p . We set L∞− = ⋂

p>1 Lp. It is a vector space

of Wiener functionals, and is an algebra, i.e., if X, Y ∈ L∞−, then XY ∈ L∞−.Further, the set L∞− is an F -space with respect to a countable family of norms‖ ‖n = ‖ ‖Ln, n ∈ N, where N is the set of all positive integers. Here a vector spaceL is called an F -space (a version of Frechet space) if it is equipped with a countablefamily of norms ‖ ‖n, n ∈ N and is a complete metric space by the metric

d(X, Y ) =∞∑

n=1

1

2n

‖X − Y‖n1 + ‖X − Y‖n .

Hence a sequence {Xm} of L∞− converges to X ∈ L∞− with respect to the metricd if and only if ‖Xm − X‖n converges to 0 for any positive integer n. Note that ifthe family of norms are ordered in a different way, ‖ ‖n′ , n′ ∈ N, the correspondingmetric d ′ defines the equivalent topology for L∞−.

Let H be the totality of d ′-dimensional real measurable functions h(t) =(h1(t), . . . , hd ′(t)) on T such that their L2-norms |h|H := (

∫T|h(t)|2 dt)1/2 are

finite. Then H is a real Hilbert space with the inner product

〈f, g〉 =∫

T

( d ′∑

k=1

f k(t)gk(t))dt.

The space H is called a Cameron–Martin space.

5.1 Derivative and Its Adjoint on Wiener Space 169

Given h ∈ H , we want to define a linear transformation Th of a Wiener functionalF of L∞− by

ThF (w) = F(w + h),

where h(t) := ∫ t

0 h(s) ds, t ∈ T is an element of W. In order that it is welldefined, the transformation should be absolutely continuous, i.e., if F(w) = 0a.s., then F(w + h) = 0 holds a.s. The fact can be verified by Girsanov’stheorem (Theorem 2.2.2), to be shown below. We will show that Th is a continuouslinear transformation from L∞− into itself. For h ∈ H , we set W(h) =∑d ′

k=1

∫Thk(t) dWk

t . Then {W(h), h ∈ H } is a Gaussian system with mean 0 andcovariance E[W(h)W(k)] = 〈h, k〉. Now, set for t ∈ T,

Zh(t) = exp{ d ′∑

k=1

∫ t

0hk(s) dWk

s − 1

2

∫ t

0|h(s)|2 ds

}.

It is a positive Lp-martingale with mean 1 for any p ≥ 2 (Proposition 2.2.3). Then

Zh = Zh(T ) = exp{W(h)− 1

2〈h, h〉

}(5.1)

is an element of L∞−. Define another probability measure Qh by dQh = Zh dP .Then by Girsanov’s theorem (Theorem 2.2.2), w(t)− h(t) is a Wiener process withrespect to Qh. Therefore, the law of F(w + h) with respect to P coincides with thelaw of F with respect to Qh. Then we have

E[|ThF |p] = EQh[|F |p] = E[|F |pZh] ≤ E[F |pr ] 1

r E[Zr ′h ]

1r′ ,

where r ′ is the conjugate of r . Consequently, we get ‖ThF‖Lp ≤ ‖Zh‖1p

Lr′ ‖F‖Lpr ,

showing that Th is a continuous linear transformation from Lpr to Lp. In particular,if F = 0 a.s., then ThF = 0 a.s., so that Th is an absolutely continuoustransformation. Since this is valid for any r > 1 and p > 1, Th is a lineartransformation from L∞− into itself and further, the transformation is continuouswith respect to the metric d. Hence Th is a continuous linear transformation on theF -space L∞−.

Let λ ∈ R. For a given h ∈ H , we consider a family of linear transformations{Tλh, λ ∈ R} on L∞−. These satisfy the group property TλhTλ′h = T(λ+λ′)h for anyλ, λ′ ∈ R. We will define the derivative operator as the infinitesimal generator ofthe one-parameter group of linear transformations {Tλh}. Let F ∈ L∞−. Supposethat there exists an H -valued Wiener functional F ′ with |F ′|H ∈ L∞− satisfying

limλ→0

d(TλhF − F

λ, 〈F ′, h〉

)= 0, (5.2)


for any h ∈ H(0), which is a dense subset of H . Then F is said to be H -differentiableand F ′ is called the H - derivative. H -derivative of a given F is at most unique. Inthe following, we denote it as DF . We denote by D the set of all H -differentiableWiener functionals F of L∞−.

The H -valued functional DF is often written as DtF or (D(1)t F, . . . ,D

(d ′)t F ),

t ∈ T. It may be regarded as an Rd ′ -valued stochastic process. However, we should

note that DtF may not be well defined for any t a.s. P , but it is well defined a.e.t, w with respect to the product measure dt dP .

We will show differential rules for composite functionals.

Proposition 5.1.1 Let F1, . . . , Fd ∈ D. Let f (x1, . . . , xd) be a C1-function onR

d such that f and ∂xj f := ∂∂xj

f, j = 1, . . . , d are of polynomial growth. Then

f (F1, . . . , Fd) belongs to D. Further, we have

Df (F1, . . . , Fd) =d∑

j=1

∂xj f (F1, . . . , Fd) ·DFj , a.s. (5.3)

Proof We have by the mean value theorem,

1

λ

{f (TλhF1, . . . , TλhFd)− f (F1, . . . , Fd)

}

=∑

j

∂xj f (F1 + θ(TλhF1 − F1), . . . , Fd + θ(TλhFd − Fd))× TλhFj − Fj

λ,

a.s., where |θ | ≤ 1. Let λ tend to 0. Since Fi are H -differentiable, TλhFj

converge to Fj andTλhFj−Fj

λconverge to 〈DFj , h〉 in L∞−. Therefore, the above

converges to∑

j ∂xj f (F1, . . . , Fd) · 〈DFj , h〉 in L∞−. Therefore f (F1, . . . , Fd) isH -differentiable and the equality (5.3) holds. ��

For a positive integer i ≥ 2, we denote by H⊗i the i-times tensor product of H .

It is again a Hilbert space with norm | |H⊗i defined by |f |H⊗i = (∫Ti |f (t)|2 dt)

12 ,

where Ti is the i-times product of the set T and t = (t1, . . . , ti ) ∈ T

i . Suppose thatF is H -differentiable and that the functional 〈DF, h1〉 is H -differentiable for anyh1 ∈ H . If there is an H ⊗H -valued functional D2F such that |D2F |H⊗H ∈ L∞−and

〈D2F, h1 ⊗ h2〉 = 〈D〈DF, h1〉, h2〉, ∀h1, h2 ∈ H,

then we say that DF is H -differentiable and D2F is the second H -derivative.For i ≥ 3, we can define the i-th H -derivative DiF , successively. Suppose that

the (i− 1)-th H -derivative Di−1F is well defined as an H⊗(i−1)-valued Wienerfunctional such that |Di−1F |H⊗(i−1) ∈ L∞− and 〈Di−1F, h1 ⊗ · · · ⊗ hi−1〉 is H -


differentiable. If there is an H⊗i-valued functional DiF such that |DiF |H⊗i ∈ L∞−and

〈DiF, h1 ⊗ · · · ⊗ hi−1 ⊗ hi〉 = 〈D〈Di−1F, h1 ⊗ · · · ⊗ hi−1〉, hi〉,

for all h1, . . . , hi ∈ H , then DiF is called the i-th H -derivative of F . Note that forany i, the i-th H -derivative of a given F is at most unique. DiF is an element ofH⊗i . It is written as Dt1 · · ·DtiF , DtF or Di

tF , where t = (t1, . . . , ti ) ∈ Ti . It is

well defined a.e. with respect to the product measure dt dP . We may regard it as a(d ′)i-dimensional random field with parameter t = (t1, . . . , ti ) ∈ T

i .The totality of Wiener functionals F in L∞−, such that i-th H -derivative DiF

exists for any i ≤ l, is denoted by Dl and we set D∞ = ⋂l Dl . Elements of Dl

and those of D∞ are called l-times H -differentiable and infinitely H -differentiable,respectively.

We will give examples of H -differentiable Wiener functionals. Let f ∈ H andlet F = W(f ) = ∑d ′

k=1

∫Tf k(t) dWk

t . It is a linear Wiener functional. We haveTλhF = W(f )+λ〈f, h〉 a.s. Therefore F is H -differentiable and 〈DF, h〉 = 〈f, h〉holds a.s. This proves DtF = f (t) a.e. dt dP . Since it is a constant function, itshigher-order derivatives are 0, i.e., Di

tF = 0 holds if i ≥ 2.Let f1, . . . , fd ∈ H and let p;Rd → R be a polynomial. A functional F

given by p(W(f1), . . . ,W(fd)) is called a polynomial Wiener functional or simplya polynomial. We denote by P the set of all polynomials. Note that any F ∈ P hasa finite moment of any order, i.e., E[|F |p] < ∞ for any p > 1. Therefore, P is adense subset of L∞−.

We will show that the domain D∞ contains the set of polynomials P andtherefore it is a dense subset of L∞−. If F is a polynomial of the form F =p(W(f1), . . . ,W(fd)), then as in Proposition 5.1.1, F is in D∞ and satisfies

DtF =∑

j

∂xj p(W(f1), . . . ,W(fd))fj (t), a.e. dt dP . (5.4)

Further, the polynomial F is i-times H -differentiable for any i and we have

Dt1 · · ·DtiF =∑

j1,...,ji

∂xj1· · · ∂xji p(W(f1), . . . ,W(fd))fj1(t1) · · · fji (ti),

a.e. dtdP . Hence P ⊂ D∞. If F is a polynomial of degree d, then Ddt F is a constant

function and Dd+1t F = 0.

Another interesting nonlinear Wiener functional, which is H -differentiable, is asolution of a continuous SDE. In Sect. 6.1, we will show the fact and will derive itsderivative explicitly.

Concerning the differential operator D, we have a formula of the integration byparts.


Proposition 5.1.2 If F ∈ D∞, it holds that

E[〈DF, h〉] = E[FW(h)] (5.5)

for any h ∈ H . Further, if F,G ∈ D∞, it holds that

E[F 〈DG,h〉] = E[−G〈DF, h〉 +GFW(h)] (5.6)

for any h ∈ H .

Proof Let λ ∈ R and let Zλh be the functional defined by (5.1) for λh ∈ H . Thenwe have E[TλhF ] = E[FZλh]. Differentiate both sides with respect to λ at λ = 0.Since d

dλE[TλhF ]|λ=0 = E[〈DF, h〉] and

d(Zλh − 1

λ,W(h)

)→ 0

as λ → 0, we get (5.5).Next, if F,G are in D∞, then the product FG is in D∞ and satisfies D(FG) =

DF ·G+ F ·DG. Hence we get the equality (5.6). ��We will define the adjoint operator of the derivative operator D. Let Fl, l =

1, . . . , n be elements of D∞ and hl, l = 1, . . . , n be elements of H . Then∑

l Flhl

is an H -valued random variable. We may regard∑

Flhl(t) as a d ′-dimensionalstochastic process with parameter t ∈ T, a.e. dt dP . It is called a simple functionalwith parameter t ∈ T. We denote by S∞

Tthe set of all simple functionals with

parameter T. For Xt = ∑nl=1 Flhl(t) ∈ S∞

T, we set

δ(X) =n∑

l=1

FlW(hl)−n∑

l=1

〈DFl, hl〉. (5.7)

Then δ is a linear map from S∞T

to L∞−. It is called the adjoint operator of D, orthe Skorohod integral of Xt by the Wiener process. The followings are known asproperties of Skorohod integrals. See Nualart [88].

Lemma 5.1.1

1. For any X = {Xt } ∈ S∞T

, δ(X) satisfies the adjoint property

E[Gδ(X)] = E[〈DG,X〉] (5.8)

for any G ∈ D∞.2. For any X = {Xt } ∈ S∞

T, δ(X) is H -differentiable and satisfies the following

commutation relation:

Dt ′δ(X) = δ(Dt ′X)+Xt ′, a.e. dt ′ dP. (5.9)


3. The isometric property: E[δ(X)δ(Y )] is equal to

d ′∑

k=1

E[ ∫

T

Xkt Y

kt dt

]+

d ′∑

k,k′=1

E[ ∫

T

∫

T

D(k′)t Xk

sD(k′)s Y k

t ds dt]

(5.10)

for any X, Y ∈ S∞T

.

Equality (5.10) is called the Shigekawa–Nualart–Pardoux energy identity (see [89,101]).

Proof Since Gδ(X) = ∑l GFlW(hl)−∑

l G〈DFl, hl〉, we have

E[Gδ(X)] = E[∑

l

GFlW(hl)−∑

l

G〈DFl, hl〉].

Apply Proposition 5.1.2 to the above. Then the right-hand side is written as

∑

l

E[〈DG,hl〉Fl] = E[〈DG,X〉].

Therefore we get (5.8).Further, δ(X) is H -differentiable, since each term of the right-hand side of (5.7)

is H -differentiable. We have indeed,

Dtδ(X) =∑

l

DtFlW(hl)+∑

l

Flhl(t)−∑

l

Dt (〈DFl, hl〉)

= δ(DtX)+Xt, a.e. dt dP .

Therefore the equality of the commutation relation (5.9) holds.Suppose X, Y ∈ S∞

T. Then we have

E[δ(X)δ(Y )] = E[〈X,Dδ(Y )〉]= E[〈X, Y 〉] + E[〈X, δ(DY)〉]=

∑

k

E[ ∫

T

Xkt Y

kt dt

]+

∑

k,k′E[ ∫

T

∫

T

D(k′)s Xk

t D(k′)t Y k

s ds dt].

Here we used (5.8), (5.9) and (5.8) in turn. ��Note The derivative operator D is a basic tool in stochastic analysis on theWiener space. It has been discussed in various contexts. Cameron [15] introducedthe derivative D on the Wiener space and showed a version of the adjointformula (5.8). It was extended and applied to potential theory on the Wiener spaceby Gross [34, 35]. Malliavin [77] studied the derivative operator D through theOrnstein–Uhlenbeck operator on the Wiener space. Stroock [105] discussed the


finite dimensional approximation of the operator D. Ikeda–Watanabe [40, 41] andWatanabe [116] developed Malliavin’s theory by introducing norms of Sobolev-typethrough the Ornstein–Uhlenbeck operator.

Our definition of the derivative D is close to Cameron [15], Gross [34, 35] andShigekawa [100]. It can be applied directly to solutions F of stochastic differentialequations driven by Wiener processes and further we can compute the derivativesDF explicitly. It will be discussed in Sect. 6.1.

The absolute continuity of the transformation Th on the Wiener space was shownby Cameron–Martin [16]. It was extended to a wider class of transformationsby Maruyama [82] and Girsanov [33], and is called the Girsanov transformation(Theorem 2.2).

5.2 Sobolev Norms for Wiener Functionals

Let us define Sobolev norms of Lp-type for Wiener functionals, making use ofderivative operator D. It is convenient to set T

0 = H 0 = {∅} (empty set),D0F = D0

∅F = F and |D0F |H 0 = |F |. Let N be the set of all nonnegativeintegers. For m ∈ N and p ≥ 2, we will define a norm for F belonging to Dm

by ‖F‖0,p = E[|D0F |pH 0]

1p = E[|F |p] 1

p if m = 0 and

‖F‖m,p := E[ m∑

i=0

|DiF |pH⊗i

] 1p = E

[|F |p+

m∑

i=1

( ∫

Ti

|DitF |2 dt

) p2] 1

p, (5.11)

if m ≥ 1. It is straightforward to prove that ‖ ‖m,p is a norm for any m ∈ N andp ≥ 2. The system of norms {‖ ‖m,p,m ∈ N, p ≥ 2} is called Sobolev norms forWiener functionals. We denote the completion of Dm by the above norm by Dm,p.We set D∞ = ⋂

m,p Dm,p. Then D∞ ⊂ L∞−. It is an F -space with countable

norms {‖ ‖m,p,m ∈ N, p = 2, 3, . . .}. Further, P ⊂ D∞ ⊂ D∞. Elements of D∞are called a smooth Wiener functional.

Next, let X = {Xt } be an element of S∞T

. It is written as X = ∑l Flhl

(finite sum), where Fl ∈ D∞ and hl ∈ H . Its H -derivatives are defined byDiX = ∑

l DiFlhl . It is an H⊗(i+1)-valued functional. Its norm is

|DiX|H⊗(i+1) =( ∫

Ti+1|∑

l

DitFlhl(t)|2 dt dt

) 12.

We will define Sobolev norms for X ∈ S∞T

as follows. For m ∈ N and p ≥ 2, we

set ‖X‖0,p = E[|X|pH ] 1p if m = 0 and

5.2 Sobolev Norms for Wiener Functionals 175

‖X‖m,p := E[ m∑

i=0

|DiX|pH⊗(i+1)

] 1p

(5.12)

= E[( ∫

T

|Xt |2 dt) p

2 +m∑

i=1

( ∫

Ti+1|Di

tXt |2 dt dt) p

2] 1

p,

if m ≥ 1. If X ∈ S∞T

does not depend on t , denote it by F . Then the above norm‖X‖m,p coincides with

√T ‖F‖m,p. Hence these two norms are compatible with

each other.For any X ∈ S∞

T, its norms ‖X‖m,p are finite for all m,p. The completion of

S∞T

by the above norm is denoted by Dm,p

T. We set D∞

T= ⋂

m,p Dm,p

T. D∞

Tis an

F -space with respect to countable norms ‖ ‖m,p, m ∈ N, p = 2, 3, . . ..

Proposition 5.2.1 The derivative operator D is extended as a continuous linearoperator from F-space D∞ to F-space D∞

T. Further, for any positive integer m and

p ≥ 2, it holds that ‖DF‖m,p ≤ ‖F‖m+1,p for any F ∈ D∞.

Proof Let {hl} be a complete orthonormal system in H . If F ∈ D∞, DF isexpanded as

∑l Glhl , where Gl = 〈DF, hl〉 and Gl ∈ D∞. Its finite sum∑n

l=1 Glhl belongs to S∞T

. Then its limit DF belongs to D∞T

. Further, it satisfies‖DF‖m−1,p ≤ ‖F‖m,p for any m and p ≥ 2. ��

The extended DF is again called the H -derivative of F ∈ D∞.

Lemma 5.2.1 (Hölder’s inequality) If F,G ∈ D∞ and X ∈ D∞T

, then FG ∈ D∞and FX ∈ D∞

T. Let m ∈ N, p ≥ 2 and r, r ′ > 1 with 1/r + 1/r ′ = 1. Then there

exists a positive constant c such that

‖FG‖m,p ≤ c‖F‖m,pr‖G‖m,pr ′ ,

‖FX‖m,p ≤ c‖F‖m,pr‖X‖m,pr ′ (5.13)

hold for any F,G,X with the above properties.

Proof We give the proof of the latter inequality only. We first consider the casem = 0. We have by Hölder’s inequality

‖FX‖p0,p = E[|FX|pH ] ≤ E[|F |pr ] 1r E[|X|pr ′H ] 1

r′ ≤ ‖F‖p0,pr‖X‖p0,pr ′ .

We next consider the case m = 1. Since D(FX) = DF ·X + F ·DX, we have

|D(FX)|H⊗2 ≤ |DF |H |X|H + |F ||DX|H⊗2 .


Therefore, using Hölder’s inequality, we get

E[|D(FX)|pH⊗2] ≤ 2p

{E[|DF |prH ] 1

r E[|X|pr ′H ] 1r′ + E[|F |pr ] 1

r E[|DX|pr ′H⊗2 ]

1r′}

≤ 2p{‖F‖p1,pr‖X‖p0,pr ′ + ‖F‖p0,pr‖X‖p1,pr ′

}.

Therefore we get (5.13) in the case m = 1. Repeating this argument inductively, weget (5.13) for any positive integer m. ��

We will consider the L2-extension of the adjoint operator δ, making use ofSobolev norms. The adjoint operator δ is extended to a continuous linear operatorfrom D1,2

Tto D0,2 and satisfies the inequality ‖δ(X)‖0,2 ≤ ‖X‖1,2 for any X ∈ D1,2

T,

in view of the isometric property (5.10). δ(X) is called again the Skorohod integralof Xt (by Wiener process).

We show that if Xt is a predictable process, the Skorohod integral coincides withthe Itô integral.

Proposition 5.2.2 Let {Ft } be the filtration generated by the Wiener processWt and let Xt be a predictable functional with respect to the filtration {Ft },belonging to D1,2

T. Then the Skorohod integral δ(X) coincides with the Itô integral∑

k

∫TXk

s dWks . Further, the isometric equality is written for short as

E[δ(X)2] =d ′∑

k=1

E[ ∫

T

(Xkt )

2 dt]. (5.14)

Proof Suppose that Xt := ∑d ′k=1

∑nm=1 F

km1(tm−1,tm](t)ek is a simple process,

where Fkm are real Ftm−1 -measurable for any m and ek, k = 1, . . . , d ′ are unit vectors

in Rd ′ . Then the k-th component of Xt is Xk

t = ∑m Fk

m1(tm−1,tm](t). It holds thatDtF

km = 0 for t > tm−1. Therefore we have 〈DFk

m, 1(tm−1.tm]〉 = 0. Consequently,we have from the formula (5.7),

δ(X) =∑

k

∑

m

Fkm(W

ktm−Wk

tm−1) =

∑

k

∫

T

Xks dW

ks .

The fact can be extended to any predictable element X belonging to D1,2T

.If Xs is predictable then we have DtXs = 0 if t ≥ s, as we have shown above.

Therefore we have D(k′)t Xk

sD(k′)s Xk

t = 0 a.e. ds dt for any k, k′. Consequently,the last integral in (5.10) is 0, and we have the isometric property E[δ(X)2] =∑

k E[∫T(Xk

t )2 dt]. ��

If Xt is not predictable, the Skorohod integral is called the anticipating stochasticintegral.


Remark We shall consider the anticipating stochastic differential equation, orSkorohod SDE. Let Wt be a one-dimensional Wiener process. We denote theanticipating stochastic integral of Xs1(0,t](s) by

∫ t

0 XsδWs . We consider stochasticdifferential equations represented by anticipating stochastic integrals;

Xt = X0 +∫ t

0σ(Xs)δWs +

∫ t

0b(Xs) ds,

where X0 ∈ D1,2 and σ, b are smooth functions on R and Xt ∈ D1,2T

. The equationis called the Skorohod equation. If we replace the above Skorohod integral by theItô integral

∫ t

0 σ(Xs) dWs in the above equation, the equation coincides with the Itôequation discussed in Chap. 3.

We are interested in the existence and the uniqueness of the solution of theSkorohod equation. If X0 is F0-measurable, the Itô equation has a unique solution.Further, it is also the solution of the Skorohod equation, since the solution Xt ofthe Itô equation is predictable and hence

∫ t

0 σ(Xs)δWs = ∫ t

0 σ(Xs) dWs holds(Proposition 5.2.2). Then the Skorohod equation also has a unique solution if X0is F0-measurable. If X0 is not F0-measurable, Itô equation has no solution (notwell defined). However, the Skorohod equation has a solution. Indeed, let Φ0,t (x)

be the stochastic flow generated by the Itô equation. Then Φ0,t (x) ∈ D1,2T

for any

x (see Sect. 6.1) and the composite process Xt = Φ0,t (X0) belongs to D1,2T

. Hencethe Skorohod integral

∫ t

0 σ(Xs)δWs is well defined and Xt satisfies the Skorohodequation. The uniqueness of the solution of the Skorohod equation appears to be adifficult problem. For details, see Nualart [88].

By the isometric property of the Skorohod integral δ(X), we have the inequality‖δ(X)‖0,2 ≤ ‖X‖1,2. The inequality can be extended to higher derivatives.

Proposition 5.2.3 For any m ∈ N, there exists a positive constant cm such that theinequality ‖δ(X)‖m,2 ≤ cm‖X‖m+1,2 holds for any X ∈ D∞

T.

Proof To avoid complicated notations, we will consider the case of a one-dimensional Wiener space (d ′ = 1), where T is equal to the interval [0, 1]. Wewill prove the inequality of the lemma for X ∈ S∞

T. Let i be a positive integer.

Since Dsδ(X) = Xs + δ(DsX) holds, we have by induction

Disδ(X) = δ(Di

sX)+i∑

h=1

Di−1sh Xsh, (5.15)

where s = (s1, . . . , si) ∈ Ti and sh = s − {sh}. We will compute L2(ds dP )-norm

of each term of the above. Using the adjoint formula (5.8) and the commutationrelation (5.9), we have


E[ ∫

Ti

δ(DisX)2 ds

]=

∫

Ti

E[δ(DisX)2] ds

=∫

Ti

( ∫

T

E[Dtδ(DisX)Di

sXt ] dt)ds

=∫

Ti

( ∫

T

{E[δ(DtDisX)Di

sXt ] + E[|DisXt |2]} dt

)ds

=∫

Ti

( ∫

T

∫

T

E[DtDisXt ′ ·Dt ′D

isXt ] dt ′ dt

)ds +

∫

Ti

∫

T

E[|DisXt |2] dt ds

≤ ‖X‖2i+1,2.

Further, we have

E[ ∫

Ti

|Di−1sh Xsh |2 dsi dsh

]≤ ‖X‖2

i−1,2.

Summing up these inequalities for i = 1, . . . , m, we get the inequality of theproposition. ��

We will extend the inequality for arbitrary Sobolev norms. The following isan analogue of the Burkholder–Davis–Gundy inequality (Proposition 2.2.2) forSkorohod integral.

Theorem 5.2.1 Let δ be the Skorohod integral operator. For any m ∈ N and apositive even number p, there exists a positive constant cm,p such that

‖δ(X)‖m,p ≤ cm,p‖X‖m+p−1,p (5.16)

holds for any X ∈ D∞T

.

Proof It is sufficient to prove the inequality for X ∈ S∞T

, since S∞T

is dense in D∞T

.We will consider the case of a one-dimensional Wiener space (d ′ = 1), where T isequal to the interval [0, 1]. Let p ≥ 2 be a positive even number. Let X ∈ S∞

T. Then

δ(X) ∈ L∞−. By the adjoint property of δ (Lemma 5.1.1), we get

E[δ(X)p] = E[δ(X)δ(X)p−1] = E[ ∫

T

Xt1Dt1(δ(X)p−1) dt1

].

We will decrease the power p − 1 of δ(X)p−1, step by step. By the commutationrelation for δ, we have

Dt1(δ(X)p−1) = (p − 1)δ(X)p−2(Xt1 + δ(Dt1X)).

Therefore, the above is equal to


(p − 1)E[( ∫

T

X2t1dt1

)δ(X)p−2

]+ (p − 1)E

[( ∫

T

Xt1δ(Dt1X) dt1

)δ(X)p−2

].

Apply Lemma 5.1.1 again to the last term. Then it is written as

(p − 1)E[ ∫

T

∫

T

Dt1Xt2Dt2(Xt1δ(X)p−2) dt1 dt2

]

= (p − 1)E[ ∫

T

∫

T

Dt1Xt2

{Dt2Xt1δ(X)p−2

+ (p − 2)Xt1Dt2δ(X) · δ(X)p−3}dt1 dt2

]

= (p − 1)E[ ∫

T

∫

T

Dt1Xt2Dt2Xt1 dt1 dt2δ(X)p−2]

+ (p − 1)(p − 2)E[ ∫

T

∫

T

Dt1Xt2Xt1Xt2 dt1 dt2δ(X)p−3]

+ (p − 1)(p − 2)E[ ∫

T

∫

T

Dt1Xt2Xt1δ(Dt2X) dt1 dt2δ(X)p−3].

Further, if p ≥ 3 the last term is rewritten as

(p − 1)(p − 2)E[ ∫

T

∫

T

∫

T

Dt3(Dt1Xt2Xt1δ(X)p−3)Dt2Xt3 dt1 dt2 dt3

].

Repeating this argument, we find that E[δ(X)p] is equal to sums of the followingterms:

E[ ∫

Tq

Dj1t1X

i1t1· · ·Djq

tq Xiqtqdt

],

p

2≤ q ≤ p.

Here il are nonnegative integers satisfying il = 1, 2 and i1 + · · · + iq = p; jl arenonnegative integers satisfying 0 ≤ jl ≤ p − 1 and j1 + · · · + jq ≤ p. Further,t = (t1, . . . , tp) and tl are subsets of t \ {tl}. If jl = 0, tl is an empty set and it holds

that Djltl X

iltl= X

iltl

. Then, by Hölder’s inequality, the above integral is dominated by

∏

l;il �=0,jl �=0

E[( ∫

Tjl

∫

T

|Djltl X

iltl| 2il dtl dtl

) p2] il

p ×∏

l;il �=0,jl=0

E[( ∫

T

|Xiltl| 2il dtl

) p2] il

p.

(5.17)It holds that for il = 1, 2 and jl �= 0,

E[( ∫

Tjl

∫

T

|Djltl X

iltl| 2il dtl dtl

) p2] il

p ≤ E[( ∫

Tjl

|Djltl X|2H dtl

) p2] il

p

≤ ‖X‖iljl ,p ≤ ‖X‖ilp−1,p.


Further, we have

E[( ∫

T

|Xiltl| 2il dtl

) p2] il

p ≤ ‖X‖il0,p ≤ ‖X‖ilp−1,p.

Then (5.17) is dominated by∏

l;il �=0 ‖X‖ilp−1,p = ‖X‖pp−1,p. Consequently, there

exists a positive constant c′p such that E[δ(X)p] ≤ c′p‖X‖pp−1,p. This proves theinequality ‖δ(X)‖0,p ≤ c0,p‖X‖p−1,p.

We will next consider the case where m ≥ 1 and p is a positive even number. Bythe definition of Sobolev norms, we have

‖δ(X)‖pm,p = E[δ(X)p] +m∑

i=1

E[( ∫

Ti

|Disδ(X)|2 ds

) p2]. (5.18)

The first term of the right-hand side has been computed. We shall consider otherterms. Note the equality (5.15). Then there is a positive constant cp such that

E[( ∫

Ti

|Disδ(X)|2 ds

) p2]

(5.19)

≤ cp

{E[( ∫

Ti

|δ(DisX)|2 ds

) p2]+

i∑

h=1

E[( ∫

Ti−1×T

|Di−1sh

Xsh |2 dsh dsh

) p2]}

.

We have

E[( ∫

Ti

|δ(DisX)|2 ds

) p2]= E

[ ∫

Ti

· · ·∫

Ti

δ(Dis1X)2 · · · δ(Di

s p2

X)2 ds1 · · · ds p2

].

By a computation similar to that for E[δ(X)p], we find that the above is equal to thesum of terms

E[ ∫

T

ip2

∫

Tq

Dj1t1(Di

s1Xt1)

i1 · · ·Djqtq (D

is p

2

Xtp)iq ds1 · · · ds p

2dt

],

where t = (t1, . . . , tq) and q = p2 , . . . , p. By Hölder’s inequality, the above is

dominated by

∏

l;il �=0,jl �=0

E[( ∫

Tjl

∫

Ti

∫

T

|Djltl D

isiXtl |

il2 dtl dsi dtl

) p2] il

p

×∏

l;il �=0,jl=0

E[( ∫

Tl

∫

T

|DisiXtl |

il2 dsi dtl

) p2] il

p. (5.20)


It holds that

E[( ∫

Tjl

∫

Ti

∫

T

|Djltl D

isiXtl |

il2 dtl dsi dtl

) p2] il

p ≤ ‖X‖ili+jl ,p≤ ‖X‖ili+p−1,p,

E[( ∫

Ti

∫

T

|DisiXtl |

il2 dsl dtl

) p2] il

p ≤ ‖X‖ili,p ≤ ‖X|ili+p−1,p.

Therefore, (5.20) is dominated by∏

l;il �=0 ‖X‖ili+p−1,p = ‖X‖pi+p−1,p . Conse-quently there exists a positive constant c′′ such that

E[( ∫

Ti

|δ(DisX)|2 ds

) p2]≤ c′′‖X‖pi+p−1,p. (5.21)

We shall consider the last term of (5.19). It holds that

E[( ∫

Ti−1×T

|Di−1si Xsh |2 dsi dsh

) p2]≤ ‖X‖pi−1,p. (5.22)

Substitute (5.21) and (5.22) in (5.19). Then we get

E[( ∫

Ti

|Disδ(X)|2 ds

) p2]≤ c‖X‖pi+p−1,p.

Then we obtain from (5.18) the inequality ‖δ(X)‖pm,p ≤ c′m,p‖X‖pm+p−1,p. Thisproves ‖δ(X)‖m,p ≤ cm,p‖X‖m+p−1,p. ��

We will summarize properties of the adjoint operator. For a real number r , wedenote by [r] the largest integer which is dominated by r . If r > 0, 2[r/2] is equal tothe largest even number which is dominated by r . Further, if r > 0, −2[−r/2] is thesmallest even number which is bigger than or equal to r . We set 〈r〉 = −2[−r/2].Theorem 5.2.2

1. Skorohod integral operator δ is extended to a continuous linear operator fromF-space D∞

Tto F-space D∞. Further, for any m ∈ N and p ≥ 2, there exists a

positive constant cm,p such that

‖δ(X)‖m,p ≤ cm,p‖X‖m+〈p〉−1,〈p〉 (5.23)

holds for any X ∈ D∞T

.2. Adjoint formula (5.8) holds for any X ∈ D∞

Tand G ∈ D∞.

3. Commutation relation (5.9) holds for any X ∈ D∞T

.4. Isometric property (5.10) holds for any X, Y ∈ D∞

T.


Proof We will prove (5.23). Since the norm ‖X‖m,p is nondecreasing with respectto p, we have by Theorem 5.2.1,

‖δ(X)‖m,p ≤ ‖δ(X)‖m,〈p〉 ≤ c‖X‖m+〈p〉−1,〈p〉.

Assertions (2)–(4) are immediate from Lemma 5.1.1. ��Derivative operator D and its adjoint δ can be applied to complex-valued

functionals. Let F = F1 + iF2, where Fk, k = 1, 2 are real smooth Wienerfunctionals and i = √−1. Then we define DF by DF1+ iDF2. Let X = X1+ iX2,where Xk, k = 1, 2 are real smooth H -valued functionals. Then we define δ(X) byδ(X1)+ iδ(X2). Norms ‖F‖m,p and ‖X‖m,p for these complex-valued functionalsare defined in the same way. F is called smooth if F1 and F2 are smooth. The set ofsmooth functionals is denoted again as D∞. Complex H -valued smooth functionalsare defined similarly and we denote by D∞

Tthe set of all complex H -valued smooth

functionals. Then Theorems 5.2.1 and 5.2.2 are valid for complex functionals.

Note For arguments of the Malliavin calculus based on the Ornstein–Uhlenbecksemigroup, we refer to Malliavin [77], Ikeda–Watanabe [40], Watanabe [116], andfurther, books of Shigekawa [102], Nualart [88] and Matsumoto–Taniguchi [83].Sobolev-type norms for Wiener functional were defined in [40] and [116] in adifferent context. In these works, norms of a Wiener functional F are defined for

all real m and p > 1 by E[|(I − L)m2 F |p] 1

p , where L is the Ornstein–Uhlenbeckoperator. To avoid confusion of notations, we will denote their norms as ‖F‖′m,p.They showed that if m is a positive integer,

∑0≤m′≤m ‖F‖′

m′,p (j ′ are nonnegativeintegers) is equivalent to our norm ‖F‖m,p. The fact is called Meyer’s equivalence[102].

Further, with respect to the estimate of the adjoint operator δ, they assert theinequality ||δ(X)‖′m,p ≤ c′‖X‖′m+1,p. In the case where m is a positive integer, it isequivalent to

||δ(X)‖m,p ≤ c‖X‖m+1,p, (5.24)

which is stronger than the inequality of our Theorem 5.2.1. However, its proof isnot clear to the author. In this monograph, we have given a new direct proof for aslightly weaker assertion (Theorem 5.2.1).

Studies of the Malliavin calculus for non-classical Wiener spaces (curved Wienerspace etc.) may be found in Driver’s survey work on Wiener space [23].

5.3 Nondegenerate Wiener Functionals 183

5.3 Nondegenerate Wiener Functionals

Let F = (F 1, . . . , F d) be a d-dimensional Wiener functional. If its compo-nents F i, i = 1, . . . , d are infinitely H -differentiable, F is called infinitelyH -differentiable. We denote by (D∞)d the set of all such d-dimensional functionals.If each component F i belongs to D∞, F is said to belong to (D∞)d . Its Sobolevnorm is defined by the sum of Sobolev norms of components Fj , j = 1, . . . , d .It holds that (D∞)d ⊂ (D∞)d . Elements of the latter are called smooth Wienerfunctional. The space (D∞

T)d and the norm of its element Xt = (X1

t , . . . , Xdt ) are

defined similarly.Let F = (F 1, . . . , F d) be an element of (D∞)d . Let G be an element of D∞.

We consider the law of F weighted by G:

μG(dx) = E[1dx(F )G].

In this section, we study the existence of the smooth density of the law μG(dx) withrespect to the Lebesgue measure. For this purpose, we consider its characteristicfunction. It is given by

ψG(v) =∫

Rd

ei(v,x)μG(dx) = E[ei(v,F )G], (5.25)

where i = √−1 is pure imaginary. It is a C∞-function of v. Indeed, let F j =(F 1)j1 · · · (F d)jd , where j = (j1, . . . , jd) is a multi-index of nonnegative integersjl, l = 1, . . . , d. Since |F | ∈ L∞−, the functional ei(v,F )F jG is integrable for anyj and we can change the order of the derivatives ∂

jv ≡ ∂

j1v1 · · · ∂jdvd and integral. Then

we have

∂ jvE[ei(v,F )G] = i|j|E[ei(v,F )F jG].

This shows that the characteristic function ψG(v) is a C∞-function and ∂jvψG(v)

coincides with the above function.In Chap. 1, we discussed the existence of the smooth density of a weighted law

μG. Proposition 1.1.2 tells us that if |ψG(v)||v|n is integrable for some positiveinteger n, then the weighted law μG(dx) has a Cn-density function fG(x). Now,in order to show the integrability of |ψG(v)||v|n, we need to assume that F isnondegenerate, which will be defined using the Malliavin covariance. For a smoothWiener functional F = (F 1, . . . , F d) we define a matrix RF called the Malliavincovariance by

RF =(〈DFi,DFj 〉

)=

∫

T

DtF(DtF )T dt. (5.26)


Let Sd−1 = {v ∈ Rd ; |v| = 1}. For θ ∈ Sd−1, we set RF (θ) = (θ, RF θ). The

functional F is called nondegenerate, if RF (θ) are invertible a.s. for any θ ∈ Sd−1and inverses satisfy

supθ

E[RF (θ)−p] < ∞ (5.27)

for any p, where the supremum for θ is taken in the set Sd−1. Since the norm ‖ ‖0,pcoincides with Lp-norms, the above condition is equivalent to that the inequalitysupθ ‖RF (θ)−1‖0,p < ∞ holds for any p ≥ 2.

Remark We give a simple example of a nondegenerate Wiener functional. Let F =(F 1, . . . , F d) be a linear functional of the Wiener process Wt given by

F i =∑

k

∫

T

hik(s) dWks , i = 1, . . . , d,

where hik(s), i = 1, . . . , d, k = 1, . . . , d ′ are square integrable functions on T. Wesaw in Sect. 5.1 that the above F is H -differentiable and its derivative is computedas DtF = (

hi,k(t)) = H(t) (d × d ′-matrix). Then its Malliavin covariance is equal

to a constant (non-random) matrix

RF =(∑

k

∫

T

hik(t)hjk(t) dt)=

∫

T

H(t)H(t)T dt.

Therefore F is nondegenerate if and only if the above matrix RF is invertible.The above F is Gaussian distributed and its covariance matrix is computed

as∫TH(t)H(t)T dt . Therefore, the Malliavin covariance RF coincides with the

covariance matrix of F . It is well known that a Gaussian distribution has a C∞-density if its covariance matrix is nondegenerate (positive definite). The densityfunction is given by (1.8) with A = RF and b = 0.

Now, we want to show that the law of any nondegenerate Wiener functional hasa C∞-density. We first discuss the inverse of the Malliavin covariance. We need alemma.

Lemma 5.3.1 Suppose that G ∈ D∞ is invertible and the inverse G−1 belongs toL∞−. Then G−1 belongs to D∞. Further, for any m,p, there exists cm,p > 0 suchthat the inequality

‖G−1‖m,p ≤ cm,p(1 + ‖G‖m,2mp)m(1 + ‖G−1‖0,2(m+1)p)

m+1 (5.28)

holds for any G with the above property.


Proof If G−1 ∈ L∞−, then Tλh(G−1) ∈ L∞−. Then Tλh(G) is invertible and

satisfies Tλh(G−1) = 1/Tλh(G) a.s. Hence we have

Tλh(G−1)−G−1

λ= −1

Tλh(G)G

Tλh(G)−G

λ, a.s.

Let λ tend to 0. The right-hand side converges to −1G2 〈DG,h〉 in Lp for any p > 1.

This means that G−1 is H -differentiable and D(G−1) = −1G2 DG. Then, repeating

this argument inductively, we find that the i-th H -derivative of G−1 exists and isequal to

DitG

−1 =∑

(−1)rGi−rD

i1t1G · · ·Dir

tr G

Gi+1, t = (t1, . . . , tr ),

where the sum is taken for positive integers r and ik such that r ≤ i and i1+· · ·+ir =i. Using Hölder’s inequality, we get

E[( ∫ ∣∣∣

Gi−rDi1t1G · · ·Dir

tr G

Gi+1

∣∣∣2dt

) p2]

(5.29)

≤ E[( ∫ ∣∣Gi−rD

i1t1G · · ·Dir

tr G∣∣2 dt

)p] 12E[|G|−2(i+1)p

] 12

≤ c′‖G‖ipi,2ip‖G−1‖(i+1)p0,2(i+1)p

≤ c′(1 + ‖G‖i,2ip)ip(1 + ‖G−1‖0,2(i+1)p)(i+1)p.

Now for a given m ∈ N, the above inequality is valid for any i = 0, 1, 2, . . . , m.Further, the last term of the above is nondecreasing with respect to i and p.Therefore, summing up (5.29) for i = 0, . . . , m, we get the inequality

‖G−1‖pm,p ≤ cm,p(1 + ‖G‖m,2mp)mp(1 + ‖G−1‖0,2(m+1)p)

(m+1)p.

This proves the inequality of the lemma. ��Proposition 5.3.1 If F of (D∞)d is nondegenerate, RF (θ)−1 belongs to D∞ forany θ . Further, for any m,p, there exists a positive constant cm,p such that

supθ

‖RF (θ)−1‖m,p ≤ cm,p(1 + ‖DF‖2m,4mp)

m(1 + supθ

‖RF (θ)−1‖0,2(m+1)p)m+1

(5.30)holds for any nondegenerate F .


Proof We apply Lemma 5.3.1 for G = RF (θ). It is sufficient to show that any m,p,there exists a positive constant cm,p such that

supθ

‖RF (θ)‖m,p ≤ cm,p‖DF‖2m,2p. (5.31)

The inequality is immediately verified if m = 0. We consider the case m ≥ 1. Leti ≤ m. Since RF (θ) = ∫

T(θ,DtF )2 dt , and

Dit (θ,DtF )2 =

∑

i′,i′′∈N,i′+i′′=i

Di′t′ (θ,DtF ) ·Di′′

t′′ (θ,DtF ),

we have

|DitR

F (θ)| ≤∑

i′,i′′

∫

T

|Di′t′ (θ,DtF ) ·Di′′

t′′ (θ,DtF )| dt

≤∑

i′,i′′

( ∫

T

|Di′t′ (θ,DtF )|2 dt

) 12( ∫

T

|Di′′t′′ (θ,DtF )|2 dt

) 12.

Therefore we get

‖RF (θ)‖pm,p ≤∑

i≤m

E[( ∫

Ti

|DitR

F (θ)|2 dt) p

2]

≤∑

i′,i′′≤m

E[( ∫

Ti+1|Di′

t′ (θ,DtF )|2 dt dt)p] 1

2E[( ∫

Ti+1|Di′′

t′′ (θ,DtF )|2 dt dt)p] 1

2

≤∑

i′,i′′≤m

‖DF‖pi′,2p‖DF‖p

i′′,2p ≤ c′‖DF‖2pm,2p,

for any θ . Then the assertion of the lemma follows. ��Now, we want to show that if F is nondegenerate, its weighted characteristic

function ψG(v) is rapidly decreasing.

Theorem 5.3.1 For any N ∈ N, there exist m ∈ N, p ≥ 2 and c > 0 such that theinequality

∣∣∣E[ei(v,F )G

]∣∣∣ ≤ c

|v|N(‖DF‖m,p sup

θ

‖RF (θ)−1‖m,p

)N‖G‖m,p (5.32)

holds for all |v| ≥ 1 for any nondegenerate Wiener functional F of (D∞)d andG ∈ D∞.


Proof For a nondegenerate smooth functional F , we will define a complex H -valued Wiener functional with parameter v ∈ R

d \ {0} by

Xt = XF,vt = −i(v,DtF )

|v|2 · RF (θ)−1,

where θ = v/|v|. Then Xt ∈ D∞T

by Proposition 5.3.1 and Lemma 5.2.1. Further,using the adjoint equality (5.8) for the Skorohod integral, we have

E[ei(v,F )δ(XG)

] = E[ ∫

T

Dt(ei(v,F ))XtG dt

](5.33)

= E[ ∫

T

ei(v,F ) · i(v,DtF )XtG dt]

= E[ei(v,F )G],

since∫i(v,DtF )Xt dt = 1. We will repeat this argument. Setting L(G) = δ(XG),

we have the iteration formula

E[ei(v,F )G] = E[ei(v,F )L(G)] = · · · = E[ei(v,F )LN(G)]

for N = 1, 2, . . .. Consequently, using Theorem 5.2.1 and Hölder’s inequality, weget

|E[ei(v,F )G]| ≤ ‖LN(G)‖0,2 ≤ ‖δ(XLN−1(G))‖0,2

≤ c1‖XLN−1(G)‖1,2 ≤ c2‖X‖1,4‖LN−1(G)‖1,4.

Repeat this argument for ‖LN−1(G)‖1,4. Then for any N , there exist increasingsequences ml, pl, l = 1, . . . , N and a positive constant cN such that

|E[ei(v,F )G]| ≤ cN‖X‖m1,p1 · · · ‖X‖mN,pN‖G‖mN,pN

≤ cN‖X‖NmN,pN‖G‖mN,pN

.

By Hölder’s inequality, we have

‖X‖m,p ≤ cm,p

|v|2 ‖i(v,D)F‖m,2p‖RF (θ)−1‖m,2p

≤ cm,p

|v| ‖DF‖m,2p supθ

‖RF (θ)−1‖m,2p.

Therefore we get the inequality of the theorem. ��Theorem 5.3.2 Suppose that a d-dimensional smooth Wiener functional F isnondegenerate. Then for any G ∈ D∞, the law of F weighted by G has a rapidlydecreasing C∞-density. Further, the density function fG(x) and its derivatives are


given by the Fourier inversion formula; for any multi-index j = (j1, . . . , jd) ofnonnegative integers, we have

∂ jfG(x) = (−i)|j|

(2π)d

∫

Rd

e−i(v,x)ψG(v)vj dv, (5.34)

where ψG(v) = E[ei(v,F )G].Proof We saw in Theorem 5.3.1 that the function ψG(v) is a rapidly decreasingC∞-function of v ∈ R

d . Therefore |ψG(v)||v|n0 is integrable for any n0 ∈ N. Thenthe weighted law μG(dx) has a Cn0 -density fG(x). Since this is valid for any n0,fG(x) is actually a C∞-function. It satisfies the formula (5.34) by Proposition 1.1.2.

Further, ψG(v) is a C∞-function of v. Then the density fG(x) is a rapidlydecreasing function. Therefore fG(x) is a rapidly decreasing C∞-function. ��

We can replace the nondegenerate condition of a smooth Wiener functional Fby Malliavin’s condition using the determinant of the Malliavin covariance RF . Letλ1 ≥ λ2 > · · · ≥ λd ≥ 0 be eigen-values of the matrix RF . If λd > 0 a.s and itsatisfies E[λ−p

d ] < ∞ for all p > 1, then we have supθ∈Sd−1E[(θ, RF θ)−p] <

∞ and hence F is nondegenerate. Note that the above is equivalent to thatE[(λ1 · · · λd)

−p] < ∞ holds for all p > 1. Therefore, if | detRF | > 0 a.s. andE[| detRF |−p] < ∞ holds for all p > 1, then F is nondegenerate. Therefore weget a cerebrated result in the Malliavin calculus for the Wiener space.

Theorem 5.3.3 (Malliavin [77], Malliavin–Thalmaier [80]) Let F be a d-dimensional smooth Wiener functional. Assume that | detRF | > 0 a.s. and(detRF )−1 ∈ L∞−. Then F is nondegenerate and the law of F weighted byG ∈ D∞ has a rapidly decreasing C∞-density.

The Malliavin criterion for the smooth density (Theorem 5.3.3) has been appliedto diffusion processes whose generator is elliptic or further, hypo-elliptic. SeeMalliavin [77], Kusuoka–Stroock [69, 70] etc. We will discuss the problem inChap. 6.

Remark (Formula of integration by parts) In literatures on Malliavin calculus, theformula of the integration by parts is often obtained and it is applied to the proof ofthe smooth density. We will derive the formula directly, using the property of adjointoperator δ stated in Theorem 5.2.2. Let F = (F 1, . . . , F d) be a nondegeneratesmooth functional. Let f be a C∞

b function on Rd . Then f (F ) is a one-dimensional

smooth functional. By Proposition 5.1.1, we have D(f (F )) = ∑di=1 ∂xi f (F )DF i.

Let (γij ) be the inverse matrix of the Malliavin covariance RF . Then we have

∂xi f (F ) = 〈D(f (F )),∑

j

γijDF j 〉T


for any i = 1, . . . , d. Let G ∈ D∞. Then G∑

j γijDF j ∈ D∞T

, since γij ∈ D∞ byProposition 5.3.1. Then we get

E[∂xi f (F ) ·G] = E[〈D(f (F )),G∑

j

γijDF j 〉]

= E[f (F ) · δ(G∑

j

γijDF j )].

Here we use the adjoint formula (5.8). Set li (G) = δ(G∑

j γijDF j ). It is a con-tinuous linear operator from D∞ into itself by Theorem 5.2.2 and Proposition 5.3.1.Then we get the formula of the integration by parts:

E[∂xi f ◦ F ·G] = E

[f ◦ F · li (G)

]. (5.35)

Further, by induction, the formula is extended as

E[∂xi1

· · · ∂xin f ◦ F ·G] = E[f ◦ F · li1,...,in (G)

], (5.36)

where li1,...,in is a continuous linear operator from D∞ into itself by Theorem 5.2.2.We will give an alternative proof of the smooth density for the law of F , by

using the above formula of the integration by parts. Set f (x) = ei(v,x), wherev = (v1, . . . , vd) and x = (x1, . . . , xd). Then it holds that ∂xi1

· · · ∂xin f (x) =invi1 · · · vinei(v,F ). Therefore, we get by (5.36)

invi1 · · · vinE[ei(v,F )G] = E[ei(v,F )li1,...,in (G)].

Therefore if |vi1 · · · vin | �= 0, we have


]∣∣∣ ≤ 1

|vi1 · · · vin |‖li1,...,in (G)‖0,2

for any n. Therefore the weighted characteristic function ψG(v) = E[ei(v,F )G] isa rapidly decreasing function. Further, since F has finite moments of any order,ψG(v) is a C∞-function. Then weighted law of F by G has a rapidly decreasingC∞-density.

In this monograph, we did not use the formula of the integration by parts for theproof of Theorem 5.3.3, since the formula should not hold for the Poisson space, aswe will discuss shortly.


5.4 Difference Operator and Adjoint on Poisson Space

We are also interested in the problem whether the law of a functional of a Poissonrandom measure has a smooth density. For the problem, we will study the Malliavincalculus on the space of Poisson random measures.

Let T = [0, T ] and Rd ′0 = R

d ′ \ {0}. Let U = T × Rd ′0 and B(U) be its Borel

field. Let Ω be the set of all integer-valued measures ω on U and let B(Ω) be thesmallest σ -field of Ω with respect to which ω(E), E ∈ B(U) are measurable. LetP be a probability measure on (Ω,B(Ω)) such that N(dt dz) := ω(dt dz) is aPoisson random measure with intensity measure n(dt dz) = dtν(dz), where ν is aLévy measure on R

d ′0 . The triple (Ω,B(Ω), P ) is called a space of Poisson random

measure with the intensity n.We will define two transformations Ω → Ω following Picard [92]. For each

u = (t, z) ∈ U, we define two maps ε−u , ε+u ;Ω → Ω by

ε−u ω(E) = ω(E ∩ {u}c), ∀E ∈ B(U),

ε+u ω(E) = ω(E ∩ {u}c)+ 1E(u), ∀E ∈ B(U).

Then equality εθ1u ◦ ε

θ2u = ε

θ1u holds for any u and θ1 = ± and θ2 = ±. For any

ω ∈ Ω , ε+u ω = ω holds for almost all u with respect to the measure N , in view ofthe definition of the measure N . Further, for almost all ω ∈ Ω , ε−u ω = ω holds foralmost all u with respect to the measure n, since the set of u such that ε−u ω �= ω

is at most countable and countable sets are null sets with respect to the measuren(du) = dtν(dz). Let us remark that the Poisson random measure ω ∈ Ω has nopoint mass almost surely, i.e., P(ω({u}) = 0) = 1 for any {u}, since

E[ω({u})] = E[N({u})] = n({u}) = 0.

Then the equality ε+u ω(E) = ω(E)+ δu(E) holds for almost all ω.Let S be a complete metric space. An S-valued B(Ω)-measurable function F(ω)

is called an S-valued Poisson functional. If S is the space of real numbers, it is calledsimply a Poisson functional. Transformations εθu, θ = ±1 induce transformationsfor a Poisson functional F(ω). We denote F(εθuω) by F ◦ εθu(ω). Let Zu, u ∈ U bea family of real valued Poisson functionals. We assume that it is measurable withrespect to (u, ω) (random field). From the definitions of transformations ε+u and ε−u ,equalities

∫

U

Zu ◦ ε+u N(du) =∫

U

ZuN(du), (5.37)

∫

U

Zu ◦ ε−u n(du) =∫

U

Zun(du) (5.38)

hold for almost all ω, for any positive random field Zu, u ∈ U.

5.4 Difference Operator and Adjoint on Poisson Space 191

Lemma 5.4.1 (Picard [91]) If a positive random field Zu, u ∈ U satisfies Zu◦ε+u =Zu ◦ ε−u for any u almost surely, then

E[ ∫

U

ZuN(du)]= E

[ ∫

U

Zun(du)]. (5.39)

Proof Let I be a sub σ -field of B(U)× B(Ω) generated by sets A×B, where A ∈B(U) and B ∈ BAc(Ω). Here BAc(Ω) is the σ -field generated by N(C), C ⊂ Ac.If Zu satisfies Zu ◦ ε+u = Zu ◦ ε−u for all u almost surely, then Zu is I-measurable.In fact, let Zu be a step functional given by

Zu(ω) =n∑

m=1

φm(ω)1Am(u),

where {A1, . . . , An} is a measurable partition of U and φm,m = 1, . . . , n arebounded positive B(Ω)-measurable functionals. If Zu ◦ ε+u = Zu ◦ ε−u holds for anyu ∈ U, then φm◦ε+u = φm◦ε−u holds for any u ∈ Am, m = 1, . . . , n. Hence each φm

is BAcm(Ω)-measurable. Therefore, Zu is I-measurable. Further, since

∫ZuN(du)

= ∑m φmN(Am) and, φm and N(Am) are independent for any m, we have

E[ ∫

U

ZuN(du)]=

∑

m

E[φm]E[N(Am)]

= E[∑

m

φmn(Am)]= E

[ ∫

U

Zun(du)].

Now, for a given I-measurable positive random field Zu, there exists a sequenceof positive I-measurable step functionals Zn

u such that Znu ↑ Zu as n ↑ ∞ for

all u almost surely. Since E[∫ ZnuN(du)] = E[∫ Zn

un(du)] holds for any Znu , the

equality E[∫ ZuN(du)] = E[∫ Zun(du)] holds for the limit random field Zu. ��Corollary 5.4.1 For any positive random field Zu, we have

E[ ∫

U

ZuN(du)]= E

[ ∫

U

Zu ◦ ε+u n(du)]. (5.40)

Proof The functional Yu := Zu ◦ ε+u satisfies Yu ◦ ε+u = Yu ◦ ε−u for all u almostsurely. Indeed, we have

(Zu ◦ ε+u ) ◦ ε−u (ω) = Zu(ε+u ◦ ε−u ω) = Zu(ε

+u ω) = Zu ◦ ε+u (ω),

(Zu ◦ ε+u ) ◦ ε+u (ω) = Zu(ε+u ◦ ε+u ω) = Zu(ε

+u ω) = Zu ◦ ε+u (ω).


Therefore equality (5.39) holds for Yu. Then, using (5.37) we get

E[ ∫

U

ZuN(du)]= E

[ ∫

U

Zu ◦ ε+u N(du)]= E

[ ∫

U

Zu ◦ ε+u n(du)].

��We should remark that transformations ε+u may not be absolutely continuous, i.e.,

P(A) = 0 may not imply P(ε+u A) = 0 for some A ∈ F . Therefore, if a Poissonfunctional F is defined a.s. P , F ◦ ε±u may not be defined for any u a.s. However,F ◦ ε+u is well defined a.e. (u, ω) with respect to n(du)P (dω). Indeed, if F = 0holds a.s., then E[∫ IεFN(du)] = 0 holds, where 1ε(u) = 1|z|>ε(u). Then we haveE[∫ F ◦ ε+u 1εn(du)] = 0 in view of Corollary 5.4.1. Therefore F ◦ ε+u = 0 holdsa.e. n(du)P (dω).

For a bounded Poisson functional F , we define difference operator {Du, u ∈U} by

DuF = F ◦ ε+u − F. (5.41)

It is well defined a.e. n(du)P (dω). If u = (t, z), we denote D(t,z)F by Dt,zF .

For u = (u1, . . . , uj ) ∈ Uj , we set ε+u = ε+u1

◦ · · · ε+uj and Du = Dju =

Du1 · · · Duj . DjuF is well defined a.e. dnj dP , where nj is the j -fold product of

the measure n given by nj (du1 · · · duj ) = n(du1) × · · · × n(duj ). The functional

DjuF is invariant by the permutation; let σ be a permutation of {1, . . . , j}. For

u = (u1, . . . , uj ), set uσ = (uσ1 , . . . , uσj ). Then it holds that F ◦ ε+u = F ◦ ε+uσa.e.

dnj dP . Further, DjuF = D

juσ

F holds a.e. dnj dP . Further, we should remark thatif ui = uj (i �= j) holds for some u = (u1, . . . , uj ), then DuF = 0 a.e.

If j = 0, we regard U0 is an empty set ∅, ε+∅ is the identity transformation and

we set D∅F = F .For u = (r, z) ∈ U, we set γ (u) = |z| ∧ 1. It is a positive function of u =

(r, z), since |z| > 0 holds for any u. For u = (u1, . . . , uj ) ∈ Uj , we set γ (u) =

γ (u1) · · · γ (uj ) and |u| = max1≤i≤j γ (ui). A bounded Poisson functional F iscalled smooth if

supu∈A(1)j

E[|DjuF |p]

γ (u)p< ∞

holds for any j ∈ N and p ≥ 2. Here, ‘sup’ means the essential supremum withrespect to the measure nj . The set of all smooth functionals is denoted by D∞. Ad-dimensional functional F = (F 1, . . . , F d) is called smooth if every componentF i is smooth. We denote by (D∞)d the set of all d-dimensional smooth functionals.

Remark Let us compare the above Du with the derivative operator Dt on the Wienerspace. Recall that the derivative operator D in the Wiener space is defined by (5.2).Heuristically, we may write it as


DtF = limλ→0

F(w + λδt )− F(w)

λ, a.e. dt P (dw),

where δt is the delta measure at the point {t}. Since ε+u ω = w+δu holds if ω({u}) =0, Du may be regarded as a discretization of the derivative operator Dt .

Example We give an example in which DF is computed simply. Let F bea linear functional of a Poisson random measure, which is written as F =∫Uh(s, z)ND(ds dz), where h(u) is a bounded function. We have

F ◦ ε+(t,z) =∫

U

h(s, y){ND(ds dy)+ δt,z(ds dy)}

=∫

U

h(s, y)ND(ds dz)+ h(t, z) = F + h(t, z)

a.e. dn dP . This shows D(t,z)F = h(t, z) a.e. dn dP . Since it is a constant function,

higher differences of F are 0, i.e., DjuF = 0 holds a.e. nj (du)P (dω) for j ≥ 2.

Another interesting Poisson functional that we can compute F ◦ ε+u and DuF isa solution of an SDE with jumps discussed in Chap. 3. It will be studied in Sect. 6.4.

Let Zu, u ∈ U be a real valued Poisson random field. We denote by D∞U

the setof all Zu such that

Zu

γ (u),

DjvZu

γ (v)γ (u), j = 1, 2, . . .

are bounded a.e. n(du)P (dω) and nj+1(dv du)P (dω), respectively. Further, D∞U0

denotes the set of Zu ∈ D∞U

such that Zu ◦ ε−u are supported by a compact subsetK of U a.e. n(du)P (dω). For Z ∈ D∞

U0, we define the Skorohod integral of Zu (by

Poisson random measure) by

δ(Z) =∫

U

Zu ◦ ε−u N(du) =∫

U

Zu ◦ ε−u (N(du)− n(du)). (5.42)

Then δ(Z) ∈ Lp for any p ≥ 2. In fact if K is a compact subset of U, N(K) isPoisson distributed with intensity n(K) < ∞ and hence E[N(K)p] < ∞ holds forany p ≥ 2.

We will show that the operator δ is the adjoint of D and that it has propertiessimilar to those of δ (Skorohod integral by Wiener process).


Lemma 5.4.2

1. (Picard [91]) For any Z ∈ D∞U0

, δ satisfies the adjoint property

E[Gδ(Z)] = E[ ∫

U

(DuG)Zun(du)]

(5.43)

for any bounded Poisson functional G.2. For any Z ∈ D∞

U0, the commutation relation holds:

Du′ δ(Z) = δ(Du′Z)+ Zu′ , a.e. n(du′)P (dω). (5.44)

3. The isometric property

E[δ(Z)δ(Y )] = E[ ∫

U

ZuYun(du)]+ E

[ ∫∫

U2DuZvDvYun(du)n(dv)

]

(5.45)holds for any Z, Y ∈ D∞

U0.

Proof It holds that

E[Gδ(Z)] = E[ ∫

U

GZu ◦ ε−u N(du)]− E

[ ∫

U

GZu ◦ ε−u n(du)].

In view of (5.40), the first term of the right-hand side is finite and is equal to

E[ ∫

U

G ◦ ε+u Zu ◦ ε−u ◦ ε+u n(du)]= E

[ ∫

U

G ◦ ε+u Zu ◦ ε−u n(du)].

Since Zu ◦ ε−u = Zu holds a.e. n(du)P (dω), we get (5.43).We will next show the commutation relation. We have for any u′ ∈ U,

Zu ◦ ε−u ◦ ε+u′ =

{Zu ◦ ε+

u′ ◦ ε−u , if u �= u′,Zu′ ◦ ε−

u′ , if u = u′.

Therefore we get

Du′ δ(Z) = δ(Z) ◦ ε+u′ − δ(Z)

=∫

Zu ◦ ε+u′ ◦ ε−u N(du)+ Zu′ ◦ ε−

u′ −∫

Zu ◦ ε−u N(du)

= δ(Du′Z)+ Zu′ ◦ ε−u′

a.e. n(du′)P (dω). Since Zu′ ◦ε−u′ = Zu′ holds a.e. n(du′)P (dω), we get the equalityof the commutation relation.


We will show the isometric property. Using the adjoint property (5.43) and thecommutation relation (5.44), we have

E[δ(Z)δ(Y )] = E[ ∫

U

ZuDuδ(Y )n(du)]

= E[ ∫

U

ZuYun(du)]+ E

[ ∫

U

Zuδ(DuY )n(du)]

= E[ ∫

U

ZuYun(du)]+ E

[ ∫

U

∫

U

DvZuDuYvn(du)n(dv)].

��We will consider the Skorohod integral δ(Z), when Zu = Zt,z is predictable

with respect to t . It turns out that δ(Z) coincides with the Itô integral. The followingproposition corresponds to Proposition 5.2.2 for the Skorohod integral δ(X) byWiener process.

Proposition 5.4.1 Let {Ft } be the filtration generated by the Poisson randommeasure N . Suppose that Zt,z ∈ D∞

U0is predictable with respect to t for all z.

Then Skorohod integrals δ(Z) coincides with Itô integral∫UZuN(du). Further, the

isometric property holds:

E[δ(Z)2] = E[ ∫

U

Z2un(du)

]. (5.46)

Proof Suppose that Zt,z is a simple predictable random field given by Zt,z =∑m φm

z 1(tm−1,tm](t), where φmz is Ftm−1 -measurable. Then it holds that Zt,z◦ε+(t,z) =

Zt,z. Therefore,

δ(Z) =∫

U

Zu(N(du)− n(du)) =∫

U

ZuN(du).

The right-hand side coincides with the Itô integral.If Zu is predictable, we have Dt,zZs,z′ = 0 for t > s. Then DvZuDuZv = 0

holds for any u = (t, z), v = (s, z′) such that t �= s a.e. Therefore the last termin (5.45) is equal to 0 if Zu is predictable. ��

Finally, we show that δ has a property, similar to the definition of δ given by (5.7).

Proposition 5.4.2 Let Zu = ∑nl=1 Flhl(u), where Fl ∈ D∞ and hl are bounded

measurable function on U with compact support. Then we have

δ(Z) =n∑

l=1

FlN(hl)−n∑

l=1

∫DuFlhl(u)n(du), a.s. (5.47)


Proof We have

δ(Flhl) =∫

Fl◦ε−u hl(u)N(du) = Fl

∫hl(u)N(du)−

∫(Fl−Fl◦ε−u )hl(u)N(du).

We shall compute the last term. It holds by (5.37) that

∫(Fl − Fl ◦ ε−u )hl(u)N(du) =

∫(Fl ◦ ε+u − Fl ◦ ε−u ◦ ε+u )hl(u)N(du) = 0.

Since Fl − Fl ◦ ε−u = Fl ◦ ε+u − Fl ◦ ε−u holds by definitions of ε+u and ε−u , we have

∫(Fl − Fl ◦ ε−u )hl(u) n(du) =

∫(Fl ◦ ε+u − Fl ◦ ε−u )hl(u)n(du)

=∫

(Fl ◦ ε+u − Fl)hl(u)n(du)

=∫

DuFlhl(u)n(du).

Summing up these computations for l = 1, 2, . . . , n, we get (5.47). ��

5.5 Sobolev Norms for Poisson Functionals

In Sect. 5.2, we defined Sobolev norms for Wiener functionals, making use ofdifferential operator Dt, t ∈ T. In this section, we will define Sobolev norms forPoisson functionals, making use of the difference operator DuF, u ∈ U. We shouldbe careful about the definition of Sobolev norms, since we want to apply it forproving the existence of the smooth density of the law of ‘nondegenerate’ Poissonfunctionals. Its definition will become more complicated than that of a Wienerfunctional, because the smooth density of a Poisson functional F will possibly berelated to the asymptotic property of DuF as γ (u) converges to 0. Then we shoulddefine Sobolev norms for DuF , restricting u to subdomains A of U, where A areopen neighborhoods of the center of the intensity measure n.

Discussions in this section and those in Sects. 5.6, 5.7, and 5.8 are close to worksby Ishikawa–Kunita–Tsuchiya [46], though notations for Sobolev norms arechanged.

We begin with a preliminary observation. Let F be s smooth Poisson functional.For a positive integer n, an L2-type Sobolev norm for F , restricted to a domainA ⊂ U such that n(A) > 0 should be given by

{E[|F |2] +

n∑

j=1

∫

Aj

E[|DjuF |2]nj (du)

} 12. (5.48)

5.5 Sobolev Norms for Poisson Functionals 197

Here Aj is the j -product set of A, included in the j -product set Uj . Further, nj isthe j -product measure of the intensity measure n defined on U

j . The norm is welldefined, since functionals Dj

uF are invariant by the permutation of u. We could writeit as ‖F‖n,2;A, but in order to avoid the confusion with Sobolev norm ‖F‖n,2 for aWiener functional, we will denote (5.48) by ‖F‖0,n,2;A. (The first suffix 0 indicatesthat the derivative operator DF is not involved. In Sect. 5.10, we will define Sobolevnorms for Wiener–Poisson functionals denoted by ‖F‖m,n,p;A.) It is compatiblewith Sobolev norms of L2-type ‖F‖m,2 for Wiener functionals.

In order to define Sobolev norms of Lp (p > 2) type, we will transform infinitemeasures nj on Aj to a probability measure m

jA on Aj in the following way. Let m

be a bounded measure on U defined by

m(E) =∫

E

γ (u)2n(du).

We define a probability measure mA on A and its product measure mjA on Aj by

mA(E) = m(A ∩ E)

m(A), m

jA(du) = mA(du1) · · ·mA(duj ),

respectively, where u = (u1, . . . , uj ) ∈ Uj . Further, we define positive functions

on A and Aj by

γA(u) = γ (u)√m(A)

, γA(u) = γA(u1) · · · γA(uj ).

Note that 1√m(A)

is the normalizing constant for the function γ (u), because we have∫Aγ (u)2n(du) = m(A). Then we can rewrite the L2-Sobolev norm (5.48) as

‖F‖0,n,2;A ={E[|F |2] +

n∑

j=1

∫

Aj

E[|DjuF |2]

γA(u)2m

jA(du)

} 12.

Let A(1) = {u ∈ U; |z| ≤ 1}. Suppose that A is a domain in A(1) which includesthe center of the Lévy measure ν and satisfies 0 < m(A) ≤ 1. For given j ∈ N, p ≥2, we define a norm of F ∈ D∞ by ‖F‖0,0,p;A = E[|F |p] 1

p if n = 0 and if n ≥ 1,

‖F‖0,n,p;A ={E[|F |p] +

n∑

j=1

∫

Aj

E[|DjuF |p]

γA(u)pm

jA(du)

} 1p. (5.49)

Remark The Sobolev norm for Wiener functionals given in (5.11) is of L2,p type;it is L2-type with respect to t and is Lp-type with respect to w. But the Sobolev


norm ‖ ‖0,n,p;A for Poisson functionals given in (5.49) is of Lp,p-type. We definedSobolev norms of Lp.p-type, because the difference rule of Du and the differentialrule of Dt are distinct. In fact, a difference rule of the operator Du is given by

Du(FG) = (DuF )G+GDuF + DuF DuG, (5.50)

while the differential rule of the operator Dt is given by a simpler form Dt(FG) =(DtF )G+ FDtG.

Let {Zu, u ∈ U} be a smooth Poisson random field of D∞U0

. We define its normsby

‖Z‖0,0,p;A =( ∫

A

E[|Zu|p]γA(u)p

mA(du)) 1

p, (5.51)

‖Z‖0,n,p;A ={‖Z‖p0,0,p;A+

n∑

j=1

∫

Aj+1

E[|DjuZu|p]

γA(u)pγA(u)pm

jA(du)mA(du)

} 1p.

When a set A is fixed in discussions, we will often drop ‘;A’ in the definitions ofSobolev norms and write them simply as

‖F‖0,n,p := ‖F‖0,n,p;A, ‖Z‖0,n,p := ‖Z‖0,n,p;A.

Lemma 5.5.1 (Hölder’s inequality) For any n ∈ N and p ≥ 2, there is a positiveconstant c not depending on the set A such that

‖FG‖0,n,p;A ≤ c‖F‖0,n,pr;A‖G‖0,n,pr ′;A, (5.52)

‖FZ‖0,n,p;A ≤ c‖F‖0,n,pr;A‖Z‖0,n,pr ′;A (5.53)

hold for any F,G ∈ D∞, Z ∈ D∞U0

, and r, r ′ > 1 with 1/r + 1/r ′ = 1.

Proof We will prove (5.53) only. By Hölder’s inequality, we have

‖FZ‖p0,0,p =∫

E[|FZu|p]γA(u)p

mA(du) ≤ E[|F |pr ] 1r

∫E[|Zu|pr ′ ]

1r′

γA(u)pmA(du)

≤ ‖F‖p0,0,pr‖Z‖p0,0,pr ′ .

Note the difference rule (5.50) for D. We can show similarly to the above

∫E[Du′FZu|p]γA(u′)pγA(u)p

mA(du′)mA(du) ≤ ‖F‖p0,1,pr‖Z‖p0,0,pr ′ ,

∫E[FDu′Zu|p]γA(u′)pγA(u)p

mA(du′)mA(du) ≤ ‖F‖p0,0,pr‖Z‖p0,1,pr ′ .

5.5 Sobolev Norms for Poisson Functionals 199

Note that m(A)1/2

γA(u′) = 1

γ (u′) ≥ 1 for u′ ∈ A. Then

∫E[Du′FDu′Zu|p]γA(u′)pγA(u)p

mA(du′)mA(du)

≤ m(A)p2

∫E[Du′FDu′Zu|p]

γA(u′)p · γA(u′)pγA(u)p mA(du′)mA(du)

≤ ‖F‖p0,1,pr‖Z‖p0,1,pr ′ .

Repeating this argument for j = 2, 3, . . ., we get (5.53) for any n. ��We will define another norms that are stronger than Sobolev norms defined

above. For n ∈ N and p ≥ 2, new norms are defined by

‖F‖�0,n,p ={E[|F |p] +

n∑

j=1

supu∈A(1)j

E[|DjuF |p]

γ (u)p

} 1p, (5.54)

‖Z‖�0,n,p ={

supu∈A(1)

E[|Zu|p]γ (u)p

+n∑

j=1

sup(u,u)∈A(1)j+1

E[|DjuZu|p]

γ (u)pγ (u)p

} 1p. (5.55)

Here, supu∈A(1) |ϕ(u)| etc. means the essential supremum of the function ϕ(u)

on the measurable space (A(1), n), which is defined by

inf{c ∈ R; |ϕ(u)| ≤ c for almost every u ∈ A(1) with respect to measure n

}.

Norms ‖F‖�0,0,p and ‖Z‖�0,0,p are defined by cutting off terms∑n

j=1 in the abovedefinitions. We may regard them as Sobolev norms of L∞,p-type, i.e., the essentialsupremum is taken for variable u ∈ U with respect to the measure n and the Lp-norm is taken with respect to the measure P . Thus these norms should be biggerthan Sobolev norms of Lp,p-type. Indeed, we have the inequalities;

‖F‖0,n,p;A ≤ ‖F‖�0,n,p, F ∈ D∞, (5.56)

‖Z‖0,n,p;A ≤ ‖Z‖�0,n,p, Z ∈ D∞U0

,

for any n, p and A.We denote by D0,n,p the completion of D∞ by the norm ‖ ‖�0,n,p and we set

D∞ = ⋂n,p D0,n,p. Elements of D∞ are called smooth Poisson functionals. Let

D0,n,pU

be the completion of D∞U0

by the norm ‖ ‖�0,n,p. We set D∞U

= ⋂n,p D0,n,p

U.

Then both D∞ and D∞U

are algebras and are F -spaces with respect to countable

norms ‖ ‖�0,n,p, n, p ∈ N.


Then for any F ∈ D∞ and Z ∈ D∞U

, norms ‖F‖0,n,p;A and ‖Z‖0,n,p;A are welldefined and are finite for any n, p and A ⊂ A(1).

It is not difficult to prove the following proposition.

Proposition 5.5.1 The operator Du is extended to a continuous linear operatorfrom F-space D∞ to F-space D∞

U. Further, it holds that ‖DF‖�0,n,p ≤ ‖F‖�0,n+1,p

for any F ∈ D∞ for any n, p.

Later, we will use another norms for Poisson functionals. For n ∈ N, p ≥ 2, wewill define norms of F by

|F |�0,n,p =(E[|F |p] +

n∑

j=1

supu∈A(1)j

E[|F ◦ ε+u |p]) 1

p. (5.57)

We will study relations of these norms.

Proposition 5.5.2 For any n, p, the norm | |�0,n,p is weaker than the norm ‖ ‖�0,n,p;there is a positive constant cn,p such that

|F |�0,n,p ≤ cn,p‖F‖�0,n,p, ∀F ∈ D∞. (5.58)

Proof Let u ∈ A(1)j . We can rewrite F ◦ ε+u as

F ◦ ε+u =∑

v⊂u

DvF = F +j∑

i=1

∑

v⊂u,�v=i

DvF. (5.59)

Therefore we have

supu

E[|F ◦ ε+u |p] ≤ 2j(E[|F |p] +

j∑

i=1

supv

E[|DivF |p]

).

Then, introducing another norms by

|F |�′0,n,p ={E[|F |p] +

n∑

i=1

supv∈A(1)j

E[|DivF |p]

} 1p,

we have the inequality (|F |�0,n,p)p ≤ (n + 1)2n(|F |�′0,n,p)p. We can show similarly

the inequality (|F |�′0,n,p)p ≤ (n+1)2n(|F |�0,n,p)p. Therefore two norms |F |�0,n,p and

|F |�′0,n,p are equivalent. Further, since |F |�′0,n,p ≤ ‖F‖�0,n,p holds, the norm |F |�0,n,pis weaker than the norm ‖F‖�0,n,p for any n, p. ��

5.6 Estimations of Two Poisson Functionals by Sobolev Norms 201

Let L0,n,p be the completion of D∞ by the norm | |�0,n,p. We set L∞ =⋂

n,p L0,n,p. Then L∞ is an F -space and the relation D∞ ⊂ L∞ holds.For a Poisson functional Xt with parameter t ∈ [0, 1), we set

|X|�0,n,p =( ∫

T

(|Xt |�0,n,p)p dt) 1

p. (5.60)

Note Sobolev norms for Poisson functionals are defined in several ways. Ishikawaet al. defined Sobolev norms for Poisson functionals in [46]. Notations of theirSobolev norms are changed in this monograph. Norms ‖ ‖n,p;A, ‖ ‖∗n,p and |‖ |‖∗n,pin [46] correspond to our norms ‖ ‖0,n,p, ‖ ‖�0,n,p and | |�0,n,p, respectively.

Hayashi–Ishikawa [37] defined another Sobolev norms. Their norm ‖ ‖n,0,p,(ρ)is close to our norm ‖ ‖0,n,p;A(ρ), but these two are not equivalent norms.

5.6 Estimations of Two Poisson Functionals by SobolevNorms

Let n(dt dz) = dtν(dz) be the intensity measure of the Poisson random measure.The set {u = (t, z); z = 0} is called the center of the intensity measure n. For agiven family of star-shaped neighborhoods {A0(ρ)} of 0 ∈ R

d ′ (see Sect. 1.2), weset A(ρ) = {u = (t, z) ∈ U; z ∈ A0(ρ)} and call {A(ρ)} the family of star-shapedneighborhoods of the center of the intensity n. We define the function ϕ(ρ) by

ϕ(ρ) := m(A(ρ)) = T

∫

A0(ρ)

|z|2ν(dz) = T ϕ0(ρ). (5.61)

Now, suppose that the Lévy measure ν satisfies the order condition of exponentα, then the function ϕ(ρ) satisfies the order condition of exponent α. We will fix0 < ρ0 ≤ 1 such that ϕ(ρ0) ≤ 1.

We are interested in estimations of Poisson functionals by Sobolev norms{‖ ‖0,n,p;A(ρ); 0 < ρ < ρ0} and their dependence on the parameter ρ. In thissection, we will give estimations of two Poisson functionals. The first one is theSkorhod integral δ(Zχρ), where Z ∈ D∞

U0and χρ is the indicator function IA(ρ) of

the set A(ρ). We first study the L2-estimate. If p = 2, the Sobolev norm is writtensimply as

‖Z‖0,n,2;A(ρ) ={ ∫

A(ρ)

E[|Zu|2]n(du)+n∑

j=1

∫

A(ρ)j+1E[|Dj

uZu|2]nj (du)n(du)} 1

2.

Then, using the isometric property (5.45), we have


E[δ(Zχρ)2] = E

[ ∫

A(ρ)

Z2un(du)

]+ E

[ ∫∫

A(ρ)2DuZvDvZun(du)n(dv)

]

≤ ‖Z‖20,1,2;A(ρ)

for any 0 < ρ < ρ0. Further, using the commutation relation for Du and δ, we canshow easily that for any positive integer n there is a positive constant cn such that

‖δ(Zχρ)‖0,n,2;A(ρ) ≤ cn‖Z‖20,n+1,2;A(ρ) (5.62)

holds for all 0 < ρ < ρ0.We want to extend the above inequality for any p ≥ 2. The following is the

counterpart to Theorem 5.2.1, where we obtained an estimation of the Skorohodintegral for Wiener functionals.

Theorem 5.6.1 ([46]) Given n ∈ N and a positive even number p ≥ 4, there existsa positive constant cn,p such that the inequality

‖δ(Zχρ)‖0,n,p;A(ρ) ≤ cn,p‖Z‖0,n+p−1,p;A(ρ) (5.63)

holds for any Z ∈ D∞U0

, where the constant cn,p does not depend on the sets{A(ρ), 0 < ρ < ρ0}.

The proof of the theorem is similar to that for Theorem 5.2.1, but details are morecomplicated. We shall first consider the estimation of E[δ(Zχρ)

p] for positive evennumber p ≥ 4. We prepare a lemma.

Let 1 ≤ q ≤ p be a positive integer. An element u = (u1, . . . , uq) of theproduct set Uq is written as {u1, . . . , uq}. For ε = (ε1, . . . , εq) ∈ {0, 1}q and u ={u1, . . . , uq}, we set Dε

u = Dε1u1 · · · Dεl

uq (= Dεluq · · · Dε1

u1).

Lemma 5.6.1 Let p ≥ 4 be an even number. Let Z ∈ D∞U0

. Then E[δ(Z)p] iswritten as sums of the following multiple integrals by nq(du), q = 1, . . . , p, wherenq(du) = n(du1) · · · n(duq).

∫

Uq

E[( p1∏

j=1

Dε1ju1 Zu1

)· · ·

( pq∏

j=1

Dεqjuq

Zuq

)]nq(du). (5.64)

Here, pi ∈ N satisfy p1 + · · · + pq = p and εij , i = 1, . . . , q, j = 1, . . . , pi aregiven by εij := (εij (1), . . . , εij (q − 1)) ∈ {0, 1}q−1. Further, ui; i = 1, . . . , q aregiven by ui := u − {ui} = {ui(1), . . . , ui(q − 1)}, which satisfy

u = {u1, . . . , uq} ⊂q⋃

i=1

pi⋃

j=1

q−1⋃

k=1

{ui(k); εij (k) = 1}. (5.65)


Proof Our discussion is close to the proof of Theorem 5.2.1. We have by the adjointformula (5.43)

E[δ(Z)p] = E[ ∫

U

Zu1Du1 δ(Z)p−1n(du1)].

We will decrease the power p−1 of δ(Z)p−1 step by step. Using the difference ruleof Du and the commutation relation (5.44), we have

Du1 δ(Z)p−1 = Du1 δ(Z)δ(Z)p−2 + δ(Z)Du1 δ(Z)p−2 + Du1 δ(Z)Du1 δ(Z)p−2

= (Zu1 + δ(Du1Z))(δ(Z)p−2 + Du1 δ(Z)p−2)+ δ(Z)Du1 δ(Z)p−2.

Substitute the above and apply the adjoint formula again. Then we see that E[δ(Z)p]is written as sums of terms

E[ ∫

U2D

j1u2Z

i1u1D

j2u1Z

i2u2D

j ′2u2D

j ′1u1 δ(Z)p−2n(du1)n(du2)

],

where i1, i2, j1, j2, j′1, j

′2 are nonnegative integers satisfying i1 + i2 = 2 and

j1, j2, j′1, j

′2 ≤ 1. Here D0

u is the identity transformation.Repeating this procedure inductively, we find that E[δ(Z)p] is written as sums

of terms stated in (5.64). For further details, see [46]. ��Proof of Theorem 5.6.1 We will first prove the inequality (5.63) in the case n = 0.By Lemma 5.6.1, E[δ(Zχρ)

p] is written as a sum of terms (5.64), replacing Z byZχρ . In this case, (5.64) is written as

∫

A(ρ)qE[( p1∏

j=1

Dε1ju1 Zu1

)· · ·

( pq∏

j=1

Dεqjuq

Zuq

)]nq(du)

= E

[ ∫

A(ρ)q

(∏p1

j=1 Dε1ju1 Zu1) · · · (

∏pq

j=1 Dεqjuq

Zuq )

γA(ρ)(u1)2 · · · γA(ρ)(uq)2 mq

A(ρ)(du)],

sincem

q

A(ρ)(du)

γA(ρ)(u1)2···γA(ρ)(uq )

2 = 1A(ρ)q nq(du). Let us consider the nominator in the

above expression. For u = (u1, . . . , uk) and ε ∈ {0, 1}k , we set

γ εA(ρ)(u) = γA(ρ)(u1)

ε1 · · · γA(ρ)(uk)εk .

We will show that there is a positive constant C = Cp (not depending on 0 < ρ < 1)such that the inequality


1

γA(ρ)(u1)2 · · · γA(ρ)(uq)2 (5.66)

≤ C(∏j γA(ρ)(u1)

ε1j

)· · ·

(∏j γA(ρ)(uq)

εqj

)γA(ρ)(u1)p1 · · · γA(ρ)(uq)

pq

holds for any u ∈ A(ρ)q . Note first that by the order condition of the Lévy measure,ϕ(ρ) ≥ cρα holds for 0 < ρ < ρ0. Then, since m(A(ρ)) = ϕ(ρ), we have for any|u| ≤ ρ < ρ0,

γA(ρ)(u) = γ (u)

ϕ(ρ)12

≤ 1

c12

ρ1− α2 ≤ c−

12 .

Now, denote by �{εij (k) = 1} the total number of triples (i, j, k) such that εij (k) =1. We set r = �{εij (k) = 1} + p − 2q. Since �{εij (k) = 1} ≥ q holds by therelation (5.65), r is a nonnegative integer. Then there are ui1 , . . . , uir ⊂ u such that

(∏j γA(ρ)(u1)

ε1j

)· · ·

(∏j γA(ρ)(uq)

εqj

)γA(ρ)(u1)

p1 · · · γA(ρ)(uq)pq

γA(ρ)(u1)2 · · · γA(ρ)(uq)2

= γA(ρ)(ui1) · · · γA(ρ)(uir ).

The last term of the above is less than or equal to c−r/2 for any u ∈ A(ρ)q .Therefore, setting C = c−r/2, we get the inequality (5.66).

The inequality (5.66) implies

E

[ ∫

A(ρ)q

|∏p1j=1 D

ε1ju1 Zu1 | · · · |

∏pq

j=1 Dεqjuq

Zuq |γA(ρ)(u1)2 · · · γA(ρ)(uq)2 m

q

A(ρ)(du)]

(5.67)

≤ CE

[∫

A(ρ)q

p1∏

j=1

|Dε1ju1 Zu1 |

γA(ρ)(u1)ε1j γA(ρ)(u1)

· · ·pq∏

j=1

|Dεqjuq

Zuq |γA(ρ)(uq)

εqj γA(ρ)(uq)m

q

A(ρ)(du)]

for any 0 < ρ < ρ0 and any Z ∈ D∞U0

. In the formula (5.67), the total number ofproduct terms is equal to p. Apply Hölder’s inequality to (5.67) with respect to theproduct measure dm

q

A(ρ) dP . Then (5.67) is dominated by

C∏

i,j

E

[ ∫

A(ρ)q

∣∣∣D

εijui

Zui

γA(ρ)(ui )εijγA(ρ)(ui)

∣∣∣p

mq

A(ρ)(du)] 1

p ≤ C‖Z‖p0,q−1,p;A(ρ).

Therefore, the absolute value of (5.64) is dominated by C′‖Z‖p0,q−1,p;A(ρ)for all

0 < ρ < ρ0 and Z ∈ D∞U0

. Note that E[δ(Zχρ)p] is written as a finite sum of


terms (5.64) for q = 1, . . . , p. Since norms ‖ ‖0,n,p;A(ρ) are nondecreasing withrespect to n, |E[δ(Zχρ)

p]| is dominated by C′′‖Z‖p0,p−1,p;A(ρ)for all 0 < ρ < ρ0

and Z ∈ D∞U0

with another constant C′′. Consequently, we get

‖δ(Zχρ)‖p0,0,p = E[δ(Zχρ)p] ≤ C′′‖Z‖p0,p−1,p;A(ρ)

. (5.68)

Next, we will consider the case n ≥ 1. It holds, by the definition of Sobolevnorms that

‖δ(Zχρ)‖p0,n,p;A(ρ)= E[|δ(Zχρ)|p] (5.69)

+n∑

j=1

∫

A(ρ)j

E[∣∣Dj

v δ(Zχρ)∣∣p]

γA(ρ)(v)pm

j

A(ρ)(dv).

The first term of the right-hand side has been computed at (5.68). We shall considerother terms. Take any integer 1 ≤ j ≤ n. Note the commutation relation of δ;Dvδ(Zχρ) = δ(DvZχρ)+ Zvχρ . Then we have

Djv δ(Zχρ) = δ(D

jvZχρ)+

∑

i

Dj−1vi Zviχρ,

where v = (v1, . . . , vj ) and vi = v − {vi}, i = 1, . . . , j . Therefore, there is apositive constant cp such that

∫

A(ρ)j

E[∣∣Dj

v δ(Zχρ)∣∣p]

γA(ρ)(v)pm

j

A(ρ)(dv) (5.70)

≤ cp

{ ∫E[∣∣δ(Dj

vZχρ)∣∣p]

γA(ρ)(v)pm

j

A(ρ)(dv)+j∑

i=1

∫E[|Dj−1

vi Zvi χρ |p]γA(ρ)(v)p

mj

A(ρ)(dv)}.

It holds that

E[δ(DjvZχρ)

p] ≤ cp

0,p‖DjvZχρ‖p0,p−1,p;A(ρ)

= cp

0,p

∑

j ′≤p−1

∫E[∣∣Dj ′

u DjvZvχρ

∣∣p]

γA(ρ)(u)pγA(ρ)(v)pm

j ′A(ρ)(du)mAρ (dv),


in view of (5.68) for the case n = 0. Then we have

∫

A(ρ)j

E[δ(D

juZχρ)

p]

γA(ρ)(v)pm

j

A(ρ)(dv)

≤ cp

0,p

∑

j ′≤p−1

∫E[∣∣Dj ′

u DjvZvχρ

∣∣p]

γA(ρ)(u)pγA(ρ)(v)pγA(ρ)(v)pm

j ′A(ρ)(du)mj

A(ρ)(dv)mA(ρ)(dv)

≤ cp

0,p

∑

j ′≤p−1

‖Zχρ‖p0,j+j ′,p;A(ρ)≤ c‖Zχρ‖p0,j+p−1,p;A(ρ)

. (5.71)

Therefore, the first term of the right-hand side of (5.70) is dominated byC‖Zχρ‖p0,j+p−1,p;A(ρ)

. We have further,

∫

A(ρ)j

E[∣∣Dj−1

vi Zvi χρ

∣∣p]

γA(ρ)(v)pmk

A(ρ)(dv) ≤ ‖Zχρ‖p0,j−1,p;A(ρ). (5.72)

Since the fact is valid for any j = 1, . . . , n, we obtain from (5.70),

‖δ(Zχρ)‖p0,n,p;A(ρ)≤ c

pn,p‖Zχρ ||p0,n+p−1,p;A(ρ)

.

This proves the assertion of the theorem. ��We will summarize properties of Skorohod integrals by Poisson random measure.

Theorem 5.6.2

1. Skorohod integral operator δ is extended to a continuous linear operator fromF-space D∞

Uto F-space D∞. Further, for any n ∈ N and p ≥ 2, there exists a

positive constant cn,p such that for any 0 < ρ < ρ0

‖δ(Zχρ)‖0,n,p;A(ρ) ≤ cn,p‖Z‖0,n+〈p〉−1,〈p〉:A(ρ) (5.73)

holds for any Z ∈ D∞U

, where 〈p〉 = −2[−p/2].2. Adjoint formula (5.43) holds for any Z ∈ D∞

Uand G ∈ D∞.

3. Commutation relation (5.44) holds for any Z ∈ D∞U

.

4. The isometric property (5.45) holds for any Y,Z ∈ D∞U

.

5. Hölder’s inequalities (5.52), (5.53) hold for any F,G ∈ D∞ and Z ∈ D∞U

.

Further, similar Hölder’s inequalities are valid for norms ‖ ‖�0,n,p.

A d-dimensional Poisson functional F = (F 1, . . . , F d) is said to belong tothe space (D∞)d , if all components Fj , j = 1, . . . , d belong to D∞. Sobolevnorms ‖F‖0,n,p;A and ‖F‖�0,n,p are defined as sums of the corresponding norms

for components Fj , j = 1, . . . , d.


Difference operator D and its adjoint δ can be extended to complex-valuedfunctionals. The space of complex-valued smooth Poisson functionals is denotedagain by D∞ and the space of complex U-valued smooth functionals is denoted byD∞U

. Proposition 5.5.1 and Theorem 5.6.1 are valid for complex-valued functionals.

We will next study the estimation of the exponential functional ei(v,DF )− 1 withrespect to Sobolev norms. In the next lemma, we are interested in the dependenceof the estimation with respect to parameter ρ.

Lemma 5.6.2 For any n ∈ N and p ≥ 2 there exists a positive constant cn,p suchthat for any F ∈ (D∞)d , v ∈ R

d and 0 < ρ < ρ0,

‖ei(v,DF ) − 1‖0,n,p;A(ρ) (5.74)

≤ cn,p|v|ϕ(ρ) 12 ‖DF‖�0,n,(n+1)p(1 + |v|ρϕ(ρ) 1

2 ‖DF‖�0,n,(n+1)p)n.

Proof The norm ‖ei(v,DF ) − 1‖0,n,p;A(ρ) is dominated by the sum of the followingterms for j = 0, . . . , n:

E

[ ∫

A(ρ)j+1

( |Dju(e

i(v,DuF ) − 1)|γA(ρ)(u)γA(ρ)(u)

)p

mj

A(ρ)(du)mA(ρ)(du)

] 1p

. (5.75)

We will compute the functional Dju(e

i(v,DuF ) − 1). It is not simple, since we haveto use the difference rule (5.50). If j = 1, we have

Du′(ei(v,DuF ) − 1) = e

i(v,DuF◦ε+u′ ) − ei(v,DuF )

= ei(v,Du′ DuF )+i(v,DuF ) − ei(v,DuF ) = ei(v,DuF )(ei(v,Du′ DuF ) − 1).

If j = 2, we have by the difference rule (5.50)

Du′′Du′(ei(v,DuF ) − 1) = ei(v,DuF )(ei(v,Du′′ DuF ) − 1)(ei(v,Du′ DuF ) − 1)

+ ei(v,DuF )ei(v,Du′ DuF )(ei(v,Du′′ Du′ DuF ) − 1)

+ ei(v,DuF )(ei(v,Du′′ DuF ) − 1)ei(v,Du′ DuF )

(ei(v,Du′′ Du′ DuF ) − 1).

Repeating this argument, we have

Dju(e

i(v,DuF ) − 1) =∑

eiZj0,...,jju,u

j∏

l=0

(ei(v,Djlul DuF ) − 1),

where jl, l = 0, . . . , j are nonnegative integers satisfying jl ≤ j and j0+· · ·+jj ≥j . ul , l = 0, . . . , j are subsets of u satisfying

⋃l ul = u. Further, Z

j0,...,jju,u are sums


of real functionals written as (v, Dj ′lu′lDuF ), j ′l ≤ jl − 1,u′l ⊂ ul . The summation

is taken for all such j0, . . . , jj and u0, . . . ,uj . Then it holds that |eiZj0,...,jju,u | = 1.

Therefore, (5.75) is dominated by a sum of terms

E

[ ∫

A(ρ)j+1

(∏j

l=0 |ei(v,Djlul DuF ) − 1|

γA(ρ)(u)γA(ρ)(u)

)p

mj

A(ρ)(du)mA(ρ)(du)

] 1p

≤ ϕ(ρ)j+1

2 sup(u,u)∈A(ρ)j+1

E

[(∏j


γ (u)γ (u)

)p] 1p

. (5.76)

In the above we used the equality γAρ (u)γA(ρ)(u) = ϕ(ρ)−j+1

2 γ (u)γ (u). Further,we have the inequality

∏j


γ (u)γ (u)≤ |ei(v,Dj0

u0 DuF ) − 1|γ (u0)γ (u)

j∏

l=1

ρ|ei(v,Djlul DuF ) − 1|

γ (ul )γ (u),

if γ (u) ≤ ρ. In fact, since u ⊂ ⋃j

l=0 ul , we have γ (u) ≥ ∏j

l=0 γ (ul ) and further,ρ

γ (u)≥ 1 holds for u ∈ A(ρ). Therefore, applying Hölder’s inequality, (5.76) is

dominated by

ϕ(ρ)j+1

2 sup(u,u)∈A(ρ)j+1

E

[( |ei(v,Dj0u0 DuF ) − 1|

γ (u0)γ (u)

)pj∏

l=1

(ρ|ei(v,Djlul DuF ) − 1|

γ (ul )γ (u)

)p] 1

p

≤ ϕ(ρ)j+1

2 I0 · I1 · · · Ij , (5.77)

where

I0 := sup(u0,u)∈A(ρ)j+1

E

[( |ei(v,Dj0u0 DuF ) − 1|

γ (u0)γ (u)

)(j+1)p] 1

(j+1)p

,

Il := sup(ul ,u)∈A(ρ)j+1

E

[(ρ|ei(v,Djlul DuF ) − 1|

γ (ul )γ (u)

)(j+1)p] 1

(j+1)p

, l = 1, . . . , j.

Since |ei(v,Djlu0 DuF ) − 1| ≤ |(v, Dj0

u0DuF )| holds, we have

I0 ≤ |v|‖DF‖�0,j0,(j+1)p, Il ≤ ρ|v|‖DF‖�0,jl ,(j+1)p, l = 1, . . . , j.

Therefore, (5.76) is dominated by

5.7 Nondegenerate Poisson Functionals 209

c|v|(j+1)ρjϕ(ρ)j+1

2 (‖DF‖�0,j,(j+1)p)(j+1). (5.78)

Hence (5.75) is also dominated by (5.78) (with a different constant c). Finally,summing up the above inequality for j = 0, . . . , n, we get the inequality of thelemma. ��

5.7 Nondegenerate Poisson Functionals

Let F be an element of (D∞)d . We are interested in the existence of the smoothdensity of the law of F . Conditions needed could be related to ‘Malliavin covari-ance’. The Malliavin covariance of a smooth d-dimensional Poisson functionalF ∈ (D∞)d conditioned to the set A(ρ) is defined by

RFρ = 1

ϕ(ρ)

∫

A(ρ)

DuF (DuF )T n(du), (5.79)

where ϕ(ρ) = n(A(ρ)). Set RFρ (θ) = (θ, RF

ρ θ). F is called nondegenerate if

RFρ (θ) are invertible a.s., inverses belong to L∞ for any θ ∈ Sd−1 and satisfy

sup(ρ,θ)

|RFρ (θ)−1|�0,n,p < ∞, ∀n, p,

where the supremum is taken for all 0 < ρ < ρ0 and θ ∈ Sd−1.For a Wiener functional G, we saw in Lemma 5.3.1 that if G ∈ D∞ is invertible

and the inverse G−1 belongs to L∞−, then G−1 is a smooth Wiener functionalbelonging to D∞. We remark the similar fact for Poisson functionals G.

Lemma 5.7.1 Suppose that a Poisson functional G ∈ D∞ is invertible and G−1

belongs to L∞. Then G−1 ∈ D∞. Further, for any n, p, there is a positive constantcn,p such that

‖G−1‖�0,n,p ≤ cn,p(1 + ‖G‖�0,n,2qp)q(1 + |G−1|�0,n,2qp)q (5.80)

holds for any G with the above property, where q = 2n − 1.

The proof is not simple. It will be given in a more general form in Theo-rem 5.10.2.

By the above lemma, the nondegenerate condition for a Poisson functional F ∈(D∞)d is equivalent to the condition

sup(ρ,θ)

‖RFρ (θ)−1‖�0,n,p < ∞, ∀n, p.


In applications, however, it is not easy to handle the above nondegeneratecondition. We are interested in the limit of the Malliavin covariances RF

ρ as ρ → 0.For this, let us introduce a regularity condition for a Poisson functional. It isconvenient to set Dt,0F = 0. Then the operator Dt,zF (= D(t,z)F ) is defined forany (t, z). A Poisson functional F is called regular if it satisfies the following.

Condition (R) Dt,zF is twice continuously differentiable with respect to z =(z1, . . . , zd

′) ∈ R

d ′ a.s. Set

Λt(F ) =d ′∑

i=1

sup|z|≤1

|∂zi Dt,zF | +d ′∑

i,j=1

sup|z|≤1

|∂zi ∂zj Dt,zF |. (5.81)

Then |Λ(F)|�0,n,p < ∞ holds for any n and p.

We shall rewrite the Malliavin covariance of F in the case where F ∈ (D∞)d

is regular. Since the measure n(dt dz) is equal to dtν(dz), the covariance can bewritten as

RFρ =

∫

T

( ∫

A0(ρ)

Dt,zF Dt,zFT νA0(ρ)(dz)

)dt,

where νA0(ρ) is a measure on A0(ρ) given by ϕ(ρ)−1ν(dz). Set

∂Dt,0F = (∂z1Dt,zF, . . . , ∂zd

′ Dt,zF )|z=0. (5.82)

We assume that the family of matrices Γρ =( ∫

A0(ρ)zizj νA0(ρ)(dz)

)converges to

a matrix Γ0 as ρ → 0. Then RFρ converges to KF , where

KF =∫

T

∂Dt,0FΓ0(∂Dt,0F)T dt. (5.83)

The functional KF is called the Malliavin covariance of F at the center. Further,F is called nondegenerate at the center if KF (θ) ≡ (θ,KF θ) are invertible for anyθ ∈ Sd−1 and inverses satisfy

supθ

|KF (θ)−1|�0,n,p < ∞, ∀n, p. (5.84)

Here the supremum for θ is taken in the set Sd−1.

Example We give a simple example where the Malliavin covariance at the center iscomputed directly. Let F = (F 1, . . . , F d) be a linear functional of Poisson randommeasure written as F i = ∫

Uhi(s, z)ND(ds dz). We saw in Sect. 5.4 the equality

D(t,z)F = h(t, z) a.e. Therefore,


RFρ =

∫

T

( ∫h(t, z)h(t, z)T νA0(ρ)(dz)

)dt.

If h(s, z) is differentiable in z, we have ∂Dt,0F = ∂h(t, z)|z=0 = (hij (t)) ≡H(t) a.e. Consequently, the Malliavin covariance of F at the center is given by

KF =∫

T

H(t)Γ0H(t)T dt.

It is a non-random matrix. Then KF ◦ ε+u = KF holds a.s. for any u. Therefore,if the matrix KF is positive definite (invertible), then |KF (θ)−1|�0,n,p = KF (θ)−1.

Hence F is nondegenerate at the centerThe law of F is infinitely divisible and its characteristic function is given by

ψ(v) = exp{ ∫

T

( ∫

Rd′ei(v,h(s,z)) − i(v, h(s, z)1D(z))ν(dz)

)ds

}.

We can show similarly to the proof of Lemma 1.2.1 the inequality

−R( ∫

T

( ∫

Rd′0

{ei(v,h(s,z))−1−i(v, h(s, z))1D(z)

}ν(dz)

)ds

)≥

∑

i,j

vi

|v| α2vj

|v| α2 Kijε ,

where Kε = ∫TH(s)ΓεH(s)T ds. Since {Kε} are uniformly positive definite by

our assumption, inequality |E[ei(v,F )]| ≤ c2 exp{−c1|v|2−α} holds with positiveconstants c1 and c2, as in Lemma 1.2.1. Then the law of F has a C∞-density.

We want to show that if F ∈ (D∞)d is regular and is nondegenerate at the center,its law has a smooth density: However, a direct proof of the fact is difficult. As afirst step, we will introduce another technical nondegenerate condition. Associatedwith F ∈ (D∞)d , we define a family of Poisson functionals with parameter (ρ, v) ∈(0, 1)× (Rd \ {0}):

QF (ρ, v) = 1

|v|2ϕ(ρ)∫

A(ρ)

|ei(v,DuF ) − 1|2n(du). (5.85)

Let δ be a constant satisfying 1 < δ < 2/α, where α is the exponent of the Lévymeasure. We fix it and define for (ρ, θ) ∈ (0, ρ0)× Sd−1,

QFρ (θ) = QF (ρ, ρ−

1δ θ). (5.86)

A Poisson functional F of (D∞)d is called nondegenerate with respect to QFρ or

simply δ-nondegenerate if {QFρ (θ)} are invertible a.s. for any (ρ, θ) and inverses

satisfy


sup(ρ,θ)

|QFρ (θ)

−1|�0,n,p < ∞, ∀n, p, (5.87)

where the supremum is taken for 0 < ρ < ρ0 and θ ∈ Sd−1.We have defined three types of nondegenerate condition for a d-dimensional

smooth Poisson functional F . In the remainder of this section, we will show that ifF is δ-nondegenerate for some 1 < δ < 2/α, its law has a rapidly decreasing C∞-density. In the next section, we will show that these three nondegenerate conditionsare equivalent.

The next one corresponds to Proposition 5.3.1 for Wiener functionals.

Proposition 5.7.1 For any n, p, there exists a positive constant cn,p such that forany δ-nondegenerate F of (D∞)d ,

‖QFρ (θ)

−1‖�0,n,p (5.88)

≤ cn,p

(1 + ‖DF‖�

0,n,(n+1)2n+2p

)(n+1)2n+2(1 + |QF

ρ (θ)−1|�

0,n,2n+2p

)2n

holds for all (ρ, θ).

Proof We will show that for any n ∈ N, p ≥ 2, there exists a positive constant csuch that

sup(ρ,θ)

‖QFρ (θ)‖�0,n,p ≤ c(1 + ‖DF‖�0,n,2(n+1)p)

2(n+1). (5.89)

Note that QFρ (θ) is rewritten as

QFρ (θ) =

1

|v|2∫

A(ρ)

|ei(v,DuF ) − 1|2γ (u)2 mA(du), (5.90)

where ρ = |v|−δ . Since |ei(v,DuF ) − 1|2 ≤ |(v, DuF )|2, we have

‖QFρ (θ)‖�0,0,p = E[QF

ρ (θ)p] 1

p ≤ ‖DF‖�0,0,p. (5.91)

Further, if 1 ≤ j ≤ n and p ≥ 2,

E[∣∣Dj

uQFρ (θ)

∣∣p] 1p

γ (u)≤ 1

|v|21

ϕ(ρ)

∫

A(ρ)

E[∣∣Dj

u(|ei(v,DuF ) − 1|2)∣∣p] 1p

γ (u)γ (u)2 mA(du),

where ρ = |v|−δ . We can show similarly to the proof of Lemma 5.6.2 that there is apositive constant c′ such that the above is dominated by


c′ |v|2(j+1)ρ2j

|v|2 (1 + ‖DF‖�0,j,2(j+1)p)2(j+1) ≤ c′(1 + ‖DF‖�0,j,2(j+1)p)

2(j+1),

since |v|ρ ≤ 1. Therefore we get (5.89).Now we will apply (5.80) for G = QF

ρ (θ). Then (5.88) follows. ��We are now in the position of establishing the smooth density of the law of a

δ-nondegenerate Poisson functional.

Theorem 5.7.1 ([46]) For any N ∈ N, there exist n ≥ 1, p ≥ 2 and a positiveconstant c such that for any δ-nondegenerate F of (D∞)d ,


]∣∣∣ (5.92)

≤ c

|v|(1− αδ2 )N

((1 + ‖DF‖�0,n,2(n+1)p)

n+1 sup(ρ,θ)

‖QFρ (θ)

−1‖�0,n,2p)N‖G‖�0,n,p

hold for all |v| ≥ 1 and G ∈ D∞.

Proof We define a complex valued random field {Zu = ZF,vu , u ∈ U} associated

with a given δ-nondegenerate functional F = (F 1, . . . , F d) and v ∈ Rd with |v| ≥

1:

Zu = ZF,vu = 1

|v|2ϕ(ρ) · χρ(u)(e−i(v,DuF ) − 1) ·QF (ρ, v)−1, (5.93)

where ρ = |v|−δ . It is an element of D∞T

for any F, v. We will apply the adjointequation (5.43) for Zu = ZuG and ei(v,F ). Then we have

E[ei(v,F )δ(ZG)

] = E[ ∫

Du(ei(v,F ))ZuGn(du)

](5.94)

= E[ ∫

ei(v,F )(ei(v,DuF ) − 1)Zun(du) ·G]

= E[ei(v,F )G

∫(ei(v,DuF ) − 1)Zun(du)

]

= E[ei(v,F )G],

since∫(ei(v,DuF ) − 1)Zun(du) = 1. Therefore, setting L(G) = δ(ZG), we have

the iteration formula

E[ei(v,F )G] = E[ei(v,F )L(G)] = · · · = E[ei(v,F )LN(G)], (5.95)

for N = 1, 2, . . .. Consequently, we get by using Theorem 5.6.1 and Hölder’sinequality


|E[ei(v,F )G]| ≤ ‖LN(G)‖0,0,2 = ‖δ(ZLN−1(G))‖0,0,2

≤ c‖ZLN−1(G)‖0,1,2 ≤ c′‖Z‖0,1.4‖LN−1(G)‖0,1,4,

where ‖ ‖0,n,p = ‖ ‖0,n,p;A(ρ). Repeating this argument, there are increasingsequences cN, nN, pN,N = 1, 2, . . . such that the inequality

|E[ei(v,F )G]| ≤ cN‖Z‖0,n1,p1 · · · ‖Z‖0,nN ,pN‖G‖0,nN ,pN

≤ cN‖Z‖N0,nN ,pN‖G‖0,nN ,pN

(5.96)

holds, since norms ‖ ‖0,n,p are nondecreasing with respect to n, p.We will compute ‖Z‖0,n,p. Apply Hölder’s inequality to (5.93). Then we have

‖Z‖0,n,p ≤ c

|v|2ϕ(ρ)‖χρ(e−i(v,DF ))−1)‖0,n,2p‖QF (ρ, v)−1‖0,n,2p.

Set ρ = |v|−δ . Then we have from Lemma 5.6.2

‖χρ(e−i(v,DF ) − 1)||0,n,2p ≤ c|v|ϕ(ρ) 1

2 (1 + ‖DF‖�0,n,2(n+1)p)n+1.

Further, we have

‖QF (ρ, v)−1‖0,n,2p ≤ sup(ρ,θ)

‖QFρ (θ)

−1‖�0,n,2p.

Furthermore, 1

|v|ϕ(|v|−δ)12≤ c′

|v|(1− αδ2 )

holds for any |v| ≥ 1, since ϕ(|v|−δ) ≥ c|v|−αδ

by the order condition. Therefore we get

‖Z‖0,n,p ≤ c′

|v|(1− αδ2 )

(1 + ‖DF‖�0,n,2(n+1)p)n+1 sup

(ρ,θ)

‖QFρ (θ)

−1‖�0,n,2p.

Substitute the above in (5.96), then we get the inequality of the theorem. ��As an immediate consequence of Theorem 5.7.1, we have the next corollary.

Corollary 5.7.1 If F of (D∞)d is δ-nondegenerate, the law of F weighted byG ∈ D∞ has a rapidly decreasing C∞-density. The density function and itsderivatives are given by the Fourier inversion formula (5.34).

5.8 Equivalence of Nondegenerate Conditions

In Sect. 5.7, we defined three types of nondegenerate Poisson functionals. In thissection, we show the equivalence of these nondegenerate conditions.

5.8 Equivalence of Nondegenerate Conditions 215

Theorem 5.8.1 ([46]) Let F be a regular functional of (D∞)d .

1. If F is δ-nondegenerate for some 1 < δ < 2/α, then F is nondegenerate at thecenter. Further, for any n, p the inequality

supθ

|KF (θ)−1|�0,n,p ≤ sup(ρ,θ)

|QFρ (θ)

−1|�0,n,p (5.97)

holds for all regular δ-nondegenerate functionals F ∈ (D∞)d .2. If F is nondegenerate at the center, F is δ-nondegenerate for any 1 < δ < 2/α.

Further, for any n, p there is a positive constant c such that the inequality

|QFρ (θ)

−1|�0,n,p ≤ cCδ′n,p(F )|KF (θ)−1|�0,n,2p (5.98)

holds for any ρ, θ and F mentioned above. Here

Cδ′n,p(F ) ≤ (1 + |Λ(F)|�0,n,8(2+δ′)p)

2(2+δ′)(1 + |KF (θ)−1|�0,n,8p)12 , (5.99)

where Λt(F ) is the Poisson functional given by (5.81) and δ′ is the conjugateof δ.

Proof of the first half. Suppose that F is δ-nondegenerate. Consider the integrandof QF

ρ (v). It holds that

|ei(v,DuF ) − 1|2 ≤ |(v, DuF )|2 = v∂Dt,0FzzT ∂Dt,0FT vT +O(|z|3),

a.e. dndP . Here we used the Taylor expansion Dt,zF = ∂Dt,0Fz + O(|z|2).Integrating the above by n(du) on the set T× A0(ρ), we get

1

|v|2ϕ(ρ)∫

T

∫

A0(ρ)

|ei(v,Dt,zF ) − 1|2dtν(dz)

= v

|v|∫

T

(∂Dt,0F)Γρ(∂Dt,0F)T dt (v

|v| )T +O(ρ), a.s.

Therefore setting θ = v/|v|, we have

lim supρ→0

QFρ (θ) ≤ KF (θ), or lim inf

ρ→0QF

ρ (θ)−1 ≥ KF (θ)−1, a.s.

Operating the transformation ε+u on both sides of the above, we get again

lim infρ→0

QFρ (θ)

−1 ◦ ε+u ≥ KF (θ)−1 ◦ ε+u , a.e. dnj dP.


Therefore, we have |KF (θ)−1|�0,n,p ≤ supρ |QFρ (θ)

−1|�0,n,p for all θ ∈ Sd−1. Thenthe first assertion of the theorem follows. ��

The proof of the latter half of the theorem is long. Our discussion is close to [46].It will be completed after discussing Lemmas 5.8.1, 5.8.2, 5.8.3, and 5.8.4. We willintroduce two other matrix-valued Poisson functionals. We will take any δ satisfying1 < δ < 2/α. Let F be a regular functional of (D∞)d . For 0 < ρ < ρ0 and v �= 0we define QF

ρ (θ) by (5.86) and further,

SFρ = 1

ϕ(ρ)

∫

A(ρ)∩{|DuF |≤ρ1δ }

DuF (DuF )T n(du). (5.100)

KFρ = 1

ϕ(ρ)

∫

T

∫

A0(ρ)

(∂Dt,0F)z · zT (∂Dt,0F)T n(dt dz). (5.101)

We set further,

SFρ (θ) = (θ, SF

ρ θ), KFρ (θ) = (θ,KF

ρ θ).

Discussions will be divided into the following three steps:

1. We compare two norms |QFρ (θ)

−1|�0,n,p and |SFρ (θ)−1|�0,n,p. See Lemma 5.8.1.

2. We compare two norms |KFρ (θ)−1|�0,n,p and |KF (θ)−1|�0,n,p. See Lemma 5.8.2.

3. We compare two norms |SFρ (θ)−1|�0,n,p and |Kρ(θ)

−1|�0,n,p. See Lemmas 5.8.3and 5.8.4.

Lemma 5.8.1 Suppose that SFρ (θ) are invertible a.s. for any ρ, θ . Then QF

ρ (θ) arealso invertible a.s. Further, for any n, p there is a positive constant c1 such that theinequality

|QFρ (θ)

−1|�0,n,p ≤ c1|SFρ (θ)−1|�0,n,p (5.102)

holds for any ρ, θ .

Proof Note that the inequality |ei(v,DuF ) − 1|2 ≥ c|(v, DuF )| holds with someconstant c > 0 on the set |DuF | < |v|−1. Then we have inequalities

QFρ (θ) ◦ ε+u ≥ cSF

ρ (v) ◦ ε+u , a.e. dnj dP.

Therefore, if SFρ (v) ◦ ε+u are invertible a.e., then QF

ρ (θ) ◦ ε+u are also invertible a.e.and satisfy

E[(QF

ρ (θ) ◦ ε+u )−p] ≤ c−pE

[(SF

ρ (θ) ◦ ε+u )−p], a.e. dnj .

Taking the essential supremum with respect to u and summing up these forj = 0, . . . , n, we get the inequality of the lemma. ��


Lemma 5.8.2 Suppose that KF (θ) are invertible a.s. for any θ ∈ Sd−1. ThenKF

ρ (θ) are also invertible a.s. for any 0 < ρ < ρ0, θ ∈ Sd−1. Further, for anyn, p there exist two positive constants c2 < c3 such that the inequality

c2|KF (θ)−1|�0,n,p ≤ |KFρ (θ)−1|�0,n,p ≤ c3|KF (θ)−1|�0,n,p (5.103)

holds for any ρ, θ .

Proof Let λρ and Λρ be the minimum and the maximum eigenvalues of Γρ ,respectively. Since lim infρ→0 Γρ ≥ Γ0 and Γ0 is nondegenerate, there exists0 < ρ1 < ρ0 such that 0 < inf0<ρ<ρ1 λρ ≤ sup0<ρ<ρ1

Λρ < ∞. For ρ1 <

ρ < ρ0, we have Γρ ≥ ϕ(ρ1)ϕ(ρ)

Γρ1 . Therefore we have 0 < c′ = inf0<ρ<ρ0 λρ ≤sup0<ρ<ρ0

Λρ = C′ < ∞. Then for any 0 < ρ < ρ0 we have the inequality

c′

Λ0KF (θ) ◦ ε+u ≤ KF

ρ (θ) ◦ ε+u ≤ C′

λ0KF (θ) ◦ ε+u , a.e. dnj dP,

where λ0,Λ0 are the minimal and the maximal eigen values of Γ0. Then theinequality of the lemma holds. ��

Now, assume that F is nondegenerate at the center. Let u ∈ A(1)j . We want toshow the invertibility of {SF

ρ (θ)◦ε+u } and then the estimation of E[(SF

ρ (θ)◦ε+u )−p].

We will define a nonnegative functional ! = !F (θ) ◦ ε+u as the infimum of 0 < ρ <

ρ0 which satisfies

∣∣SFρ (θ) ◦ ε+u −KF

ρ (θ) ◦ ε+u∣∣ >

1

2KF

ρ (θ) ◦ ε+u .

(We set ! = 1 if the above is an empty set.) Then on the set {! < ε}, the inequality

sup0<ρ<ε

|SFρ (θ) ◦ ε+u −KF

ρ (θ) ◦ ε+u |KF

ρ (θ) ◦ ε+u>

1

2

holds.

Lemma 5.8.3 For any p ≥ 2, there is a positive constant c4 such that the inequality

P(!F (θ) ◦ ε+u < ε) ≤ c4Aδ′p (F )εp, ∀0 < ε < ρ0, θ ∈ Sd−1, u ∈ A(1)j

(5.104)holds for all F of (D∞)d which are regular and nondegenerate at the center. Here

Aδ′p (F ) = E

[ ∫

T

(1 + |Λt(F ) ◦ ε+u |)2p(2+δ′) dt] 1

2E[KF (θ)−2p ◦ ε+u ]

12 , (5.105)

where δ′ is the conjugate of δ.


Proof We will first rewrite KFρ . Since Γρ is given by (1.21), the matrix KF

ρ iswritten as

KFρ =

1

ϕ(ρ)

∫

T

∫

A0(ρ)

(∂Dt,0F)zzT (∂Dt,0F)T n(dt dz).

Therefore,

KFρ (θ) = 1

ϕ(ρ)

∫

T

∫

A0(ρ)

(θ, ∂Dt,0Fz)2n(dt dz),

where u = (t, z).Now we will fix F, θ,u. In the following arguments, we will write DuF ◦ ε+u ,

SFρ (θ) ◦ ε+u , KF

ρ (θ) ◦ ε+u , Λt(F ) ◦ ε+u etc. as DuF , Sρ,Kρ , Λt etc., dropping F, θ

and ε+u . We define E(ρ) = {(t, z) ∈ A(ρ); |Dt,zF | ≤ ρ1/δ}. Then the inequality

∣∣Sρ −Kρ

∣∣ ≤ 1

ϕ(ρ)

∫

T

∫

A0(ρ)

∣∣∣(θ, Dt,zF )2 − (θ, ∂Dt,0Fz)2∣∣∣ 1E(ρ)(t, z)n(dt dz)

+ 1

ϕ(ρ)

∫

T

∫

A0(ρ)

(θ, ∂Dt,0Fz)21E(ρ)c (t, z)n(dt dz)

= I1(ρ)+ I2(ρ)

holds. Therefore we have

P(! < ε) ≤ P(

supρ<ε

I1(ρ)

Kρ

≥ 1

4

)+ P

(supρ<ε

I2(ρ)

Kρ

≥ 1

4

)(5.106)

≤ P( supρ<ε I1(ρ)

K≥ c

4

)+ P

( supρ<ε I2(ρ)

K≥ c

4

),

where c is a positive constant such that Kρ ≥ cK .We want to compute the right-hand side. We first consider I1(ρ). In view of

Condition (R), the integrand of I1(ρ) is estimated as

|(θ, Dt,zF )2 − (θ, ∂Dt,0Fz)2| ≤ |z|3Λ2t ,

where Λt = Λt(F ) ◦ ε+u . Therefore, we have I1(ρ) ≤ ρ∫TΛ2

t dt . Then

E[

supρ<ε

I1(ρ)2p

]≤ c1ε

2pE[ ∫

T

|Λt |4p dt].

Therefore, by Tchebyschev’s inequality


P( supρ<ε I1(ρ)

K≥ c

4

)≤ c′1εpE

[ ∫

T

|Λt |4p dt] 1

2E[K−2p] 1

2 .

We will next consider I2. Let δ′ be the conjugate of δ. Since m(dt dz) =|z|2n(dt dz) holds for |z| ≤ ρ0, we have by Tchebyschev’s inequality,

I2(ρ) ≤ 1

ϕ(ρ)

∫

T

∫

A0(ρ)

|∂Dt,0F |21E(ρ)cm(dt dz)

≤ 1

ρδ′/δ1

ϕ(ρ)

∫

T

∫

A0(ρ)

|∂Dt,0F |2|Dt,zF |δ′m(dt dz)

≤ 1

ρδ′/δ1

ϕ(ρ)

∫

T

∫

A0(ρ)

|∂Dt,0F |2+δ′ |z|δ′m(dt dz)

≤ ρ

∫

T

Λ2+δ′t dt.

Therefore, we have E[supρ<ε I2(ρ)2p] ≤ c2ε

2pE[ ∫

T|Λt |2p(2+δ′) dt

]. Then,

P( supρ<ε I2(ρ)

K>

c

4

)≤ c′2εpE

[ ∫

T

|Λt |2p(2+δ′) dt] 1

2E[K−2p] 1

2 .

Therefore, in view of (5.106), P(! < ε) is dominated by

εp(c′1E

[ ∫

T

|Λt |4p dt]1/2 + c′2E

[ ∫

T

|Λt |2p(2+δ′) dt] 1

2)E[K−2p] 1

2

≤ εpc′3E[ ∫

T

(1 + |Λt(F ) ◦ ε+u |)2p(2+δ′) dt] 1

2E[KF (θ)−2p ◦ ε+u ]

12 .

Then we get the assertion of the lemma. ��Lemma 5.8.4 If F ∈ (D∞)d is regular and nondegenerate at the center, thenfunctionals {SF

ρ (θ), 0 < ρ < ρ0, θ ∈ Sd−1} are invertible. Further, for any j, p

there is a positive constant c5 such that the inequality

E[(SF

ρ (θ) ◦ ε+u )−p] ≤ c5B

δ′p (F )E

[(KF

ρ (θ) ◦ ε+u )−2p] 12 , a.e. dnj , (5.107)

holds for any ρ, θ and for all F ∈ (D∞)d which is regular and nondegenerate atthe center. Here

Bδ′p (F ) =

(1+E

[ ∫

T

(1+|Λt(F )◦ε+u |)8p(2+δ′) dt] 1

4)(

1+E[(KF

ρ (θ)◦ε+u )−8p] 1

4).

(5.108)


Proof We use simplified notations Sρ etc. as in the proof of the previous lemma.Let ! be the nonnegative functional defined before Lemma 5.8.3. Since Sρ,Kρ areleft continuous with respect to ρ, we have

|Sρ∧! −Kρ∧!| ≤ 1

2Sρ∧!.

Therefore we have Sρ∧! ≥ 12Kρ∧! for any 0 < ρ < ρ0. Further, since ϕ(ρ)Sρ

is nondecreasing with respect to ρ a.s., we have Sρ ≥ ϕ(!∧ρ)ϕ(ρ)

S!∧ρ if |v| ≥ 1.Therefore,

Sρ ≥ 1

2

ϕ(! ∧ ρ)

ϕ(ρ)Kρ∧! > 0, ∀ 0 < ρ < 1, a.s.

(We set ϕ(0) = 0.) Then we have by Schwartz’s inequality

E[S−pρ

] ≤ 2E[K

−2pρ∧!

] 12E[( ϕ(ρ)

ϕ(! ∧ ρ)

)2p] 12. (5.109)

It holds that E[K−2pρ∧! ] 1

2 ≤ c−pE[K−2p] 12 . We shall compute the last term. Set

a(x) = P(! ≤ x). We shall apply Lemma 5.8.3, replacing p by 4p. Then it holdsthat a(x) ≤ Aδ′

4p(F )x4p for any x. Since ϕ(x) ≥ c1xα holds by the order condition,

we have

E[( 1

ϕ(! ∧ ρ)

)2p1!<ρ

]=

∫ ρ

0

1

ϕ(x)2p a(dx) ≤ c1Aδ′4p(F ).

Therefore,

E[( ϕ(ρ)

ϕ(! ∧ ρ)

)2p] 12 ≤ P(! ≥ ρ)+ ϕ(ρ)pE

[( 1

ϕ(! ∧ ρ)

)2p1!<ρ

] 12

≤ c2

(1 +

√Aδ′

4p(F )), ∀0 < ρ < ρ0. (5.110)

Consequently (5.109) and (5.110) imply the inequality of the lemma. ��Proof of Theorem 5.8.1 Inequality (5.102) holds by Lemma 5.8.1. Further, fromLemma 5.8.4, we get

|SFρ (θ)−1|�0,n,p ≤ c6C

δ′n,p(F )|KF (θ)−1|�0,n,2p. (5.111)

Therefore the inequality of the theorem follows. ��We shall next study the equivalence of ‘nondegenerate’ and ‘nondegenerate at

the center’.


Theorem 5.8.2 Let F be a regular functional of (D∞)d .

1. If F is nondegenerate, it is nondegenerate at the center. Further, we have for anyn, p,

supθ

|KF (θ)−1|�0,n,p ≤ sup(ρ,θ)

|RFρ (θ)−1|�0,n,p. (5.112)

2. If F is nondegenerate at the center, it is nondegenerate. Further, for any n, p

there is a positive constant c such that

sup(ρ,θ)

|RFρ (θ)−1|�0,n,p ≤ cC0

n,p(F ) supθ

|KF (θ)−1|�0,n,2p, (5.113)

where C0n,p(F ) is given by (5.99) with δ′ = 0.

Proof The first assertion can be verified similarly as the first assertion of Theo-rem 5.8.1. Next, suppose that F is nondegenerate at the center. We will compareRF

ρ (θ) and KFρ (θ). Let us replace SF

ρ (θ) by RFρ (θ) in the argument of Lemmas 5.8.3

and 5.8.4. We will define a nonnegative functional ! = !F (θ) ◦ ε+u as the infimumof 0 < ρ < ρ0 which satisfies

∣∣RFρ (θ) ◦ ε+u −KF

ρ (θ) ◦ ε+u∣∣ >

1

2KF

ρ (θ) ◦ ε+u .

Rewrite RFρ (θ) ◦ ε+u etc as RF

ρ etc., dropping ε+u . Then we have

|RFρ (θ)−KF

ρ (θ)| ≤ 1

ϕ(ρ)

∫

A(ρ)

∣∣∣(θ, Dt,zF )2 − (θ, ∂Dt,0Fz)2∣∣∣ n(dt dz).

Denote the right-hand side of the above as I1(ρ) and set I2(ρ) = 0. We can proceedwith our arguments similarly to the proof Lemma 5.8.3. Then we get the assertionof Lemma 5.8.3, by setting δ′ = 0. Further, repeating arguments of Lemmas 5.8.4and Theorem 5.8.1, we find that (5.111) should be changed to the inequality

|RFρ (θ)−1|�0,n,p ≤ c′6C0

n,p(F )|KFρ (θ)−1|�0,n,2p. (5.114)

Therefore F is nondegenerate. The inequality (5.113) follows from (5.114)and (5.103). ��

From Theorems 5.8.1 and 5.8.2, we have the following.

Theorem 5.8.3 Let F be a regular functional of (D∞)d . The following statementsare equivalent:

(i) F is δ-nondegenerate for some 1 < δ < 2/α.(ii) F is δ-nondegenerate for any 1 < δ < 2/α.


(iii) F is nondegenerate at the center.(iv) F is nondegenerate.

Note The Malliavin calculus for jump processes was studied by Bismut [10], Lean-dre [74], Bichiteler–Gravereau–Jacod [7]. Picard [93] proposed another approachto the Malliavin calculus on a Poisson space. The difference operator Du and itsadjoint δ were introduced by Picard [92, 93]. His discussion was developed furtherby Picard–Savona [95], Ishikawa–Kunita [45] and Hayashi–Ishikawa [37] etc.

Picard [92] and Picard–Savona [95] introduced another condition for smoothdensity. For the Lévy measure ν, they assumed that there exist 0 < α < 2 andc, C > 0 such that the inequality

cρα|v|2 ≤∫

0<|z|<ρ

(v, z)2ν(dz) ≤ Cρα|v|2

holds for any v ∈ Rd ′ and 0 < ρ < ρ0. The condition is equivalent to that the

Lévy measure is nondegenerate and satisfies the order condition of exponent α atthe center with respect to family of balls B0(ρ) = {z ∈ R

d ′ ; |z| < ρ}, i.e., in thecase where the star-shaped neighborhoods A0(ρ) coincide with balls B0(ρ). HenceTheorem 5.8.1 is an extension of their work.

5.9 Product of Wiener Space and Poisson Space

We will define the product of a Wiener space and a Poisson space. Let W be theset of all continuous maps w;T → R

d ′ such that w(0) = 0, and let B(W) beits Borel field. Let Ω be the set of all integer-valued measures on U = T × R

d ′0 ,

and let B(Ω) be its Borel field. We consider the product space Ω = W × Ω andB(Ω) = B(W) ⊗ B(Ω). Its sub σ -fields B(W) ⊗ {∅,Ω} and {∅,W} ⊗ B(Ω) areagain denoted by B(W) and B(Ω), respectively.

Let P be a probability measure on the product space (Ω,B(Ω)), such that itsrestriction to B(W) is a Wiener measure and its restriction to B(Ω) is a Poissonrandom measure associated with the Lévy measure ν, and further, B(W) and B(Ω)

are independent. The triple (Ω,B(Ω), P ) is called the product of the Wiener spaceand the Poisson space, or simply, Wiener–Poisson space with the Lévy measureν. We assume that at the center 0, the Lévy measure ν is nondegenerate and theintensity measure n satisfies the order condition of exponent 0 < α < 2 with respectto a given family of star-shaped neighborhoods {A(ρ), 0 < ρ < ρ0}, as in Sect. 5.6.

The first component w of (w, ω) is called a Wiener variable and the secondcomponent ω is called a Poisson variable. Let S be a complete metric space. An S-valued B(Ω)-measurable function F(w,ω) is called an S-valued Wiener–Poissonfunctional. When S = R, it is simply called a Wiener–Poisson functional. A B(W)-measurable function F(w) is called a Wiener functional and a B(Ω)-measurablefunction F(ω) is called a Poisson functional.

5.9 Product of Wiener Space and Poisson Space 223

Let p > 1. We denote by Lp(Ω) the set of all Wiener–Poisson functionalsF such that E[|F |p] < ∞. We set Lp = Lp(Ω) and L∞− = ⋂

p>1 Lp.

It is an F -space. Let F(w,ω) be a Wiener–Poisson functional in L∞−. Let H

be the Cameron–Martin space introduced in Sect. 5.1. Given h ∈ H , we definetransformations Th, h ∈ H by ThF (w,ω) = F(w+h, ω), where h(t) = ∫ t

0 h(s) ds.Then for any p > 1 and r > 1, Th is a continuous linear transformation from Lpr

to Lp. Further, it is a continuous linear transformation from the F-space L∞− toitself. The fact can be verified in the same way as the transformation Th for Wienerfunctionals discussed in Sect. 5.1.

We will define the derivative operator DF for a Wiener–Poisson functionalF ∈ L∞−. It is similar to the derivative operator for Wiener functionals. It may beregarded as a partial derivative of F(w,ω) with respect to Wiener variable w. Letλ ∈ R. Then {Tλh, λ ∈ R} is a one-parameter group of transformations on L∞−.Suppose that for F ∈ L∞− there exists an H -valued Wiener–Poisson functional F ′such that |F ′|H ∈ L∞− and (5.2) holds for any h ∈ H(0) with respect to the metricd of L∞−, where H(0) is a dense subset of H . Then F is said to be H -differentiableand F ′ is called its H -derivative. We denote F ′ by DF or DtF . We denote by D theset of all H -differentiable Wiener–Poisson functionals. If F is a Poisson functionalin L∞−, then TλhF = F holds for any λ, h. Hence F is H -differentiable and theH -derivative is 0.

For a positive integer i ≥ 2, DiF is defined as DiF = D(Di−1F) by induction,similarly to that for the Wiener functional in Sect. 5.1. It is an H⊗i-valued Wiener–Poisson functional with |DiF |H⊗i ∈ L∞−. We denote it by Di

tF, t = (t1, . . . , ti ) ∈Ti .

For the Poisson space and Poisson variables, we use the same notations as inSects. 5.4, 5.5, 5.6, 5.7, and 5.8. Let u ∈ U. Transformations ε±u ; Ω → Ω is definedby ε±u (w, ω) = (w, ε±u ω). Further, the difference operator D is defined for F inL∞− by

DuF (w,ω) = F(w, ε+u ω)− F(w,ω).

It is well defined for almost all (u,w, ω) with respect to dndP . It is written as Dt,zF

if u = (t, z). For u = (u1, . . . , uj ) ∈ Uj , we set Dj

u = Du1 · · · Duj F .

If F is a Wiener functional in L∞−, then F ◦ ε+u = F and hence DF = 0.Hence D may be regarded as a partial difference of F(w,ω) with respect to Poissonvariable ω.

A Wiener–Poisson functional F is called smooth if for any i, j ∈ N and p ≥ 2,D

juF are infinitely H -differentiable for almost all u and satisfy

supu∈A(1)j

E[( ∫

Ti |DitD

juF |2 dt

) p2]

γ (u)p< ∞.


Here, ‘sup’ means the essential supremum with respect to the measure nj . Theset of all smooth functionals is denoted by D∞. A d-dimensional functional F =(F 1, . . . , F d) is called smooth if every component F i is smooth. We denote by(D∞)d the set of all d-dimensional smooth functionals.

The next lemma is shown immediately.

Lemma 5.9.1 For F ∈ D∞, DitD

juF is well defined a.e. dti dnj dP for any i, j of

N. Further, we have DitD

juF = D

juD

itF a.e. dti dnj dP .

A Wiener–Poisson functional with parameter T given by Xt = ∑nl=1 Flhl(t),

t ∈ T with Fl ∈ D∞ and hl ∈ H, l = 1, 2, . . ., is called a simple functional withparameter T. We denote by S∞

Tthe set of all simple functionals with parameter T.

For such Xt = ∑l Flhl(t) ∈ S∞

T, we define the Skorohod integral by the Wiener

process denoted by δ(X) as follows:

δ(X) =n∑

l=1

FlW(hl)−n∑

l=1

〈DFl, hl〉. (5.115)

It is an element of L∞−.Let Zu, u ∈ U be a Wiener–Poisson random field. We denote by D∞

Uthe set of

all Zu such that

DitZu

γ (u),

DjvD

itZu

γ (v)γ (u), i, j = 0, 1, 2, . . .

are bounded a.e. dti dn dP and dti dnj+1 dP , respectively. Further, D∞U0

denotes

the set of Zu ∈ D∞U

such that Zu ◦ ε−u are supported by a compact subset K of Ua.e. dn dP . For Z ∈ D∞

U0, we define the Skorohod integral of Zu by Poisson random

measure by

δ(Z) =∫

U

Zu ◦ ε−u N(du) =∫

U

Zu ◦ ε−u (N(du)− n(du)). (5.116)

It is an element of L∞−.

Lemma 5.9.2

1. The adjoint formula

E[Gδ(X)] = E[ ∫

T

(DtG)Xt dt]

(5.117)

holds for any G ∈ D∞ and X ∈ S∞T

. Further, the adjoint formula

5.9 Product of Wiener Space and Poisson Space 225

E[Gδ(Z)] = E[ ∫

U

(DuG)Zun(du)]

(5.118)

holds for any bounded G and Z ∈ D∞U0

.

2. The operator δ satisfies the commutation relation: if X ∈ S∞T

, then

Dtδ(X) = δ(DtX)+Xt, Duδ(X) = δ(DuX), a.e. (5.119)

The operator δ satisfies the commutation relation; if Z ∈ D∞U0

, then

Duδ(Z) = δ(DuZ)+ Zu, Dt δ(Z) = δ(DtZ), a.e. (5.120)

3. δ(X), δ(Y ) and δ(Z), δ(Y ) satisfy isometric properties (5.10) and (5.45), respec-tively.

Proof Let us first show the adjoint formula (5.117). We can show that equation (5.5)holds for F ∈ D∞, similarly to the proof of Proposition 5.1.2. Then (5.6) holds forany F,G ∈ D∞. Then we can verify (5.117) similarly to the proof of Lemma 5.1.1.For the proof of the adjoint formula (5.118), let us remark that Lemma 5.4.1 is validfor positive Wiener–Poisson functional Zu. Then Corollary 5.4.1 is also valid forsuch Zu. Then the adjoint formula for δ can be verified in the same way as in theproof of Lemma 5.4.2.

The commutation relation (5.119) for δ and D and the isometric equality (5.10)can be verified in the same way as in the proof of Lemma 5.1.1. Further, thecommutation relation for D and δ and its isometric property (5.45) can be verifiedsimilarly to the proof of Lemma 5.4.2.

We show that D and δ are commutative. Suppose X = ∑l Flhl . Then we have

Duδ(X) = Du(∑

l

FlW(hl)−∑

l

〈DFl, hl〉)

=∑

l

DuFlW(hl)−∑

l

〈DuDFl, hl〉 = δ(DuX).

Hence we have Duδ = δDu. Next, if Zu = ∑Flhl(u), we have

Dt δ(Z) = Dt

∑

l

∫Fl ◦ ε−u hl(u)(N(du)− n(du))

=∑

l

∫DtFl ◦ ε−u hl(u)(N(du)− n(du)) = δ(DtZ).

Hence we have Dt δ = δDt .


Finally, isometric properties (5.10) and (5.45) are verified in the same way as inLemmas 5.1.1 and 5.4.2, respectively. ��

5.10 Sobolev Norms for Wiener–Poisson Functionals

We will define Sobolev norms for Wiener–Poisson functionals. These norms shouldbe a combination of Sobolev norms for Wiener functionals and those for Poissonfunctionals. Let A be an open subset of A(1) = {u = (t, z) ∈ U; |z| ≤ 1} satisfying0 < m(A) ≤ 1. Let mA be the conditional probability measure. Let mj

A be a j -times product measure of mA and γA(u) be positive functions which are defined inSect. 5.5. Let i, j ∈ N and p ≥ 2. Set

Ii,j,p;A(F ) =∫

Aj

E[( ∫

Ti |DitD

juF |2 dt

) p2]

γA(u)pm

jA(du), F ∈ D∞,

Ii,j,p;A(X) =∫

Aj

E[( ∫

Ti+1 |DitD

juXt |2 dt dt

) p2]

γA(u)pm

jA(du), X ∈ S∞

T,

Ii,j,p;A(Z) =∫

Aj+1

E[( ∫

Ti |DitD

juZu|2 dt

) p2]

γA(u)pγA(u)pm

jA(du)mA(du), Z ∈ D∞

U0.

If i = 0 or j = 0, we define them for F ∈ D∞ as I0,0,p;A(F ) = E[|F |p] and

Ii,0,p;A(F ) = E[( ∫

Ti

|DitF |2 dt

) p2], I0,j,p;A(F ) =

∫

Aj

E[|DjuF |p]

γA(u)pm

jA(du).

We can define those for X ∈ S∞T

and Z ∈ D∞U0

in similar ways. Then we defineSobolev norms as

‖F‖m,n,p;A =( m∑

i=0

n∑

j=0

Ii,j,p;A(F )) 1

p, F ∈ D∞, (5.121)

‖X‖m,n,p;A =( m∑

i=0

n∑

j=0

Ii,j,p;A(X)) 1

p, X ∈ S∞

T, (5.122)

‖Z‖m,n,p;A =( m∑

i=0

n∑

j=0

Ii,j,p;A(Z)) 1

p, Z ∈ D∞

U0. (5.123)

When the domain A is fixed in discussions, we will often drop ;A from these normsand write

‖F‖m,n,p;A = ‖F‖m,n,p, ‖X‖m,n,p;A = ‖X‖m,n,p, ‖Z‖m,n,p;A = ‖Z‖m,n,p.

5.10 Sobolev Norms for Wiener–Poisson Functionals 227

Next, we set

I�i,j,p(F ) = sup

u∈A(1)j

E[( ∫

Ti |DitD

juF |2 dt

) p2]

γ (u)p, F ∈ D∞,

I�i,j,p(X) = sup

u∈A(1)j

E[( ∫

Ti+1 |DitD

juXt |2 dt dt

) p2]

γ (u)p, X ∈ S∞

T,

I�i,j,p(Z) = sup

(u,u)∈A(1)j+1

E[( ∫

Ti |DitD

juZu|2 dt

) p2]

γ (u)pγ (u)p, Z ∈ D∞

U0,

if i, j ≥ 1. We will define I�i,j,p(F ) etc. for i = 0 or j = 0 as I

�0,0,p(F ) = E[|F |p]

and

I�i,0,p(F ) = E

[( ∫

Ti

|DitF |2 dt

) p2], I

�0,j,p(F ) = sup

u∈A(1)j

E[|DjuF |p]

γ (u)p.

We can define those for X ∈ S∞T

and Z ∈ D∞U0

in similar ways. Then we defineanother norms by

‖F‖�m,n,p =( m∑

i=0

n∑

j=0

I�i,j,p(F )

) 1p, F ∈ D∞, (5.124)

‖X‖�m,n,p =( m∑

i=0

n∑

j=0

I�i,j,p(X)

) 1p, X ∈ S∞

T, (5.125)

‖Z‖�m,n,p =( m∑

i=0

n∑

j=0

I�i,j,p(Z)

) 1p, Z ∈ D∞

U0. (5.126)

Let {A(ρ), 0 < ρ < ρ0} be a family of star-shaped neighborhoods of the centerof the measure n. We have inequalities

‖F‖m,n,p;A(ρ) ≤ ‖F‖�m,n,p, F ∈ D∞, (5.127)

‖X‖m,n,p;A(ρ) ≤ ‖X‖�m,n,p, X ∈ S∞T,

‖Z‖m,n,p;A(ρ) ≤ ‖Z‖�m,n,p, Z ∈ D∞U0

for any m, n, p, ρ. Completions of D∞, S∞T

and D∞U0

by norms ‖ ‖�m,n,p are

denoted by Dm,n,p, Dm,n,p

Tand Dm,n,p

U, respectively. The intersections of these

with respect to m, n, p are denoted by D∞, D∞T

and D∞U

, respectively. These


are F -spaces with respect to countable norms {‖ ‖�m,n,p;m, n, p ∈ N}. IfF,X,Z belong to the above functional spaces, respectively, then for any A,‖F‖m,n,p;A, ‖X‖m,n,p;A, ‖Z‖m,n,p;A are well defined. These are norms and are

dominated by ‖F‖�m,n,p, ‖X‖�m,n,p and ‖Z‖�m,n,p, respectively.The next proposition is an extension of Propositions 5.2.1, 5.5.1 and

Lemma 5.9.1. Proof is straightforward and it is omitted.

Proposition 5.10.1

1. The operator D is extended to a continuous linear operator from F-space D∞to F-space D∞

T. It holds that ‖DF‖�m,n,p ≤ ‖F‖�m+1,n,p for any m, n ∈ N and

p ≥ 2.2. The operator D is extended to a continuous linear operator from F-space D∞

to F-space D∞U

. It holds that ‖DF‖�m,n,p ≤ ‖F‖�m,n+1,p for any m, n ∈ N andp ≥ 2.

3. It holds that DtDuF = DuDtF for any F ∈ D∞ and DtDuY = DuDtY for anyY ∈ D∞

Tor Y ∈ D∞

U, a.e. (t, u,w, ω) with respect to dtdndP .

The next is a combination of Lemmas 5.2.1 and 5.5.1. The proof is omitted.

Lemma 5.10.1

1. (Hölder’s inequality) For any m, n ∈ N and p ≥ 2, there exists a positiveconstant c such that

‖FG‖m,n,p;A ≤ c‖F‖m,n,pr;A‖G‖m,n,pr ′;A, (5.128)

‖FX‖m,n,p;A ≤ c‖F‖m,n,pr;A‖X‖m,n,pr ′;A,

‖FZ‖m,n,p;A ≤ c‖F‖m,n,pr;A‖Z‖m,n,pr ′;A

hold for any F,G ∈ D∞, X ∈ D∞T, Z ∈ D∞

U, domain A and r, r ′ > 1 with

1/r + 1/r ′ = 1.2. Similar Hölder’s inequalities are valid with respect to norms ‖ ‖�m,n,p.

Finally the norm |F |�0,n,p for a simple Wiener–Poisson functional F is defined

by (5.57). Then the norm | |�0,n,p is weaker than the norm ‖ ‖�m,n,p for any m and

satisfies |F |�0,n,p ≤ cm,n,p‖F‖�m,n,p. Let L0,n,p be the completion of D∞ by the

norm | |�0,n,p. We set L∞ = ⋂n,p L0,n,p. Then, D∞ ⊂ L∞.

A Wiener–Poisson functional F = (F 1, . . . , F d) is said to belong to (D∞)d , ifeach component Fj belongs to D∞. Sobolev norms ‖F‖m,n,p;A and ‖F‖�m,n,p aredefined as sums of the corresponding norms for components Fj , j = 1, . . . , d .Spaces (D∞

T)d, (D∞

U)d and their norms are defined similarly. Lemma 5.6.2 is

extended in the following manner.


Lemma 5.10.2 For any m, n ∈ N and p ≥ 2, there exists a positive constant cm,n,p

such that

‖(ei(v,DF ) − 1)‖m,n,p;A(ρ) (5.129)

≤ cm,n,p|v|ϕ(ρ) 12 ‖DF‖�m,n,(n+1)p

(1 + |v|ρϕ(ρ) 1

2 ‖DF‖�m,n,(n+1)p

)n

holds for all v ∈ Rd ,0 < ρ < ρ0 and F ∈ (D∞)d .

The proof will be omitted, since it can be done similarly to the proof ofLemma 5.6.2.

We shall study estimations of operators δ and δ. By the isometric properties ofoperators δ and δ, these satisfy

E[δ(X)2] 12 ≤ ‖X‖�1,0,2, E[δ(Z)2]1/2 ≤ ‖Z‖�0,1,2

for X ∈ ST and Z ∈ D∞U0

. Therefore the operator δ is extended to a continuous

linear operator from D1,0,2T

to D0,0,2. Further, δ is extended to a continuous linear

operator from D0,1,2U

to D0,0,2.

Proposition 5.10.2 Let {Ft } be the filtration generated by the Wiener process Wt

and the Poisson random measure N(dt dz). Suppose X ∈ D1,0,2T

is predictablewith respect to the filtration. Then the Skorohod integral δ(X) coincides with theItô integral

∑k

∫TXk

s dWks . Suppose that Zt,z ∈ D0,1,2

Uis predictable with respect

to {Ft } for any z. Then the Skorohod integral δ(Z) coincides with Itô integral∫T

∫R

d′0Zt,zN(dt dz).

The proof for δ can be carried out similarly to the proof of Proposition 5.2.2. Theproof for δ can be carried out similarly to the proof of Proposition 5.4.1. Details areomitted. ��

We will summarize properties of adjoint operators δ and δ.

Theorem 5.10.1

1. The adjoint operator δ is extended to a continuous linear operator from F-spaceD∞T

to F-space D∞. Further, for any m, n ∈ N and p ≥ 2, there exists a positiveconstant c = cm,n,p not depending on domains A(ρ) such that

‖δ(X)‖m,n,p;A(ρ) ≤ cm,n,p‖X‖m+〈p〉−1,n,〈p〉;A(ρ) (5.130)

holds for any 0 < ρ < ρ0 for all X ∈ D∞T

, where 〈p〉 = −2[−p/2].The adjoint operator δ is extended to a continuous linear operator from F-

space D∞U

to F-space D∞. Further, for any m, n ∈ N and p ≥ 2, there exists apositive constant c = cm,n,p not depending on domains A(ρ) such that


‖δ(Zχρ)‖m,n,p;A(ρ) ≤ cm,n,p‖Z‖m,n+〈p〉−1,〈p〉;A(ρ) (5.131)

holds for any 0 < ρ < ρ0 for all Z ∈ D∞U

.2. For δ, the adjoint formula (5.117) holds for any X ∈ D∞

Tand G ∈ D∞. For δ,

the adjoint formula (5.118) holds for any Z ∈ D∞U

and G ∈ D∞.3. For δ, the commutation relation (5.119) holds for any X ∈ D∞

T. For δ, the

commutation relation (5.120) holds for any Z ∈ D∞U

.4. For δ, the isometric property (5.10) holds for any X, Y ∈ D∞

T. For δ, the

isometric property (5.45) holds for any Y,Z ∈ D∞U

.

Proof We will prove assertion 1 for positive even number p only. We write A(ρ) asA. The operator δ on the Wiener–Poisson space satisfies the adjoint formula (5.117).Then the inequality of Theorem 5.2.1 holds for this δ, i.e., there exists c ≥ 1 suchthat

‖δ(X)‖m,0,p;A = ‖δ(X)‖m,p ≤ c‖X‖m+p−1,p = c‖X‖m+p−1,0,p;A (5.132)

for any domain A for all X ∈ S∞T

. We shall compute ‖δ(X)‖m,n,p;A for n ≥ 1. Sincethe norm is defined by (5.121), we consider Ii,j,p;A(δ(X)) for 1 ≤ i ≤ m, 1 ≤ j ≤n. Note that Dj

uδ(X) = δ(DjuX) holds a.e. for X ∈ S∞

T. Then

Ii,j,p;A(δ(X)) =∫

A

E[(∫ |Ditδ(D

juX)|2 dt)

p2 ]

γA(u)pm

jA(du). (5.133)

Taking m = i in (5.132), we have

E[( ∫

|Ditδ(D

juX)|2 dt

) p2]≤ cp

∑

1≤i′≤p

E[( ∫

|Di+i′−1t D

juXt |2 dt dt

) p2],

a.e. u. Then we get

Ii,j,p;A(δ(X)) ≤ cp∑

1≤i′≤p

‖X‖i+i′−1,j,p;A ≤ c′‖X‖pi+p−1,j,p;A.

Sum up the above for i = 0, . . . , m, j = 0, . . . , n. Then we get

‖δ(X)‖pm,n,p;A ≤ C‖X‖p

m+p−1,n,p;A.

Finally the inequality (5.130) is extended to X ∈ D∞T

, since S∞T

is dense in D∞T

.Next consider δ(Z) for Wiener–Poisson functional Z. Theorem 5.6.1 is valid for

Wiener–Poisson functional Z. Since D and δ are commutative, we get (5.131) in thesame way.

Finally, assertions 2–4 are verified directly from Lemma 5.9.2. ��


Differential operator D, difference operator D and their adjoints δ, δ can beextended for complex-valued functionals. Theorem 5.10.1 is valid for complexvalued functionals.

The next theorem is an extension of Lemma 5.3.1 for Wiener functionals.

Theorem 5.10.2 (c.f. Lemma 3.3 in [46]) Suppose that G ∈ D∞ is invertible andthe inverse G−1 belongs to L∞. Then G−1 ∈ D∞. Further, for any m, n, p, thereexists cm,n,p > 0 such that the inequality

‖G−1‖�m,n,p ≤ cm,n,p(1 + ‖G‖�m,n,2qp)q(1 + |G−1|�0,n,2qp)q (5.134)

holds for any G with the above property, where q = (m+ 1)(2n − 1).

Proof We will fix i, j ∈ N and compute DitD

ju(G

−1). For the computation, we usedifference rules for X, Y ∈ D∞ such that Y−1 exists:

Du

(XY

)= DuX · Y −X · DuY

Y · Y ◦ ε+u(5.135)

and for v ∈ A(1)j ,

Du(X ◦ ε+v ) =∑

w⊂v

DuDwX, (5.136)

where w runs over all subsets of v. Then, for the functional G−1, we have theequality

Du1(G−1) = −Du1G

G ·G ◦ ε+u1

. (5.137)

Repeating operations Du2 , . . . , Duj and applying the formula (5.135), we have

Dju(G

−1) = Yu

(G ◦ ε+u )! = Yu · (G−1 ◦ ε+u )!, (5.138)

where (G ◦ ε+u )! := ∏v⊂u G ◦ ε+v and Yu is a linear sum of functionals of the form

Du1G · · · DusjG, (5.139)

where sj = 2j − 1 and ui (i = 1, . . . , sj ) are subsets of u (including empty set)satisfying u1 ∪ · · · ∪ usj ⊃ u. The summation is taken for all such {u1, . . . ,usj }.


Next, with respect to the derivative operator Dt, t ∈ Ti , we have the formula

Dit

(XY

)=

∑(−1)r

Dt0X · Y i−r ·Dt1Y · · ·Dtr Y

Y i+1, (5.140)

where the sum is taken for positive integers r ≤ i, t0, . . . , tr are disjoint subsets of tand �(t0)+ · · · + �(tr ) = i. It can be proved similarly to the proof of Lemma 5.3.1.Then from (5.138) and (5.140), Di

tDjuG

−1 is written as

DitD

juG

−1 = Zijt,u

(G ◦ ε+u !)i+1, (5.141)

where Zijt,u is a linear sum of functionals of the form

(Dv1G · · · DvsjG)i−r ×Dt0(Dv(0)1

G · · · Dv(0)sj

G) · · ·Dtr (Dv(r)1G · · · Dv(r)sj

G),

(5.142)where vj , v(h)

j ⊂ u for h = 0, . . . , r , t1, . . . , tr are disjoint subsets of t ={t1, . . . , ti} and �(t0)+ · · · + �(tr ) = i.

Now, we will give an estimate of the functional (5.141) with respect to the norm‖ ‖�i,j,p. We have

E[( ∫

Ti |DitD

juG

−1|2 dt) p

2]

γ (u)p= E

[( ∫Ti |Zij

t,u|2 dt) p

2 (G ◦ ε+u !)−(i+1)p]

γ (u)p≤ I1 × I2,

where

I1 = E[( ∫ |Zij

t,u|2 dt)p] 1

2

γ (u)p, I2 = E

[(G ◦ ε+u !)−(i+1)2p] 1

2 .

Consider the term I1. The functional Zijt,u is written as sums of terms Z′

t,u

which are represented by q ′ = (i + 1)sj -products of functionals DthDv(h)l

G, i.e.,

Z′t,u = ∏q ′

h,l Dti Dv(h)l

G. Further,∏

h,l γ (v(h)l ) ≤ γ (u) holds, since

⋃h,l v(h)

l ⊃ u.

Consequently, using Hölder’s inequality, we have

I 21 = E

[( ∫ |Z′t,u|2 dt

)p]

γ (u)2p ≤q ′∏

h,l

E[( ∫ |DthDv(h)l

G|2 dt)q ′p] 1

q′

γ (v(h)l )

2pq′

≤ c1(1 + ‖G‖�i,j,2q ′p)

2q ′p. (5.143)

5.11 Nondegenerate Wiener–Poisson Functionals 233

Now, since Zijt,u is a linear sum of Z′

t,u, there is a positive constant c2 such that

I1 ≤ c2(1 + ‖G‖�i,j,2qp)qp. (5.144)

On the other hand,

I2 ≤sj∏

k=0

E[|G−(i+1)2sj p ◦ ε+uk

|]1

2sj

≤ (1 + |G−1|�0,j,2sj p)sj p ≤ (1 + |G−1|�0,j,2qp)qp. (5.145)

The last inequality follows since sj < q and norms | |�0,n,p is nondecreasing withrespect to p. These two estimates (5.144) and (5.145) imply

E[( ∫

Ti |DitD

juG

−1|2 dt) p

2]

γ (u)p≤ c3(1 + ‖G‖�i,j,2qp)q(1 + |G−1|�0,j,2qp)q .

Finally, sum up the above inequalities for i = 1, . . . , m and j = 1, . . . , n andadjoin ‖G‖0,0,p. Since the right-hand sides of above inequalities are nondecreasing

with respect to m, n, we find that the seminorm ‖G−1‖�m,n,p is also bounded by

c4(1 + ‖G‖�m,n,2qp)q(1 + |G−1|�0,n,2qp)q .

Therefore, we get the inequality of the theorem. ��Note In [45], Sobolev norms for Wiener–Poisson functionals are defined. Norms| |k,l,p, ‖ ‖k,l,p and ‖ ‖∼k,l,p correspond to our ‖ ‖l,k,p;U.

An estimation of δ(Zχρ) with respect to above norms is given in [45], Theorem3.2. However, the discussion contains a gap; here it is rectified. The assertion ofTheorem 3.2 in [45] is recovered by our inequalities (5.129) and (5.131).

5.11 Nondegenerate Wiener–Poisson Functionals

We will define nondegenerate Wiener–Poisson functionals and show the existenceof smooth densities of their laws. In the first step, we will give a criterioncalled δ-nondegenerate, which is a combination of Malliavin’s criterion for Wienerfunctionals given in Sect. 5.3 and a criterion for Poisson functionals given inSect. 5.7.

Let 0 < ρ < ρ0 and θ ∈ Sd−1. Let 1 < δ < 2α

. Associated withF = (F 1, . . . , F d) ∈ (D∞)d , we will consider the following Wiener–Poissonfunctionals:


QF (ρ, v) = RF (v)+QF (ρ, v), QFρ (θ) = RF (θ)+QF

ρ (θ), (5.146)

where RF is defined by (5.26) and QF (ρ, v),QFρ (θ) are defined by (5.85), (5.86),

respectively. A functional F ∈ (D∞)d is called δ-nondegenerate (nondegeneratewith respect to QF

ρ ) if QFρ (θ) are invertible and inverses satisfy

sup(ρ,θ)

|QFρ (θ)

−1|�0,n,p < ∞, ∀n, p, (5.147)

where the supremum for (ρ, θ) is taken in the domain (0, ρ0)× Sd−1.We want to get estimations of QF

ρ (θ) and its inverse with respect to Sobolevnorms.

Proposition 5.11.1 Suppose that F ∈ (D∞)d is δ-nondegenerate.Then QF

ρ (θ)−1 ∈ D∞ for any 0 < ρ < ρ0 and θ ∈ Sd−1. Further, for any m, n, p,

there exists a positive constant cm,n,p such that for any δ-nondegenerate functionalF the following holds for all (ρ, θ):

‖QFρ (θ)

−1‖�m,n,p ≤ cm,n,p

(1 + ‖DF‖�

m,n,2n+2p+ ‖DF‖�

m,n,(n+1)2n+2p

)(n+1)2n+2

×(

1 + |QFρ (θ)

−1|�0,n,2n+2p

)2n

. (5.148)

Proof We will apply Theorem 5.10.2 for the functional G = QFρ (θ). We need the

estimate of the norm of QFρ (θ). We have

‖RF (θ)‖�m,n,p ≤ c(‖DF‖�m,n,2p)2, (5.149)

‖QFρ (θ)‖�m,n,p ≤ c′(1 + ‖DF‖�m,n,2(n+1)p)

2(n+1).

Indeed, the first inequality is immediate from the definition of the norms. The secondinequality can be verified similarly to the proof of Proposition 5.7.1, noting thatoperators D and D are commutative. The above two inequalities imply the followinginequality:

‖QFρ (θ)‖�m,n,p ≤ cm,n,p

{(‖DF‖�m,n,2p)

2+(1 + ‖DF‖�m,n,2(n+1)p)2(n+1)}.

Then there is a positive constant cm,n,p such that (5.148) holds. ��The next theorem is an extension of Theorem 5.3.1 for Wiener functionals and

of Theorem 5.7.1 for Poisson functionals.

Theorem 5.11.1 Let γ0 = 1 − αδ2 > 0. Then, for any N ∈ N, there exist

m, n ∈ N, p > 4 and a positive constant C such that for any d-dimensional δ-nondegenerate Wiener–Poisson functional F in (D∞)d ,



]∣∣∣ ≤ C

|v|Nγ0

((‖DF‖�m,n,p + (1 + ‖DF‖�m,n.(n+1)p)

n+1)

× sup(ρ,θ)

‖QFρ (θ)

−1‖�m,n,p

)N‖G‖�m,n,p (5.150)

hold for all |v| ≥ 1 and G ∈ D∞.

Proof We will define a d ′-dimensional stochastic process Xt and a random field Zu

associated with a given δ-nondegenerate smooth Wiener–Poisson functional F :

Xt = XF,vt = −i(v,DtF )

|v|2QF (ρ, v), Zu = ZF,v

u = (e−i(v,DuF ) − 1)1A(ρ)(u)

|v|2ϕ(ρ)QF (ρ, v),

(5.151)where ρ = |v|−δ . Adjoint formulas (5.117) and (5.118) are extended to complexvalued Wiener–Poisson functionals. Then we have

E[ei(v,F ){δ(XG)+ δ(ZG)}

]

=E[ ∫

Dt(ei(v,F ))XtG dt

]+ E

[ ∫Du(e

i(v,F ))ZuGn(du)]

= E[ ∫

ei(v,F )i(v,DtF )XtG dt]+ E

[ ∫ei(v,F )(ei(v,DuF ) − 1)ZuGn(du)

]

=E[ ei(v,F )G

|v|2QF (ρ, v)

∫(v,DtF )2 dt

]

+ E[ ei(v,F )G

|v|2ϕ(ρ)QF (ρ, v)

∫

A(ρ)

|ei(v,DuF ) − 1|2n(du)]

=E[ei(v,F )G

].

Therefore, setting L(G) = δ(XG)+ δ(ZG), we have the iteration formula;

E[ei(v,F )G] = E[ei(v,F )L(G)] = · · · = E[ei(v,F )LN(G)], (5.152)

for N = 1, 2, . . .. Now, note LN(G) = δ(XLN−1(G)) + δ(ZLN−1(G)). Then,using Theorem 5.10.1 and Hölder’s inequality, we have

‖LN(G)‖0,0,2 ≤ ‖δ(XLN−1(G))‖0,0,2 + ‖δ(ZLN−1(G))‖0,0,2

≤ c1(‖XLN−1(G)‖1,0,2 + ‖ZLN−1(G)‖0,1,2)

≤ c2(‖X‖1,0,4 + ‖Z‖0,1,4)‖LN−1(G)‖1,1,4,


where ‖ ‖m,n,p = ‖ ‖m,n,p;A(ρ). Repeating this argument inductively, there existmN, nN, pN and cN such that

‖LN(G)‖0,0,2 ≤ cN(‖X‖mN,nN ,pN+ ‖Z‖mN,nN ,pN

)N‖G‖mN,nN ,pN.

Further, by Hölder’s inequality

‖X‖m,n,p + ‖Z‖m,n,p

≤( 1

|v|2 ‖i(v,DF)‖m,n,2p + 1

|v|2ϕ(ρ)‖χρ(e−i(v,DF ) − 1)‖m,n,2p

)

× ‖QF (ρ, v)−1‖m,n,2p.

We have

‖i(v,DF)‖m,n,2p ≤ c|v|‖DF‖�m,n,2p, (5.153)

‖e−i(v,DF ) − 1‖m,n,2p ≤ c|v|ϕ(ρ) 12 (1 + ‖DF‖�m,n,2(n+1)p)

n+1,

‖QF (ρ, v)−1‖m,n,2p ≤ sup(ρ,θ)

‖QFρ (θ)

−1‖�m,n,2p.

The first inequality is immediate from the definition of the norm. The secondinequality follows from Lemma 5.10.2. The third one is obvious. Then we get

‖X‖m,n,p + ‖Z‖m,n,p (5.154)

≤ c( 1

|v| ‖DF‖�m,n,2p + 1

|v|ϕ(ρ) 12

(1 + ‖DF‖�m,n,2(n+1)p)n+1

)

× sup(ρ,θ)

‖QFρ (θ)

−1‖�m,n,2p,

where ρ = |v|−δ . Since 1|v| ≤ c1

|v|ϕ(|v|−δ)≤ c2|v|γ0 , the last term of the above is

dominated by

c

|v|γ0(‖DF‖�m,n,2p + (1 + ‖DF‖�m,n,2(n+1)p)

n+1) sup(ρ,θ)

‖QFρ (θ)

−1‖�m,n,2p.

Then we get the inequality of the theorem, rewriting 2p as p. ��Corollary 5.11.1 Suppose that F in (D∞)d is δ-nondegenerate. Then for any G ∈D∞, the law of F weighted by G has a rapidly decreasing C∞-density. Further, thedensity function fG(x) is given by the Fourier inversion formula (5.34).

Now we will discuss two other nondegenerate conditions. A Wiener–Poissonfunctional F is called regular if it satisfies Condition (R) introduced in Sect. 5.7.Let F be a regular functional of (D∞)d . We will define its Malliavin covariance atthe center by


KF =∫

T

DtF(DtF )T dt +∫

T

(∂Dt,0F)Γ0(∂Dt,0F)T dt, (5.155)

where ∂Dt,0F is defined by (5.82). For θ ∈ Sd−1, we set KF (θ) = (θ, KF θ). If{KF (θ), θ ∈ Sd−1} are invertible and inverses satisfy

supθ

|KF (θ)−1|�0,n,p < ∞, ∀n, p, (5.156)

then F is called nondegenerate at the center.

Theorem 5.11.2 Let F be a regular functional belonging to (D∞)d .

1. If F is δ-nondegenerate for some 1 < δ < 2/α, then F is nondegenerate at thecenter. Further, it holds for any n, p that

supθ

|KF (θ)−1|�0,n,p ≤ sup(ρ,θ)

|QFρ (θ)

−1|�0,n,p. (5.157)

2. If F is nondegenerate at the center, then F is δ-nondegenerate for any 1 < δ <

2/α. Further, for any n, p there is a positive constant c such that the inequality

sup(ρ,θ)

|QFρ (θ)

−1|�0,n,p ≤ cCδ′n,p(F ) sup

θ

|KF (θ)−1|�0,n,2p, (5.158)

holds for any F ∈ (D∞)d , which is regular and nondegenerate at the center,where Cδ′

n,p(F ) is given by (5.99) replacing KF (θ) by KF (θ).

Proof The first assertion and the inequality (5.157) can be verified in the same wayas in the first assertion of Theorem 5.8.1. We will prove the second assertion by amethod similar to the proof of Theorem 5.8.1. We set SF

ρ = RF + SFρ and KF

ρ =RF +KF

ρ , where SFρ and KF

ρ are defined by (5.100) and (5.101), respectively. Thenwe can verify the followings:

1. Suppose that SFρ (θ) are invertible a.s. for any ρ, θ . Then QF

ρ (θ) are alsoinvertible a.s. for any ρ, θ . Further, there is a positive constant c1 such that theinequality

|QFρ (θ)

−1|�0,n,p ≤ c1|SFρ (θ)−1|�0,n,p (5.159)

holds for any ρ, θ and n, p (Lemma 5.8.1).2. Suppose that KF (θ) are invertible a.s. for any θ ∈ Sd−1. Then KF

ρ (θ) are alsoinvertible a.s. for any (ρ, θ). Further, there are two positive constants c2, c3 suchthat for any (ρ, θ), the inequalities

c2|KF (θ)−1|�0,n,p ≤ |KFρ (θ)−1|�0,n,p ≤ c3|KF (θ)−1|�0,n,p (5.160)

hold for any n, p (Lemma 5.8.2).


3. If F ∈ (D∞)d is regular and nondegenerate at the center, functionals {SFρ (θ)} are

invertible. Further, it holds for any p > 1 that

E[(SF

ρ (θ) ◦ ε+u )−p] ≤ c5B

δ′p (F )E

[(KF

ρ (θ) ◦ ε+u )−2p] 12 , ∀ρ, θ,u, (5.161)

where Bδ′p (F ) is given by (5.108). Indeed, since the equality

SFρ (θ)− KF

ρ (θ) = SFρ (θ)−KF

ρ (θ)

holds, discussions of Lemma 5.8.3 can be applied to the present case and we get

P(!F (θ) ◦ ε+u < ε) ≤ c4Aδ′p (F )εp, ∀0 < ε < 1, θ ∈ Sd−1, u ∈ U

k

(5.162)where Aδ′

p (F ) is given by (5.105) (replacing KF by KF ). Then we get (5.161)similarly to the proof of Lemma 5.8.4.

Now we get from (5.161),

|SFρ (θ)−1|�0,n,p ≤ c6C

δ′n,p(F )|KF

ρ (θ)−1|�0,n,2p. (5.163)

Combining this with (5.160), the inequality of the theorem follows. ��Let us consider another family of Malliavin covariances conditioned to A(ρ).

RFρ =

∫

T

DtF(DtF )T dt + 1

ϕ(ρ)

∫

A(ρ)

DFu(DuF )T n(du). (5.164)

Suppose that RFρ (θ) are invertible and inverses satisfy

sup(ρ,θ)

|RFρ (θ)−1|�0,n,p < ∞, ∀n, p. (5.165)

Then F is said to be nondegenerate. The following can be verified similarly to thecase of Poisson functionals discussed in Sect. 5.8, Theorem 5.8.2.

Theorem 5.11.3 Let F be a regular functional belonging to (D∞)d .

1. If F is nondegenerate, it is nondegenerate at the center. Further, we have for anyn, p,

supθ

|KF (θ)−1|�0,n,p ≤ sup(ρ,θ)

|RFρ (θ)−1|�0,n,p. (5.166)

5.12 Compositions with Generalized Functions 239

2. If F is nondegenerate at the center, it is nondegenerate. Further, for any n, p

there is a positive constant c such that

sup(ρ,θ)

|RFρ (θ)−1|�0,n,p ≤ cC0

n,p(F ) supθ

|KF (θ)−1|�0,n,2p, (5.167)

for all nondegenerate F , where C0n,p(F ) is given by (5.99) with δ′ = 0 replacing

KF (θ) by KF (θ).

From Theorems 5.11.2 and 5.11.3, we have the following,

Theorem 5.11.4 Let F be a regular functional belonging to (D∞)d : The followingstatements are equivalent.

(i) F is δ-nondegenerate for some 1 < δ < 2/α.(ii) F is δ-nondegenerate for any 1 < δ < 2/α.

(iii) F is nondegenerate at the center.(iv) F is nondegenerate.

Remark We will discuss briefly the density problem in the case where the Lévymeasure ν may not satisfy the order condition. A typical case is that the Lévymeasure is of finite mass. For a Wiener–Poisson functional F ∈ (D∞)d , we considerthe Malliavin covariance RF = ∫

TDtF · DFT

t dt . The functional F is calledstrongly nondegenerate if supθ E[RF (θ)−p] < ∞ holds for any p ≥ 2. Then wecan show that for any N ∈ N, there exist m ∈ N, p ≥ 2 and c > 0 such that theinequality

∣∣∣E[ei(v,F )G]∣∣∣ ≤ c

|v|N(‖DF‖m,0,p sup

θ

‖RF (θ)−1‖m,0,p

)N‖G‖m,0,p (5.168)

holds for all |v| ≥ 1 for any strongly nondegenerate Wiener–Poisson functionalF of (D∞)d and G ∈ D∞. The proof can be carried out similarly to the proof ofTheorem 5.3.1, replacing norms ‖ ‖m,p by ‖ ‖m,0,p. Consequently, if F is a stronglynondegenerate Wiener–Poisson functional in the above sense, its law weighted byG has a rapidly decreasing C∞-density.

5.12 Compositions with Generalized Functions

Let F be a smooth Wiener–Poisson functional of (D∞)d and let f (x) be a smoothfunction on R

d . Then the composite f (F ) is a smooth Wiener–Poisson functional.In this section, we will define the composite of a generalized function f and anondegenerate smooth Wiener–Poisson functional F . The composite f (F ) will nolonger be a smooth Wiener–Poisson functional. It should be a generalized Wiener–Poisson functional.


Let us recall the Schwartz distribution or generalized function on a Euclideanspace. Let S = S(Rd) be the space of complex valued rapidly decreasing C∞-functions on R

d . It is called the Schwartz space. We define semi-norms by

|f |n =( ∑

|k|+j≤n

∫ ((1 + |x|2)j |∂kf (x)|

)2dx

) 12. (5.169)

Denote the completion of S with respect to the norm | · |n by Sn. For a negativeinteger −n, we define the norm by

|f |−n = supg∈Sn,|g|n≤1

|(f, g)|, (5.170)

where (f, g) = ∫f (x)g(x) dx. The completion of S by the norm | · |−n is denoted

by S−n. We set S∞ = ⋂n>0 Sn and S−∞ = ⋃

n>0 S−n. Then it holds that S∞ = Sand S−∞ = S ′ is the space of tempered distributions.

Let f (x) be an integrable function on Rd . We define the Fourier transform and

the inverse Fourier transform of f by

Ff (v) = f (v) =( 1

2π

) d2∫

Rd

e−i(v,x)f (x) dx,

Ff (v) = f (v) =( 1

2π

) d2∫

Rd

ei(v,x)f (x) dx,

respectively.

Proposition 5.12.1 The Fourier transform F;S → S is a linear, one to one andonto map. Further, it satisfies the following properties:

1. It holds that FFf = FFf = f for any f ∈ S .2. (Parseval’s equality) We have

∫f g dx = ∫ FfFg dv for any f, g ∈ S .

3. (Fourier inversion formula) Let i and j be multi-indexes of nonnegative integers.Then we have

(ix)i∂ jf (x) = F(∂ i(vjFf ))(x) (5.171)

for any f ∈ S .

Proof The linear property of the Fourier transform is obvious from the definition ofthe transform. Then it is sufficient to prove (1)–(3) for a real function f . If f is adensity function of a signed measure μ, we have the inversion formula FFf = f inview of (1.5). Next, take the complex conjugate for both sides. Then we have f =FFf = FFf . Therefore the first assertion follows. Further, equalities FFf =FFf = f show that the map F : S → S is one to one and onto.


By the Fubini theorem, we have

∫f (x)g(x) dx =

( 1

2π

) d2∫

f (x)( ∫

ei(v,x)g(v) dv)dx

=( 1

2π

) d2∫ ( ∫

e−i(v.x)f (x) dx)g(v) dv =

∫f (v)g(v) dv.

Finally equality (5.171) follows from (1.7). ��Next, we will define generalized Wiener–Poisson functionals. Let ‖ ‖�m,n,p be

Sobolev norms on the Wiener–Poisson space defined in Sect. 5.10. We shall define

their dual norms. For F ∈ Lp

p−1 and G ∈ Dm,n,p, we set 〈X, Y 〉 = E[XY ]. Thedual norm of ‖ ‖�m,n,p is defined by

‖F‖∗m,n,p := supG∈Dm,n,p,‖G‖�m,n,p≤1

|〈F,G〉|.

We denote by Dm,n,p∗ the completion of Lp

p−1 by the norm ‖ ‖∗m,n,p and we set

D∞∗ = ⋃m,n∈N,p>1 Dm,n,p∗ . We have D∞ ⊂ L∞− ⊂ D∞∗ . Elements of D∞∗

are called generalized Wiener–Poisson functionals. The bilinear form 〈F,G〉 isextended naturally to a generalized functional F and a smooth functional G.

Let f be a tempered distribution and let F be a nondegenerate element of (D∞)d .We will define the composition of f and F , following the idea of Watanabe [117]and Hayashi–Ishikawa [37]. The definition is somewhat different from theirs. Weprepare a lemma.

Lemma 5.12.1 Let F be a δ-nondegenerate Wiener–Poisson functional belongingto (D∞)d . Then for any N ∈ N there exist m, n ∈ N, p > 2 and a positive constantCN such that

‖f (F )‖∗m,n,p ≤ CN |f |−N (5.172)

holds for all f ∈ S .

Proof For the proof of the lemma, we will apply the Fourier transform. For agiven G ∈ D∞, consider the (signed) measure μG(A) = E[1A(F )G]. Its Fourier

transform is equal to (2π)− d2 ψG(v), where ψG(v) is the characteristic function of

the measure μG. It is rapidly decreasing with respect to v. Therefore the measureμG has a rapidly decreasing C∞-density function, which we denote by fG(y).

Let f ∈ S . Then using Parseval equality, we have

E[f (F )G] =∫

Rd

f (y)fG(y) dy =( 1

2π

) d2∫

Rd

f (v)ψG(v) dv. (5.173)


We shall compute the semi-norm |ψG|N . For a given N ∈ N, take N ′ ∈ N satisfyingγ0N

′ > d + 1 + N , where γ0 = 1 − αδ/2. Then by Theorem 5.11.1, there existm, n, p and positive constants c1 and c2 such that for any |k| ≤ N , the inequality

|∂kv ψG(v)| = |E[ei(v,F )F kG]| ≤ c1|v|−γ0N

′ ‖G‖�m,n,p

≤ c2(1 + |v|2)−γ0N ′2 ‖G‖�m,n,p

holds if |v| ≥ 2. Then for any j + |k| ≤ N , there is a positive constant cN such that

( ∫(1 + |v|2)j ||∂k

v ψG(v)|2 dv) 1

2 ≤ cN

( ∫(1 + |v|2)− d+1

2 dv) 1

2 ‖G‖�m,n,p

≤ c′N‖G‖�m,n,p.

Therefore we have |ψG|N ≤ c′N‖G‖�m,n,p.We will prove (5.172). It holds that

‖f (F )‖∗m,n,p = supG∈D∞,‖G‖�m,n,p≤1

|E[f (F )G]| (5.174)

≤( 1

2π

) d2

supG;‖G‖�m,n,p≤1

∣∣∣∫

Rd

ψG(v)f (v) dv

∣∣∣

≤( 1

2π

) d2

supG;‖G‖�m,n,p≤1

|ψG|N |f |−N.

Further,

|f |−N = sup|g|N≤1

|〈f, g〉| = sup|g|N≤1

|〈f , g〉| = sup|g|N≤1

|〈f, g〉| = |f |−N.

Here, we used the fact that F;S → S is one to one and onto, and the Percevalequality. Therefore we get the inequality (5.172). ��

By the above lemma, the composition f (F ) can be extended to any f ∈ S−n.Since S ′ = ⋃

n>0 S−n, it is extended to any tempered distribution f . We setconventionally

E[f (F ) ·G] := 〈f (F ),G〉. (5.175)

We summarize the composition as a theorem.


Theorem 5.12.1 Let F be a regular and nondegenerate element of (D∞)d . Then forany tempered distribution f , the composition f (F ) can be defined as an element ofD∞∗ . Further, for any N ∈ N, there exist m, n ∈ N, p > 2 and a positive constantCN such that

‖f (F )‖∗m,n,p ≤ CN |f |−N, ∀f ∈ S−N. (5.176)

Further, it satisfies

E[f (F ) ·G] =( 1

2π

) d2(Ff,ψG) (5.177)

for any f ∈ S−∞ and G ∈ D∞, where Ff is the Fourier transform of f .

So far we considered the composite of a tempered distribution Φ and anondegenerate Wiener–Poisson functional F . However, if F is a Wiener functional,the statement of Theorem 5.12.1 should be changed slightly. Let Dm,p∗ be thedual of the space of Wiener functionals Dm,p and let ‖ ‖∗m,p be its norm. LetD∞∗ = ⋃

m,p Dm,p∗ . Elements of D∞∗ is called generalized Wiener functionals. If

f is a tempered distribution on Rd and F is a d-dimensional nondegenerate Wiener

functional, the composition f (F ) is well defined as an element of D∞∗ . Further, theassertion of Theorem 5.12.1 is valid if we replace the inequality (5.176) by

‖f (F )‖∗m,p ≤ CN |f |−N.

We are particularly interested in the case where f is the delta function δx of thepoint x ∈ R

d .

Corollary 5.12.1 Let F be a d-dimensional smooth nondegenerate Wiener func-tional or a regular nondegenerate Wiener–Poisson functional of (D∞)d . Let G bea smooth Wiener functional or smooth Wiener–Poisson functional, respectively. LetψG(v) be the characteristic function of the law of F weighted by G. Then the densityfunction fG(x) of the law of F weighted by G is represented by

fG(x) = E[δx(F ) ·G] =( 1

2π

)d∫

Rd

e−i(v,x)ψG(v) dv, (5.178)

where δx is the delta function at the point x.

Finally, we remark that if F is nondegenerate, the conditional law P(G|F = x)

is well defined for all x without ambiguity of measure 0. Indeed, it is given by

P(G|F = x) = E[δx(F ) ·G]E[δx(F )] , (5.179)

provided that E[δx(F )] �= 0.


Note The composite of Schwartz distribution and nondegenerate Wiener functionalwas first discussed by Watanabe [116, 117], using Sobolev type norm ‖ ‖′m,p definedby Orenstein–Uhlenbeck generator (Note’ at the end of Sect. 5.2). The composite ofSchwartz distribution and nondegenerate Wiener–Poisson functional is studied inHayashi-Ishikawa [37] slightly different context.

Chapter 6Smooth Densities and Heat Kernels

Abstract We discuss the existence of the smooth density of ‘nondegenerate’diffusions and jump-diffusions determined by SDEs. We will use the Malliavincalculus studied in the previous chapter. In Sects. 6.1, 6.2, and 6.3, we considerdiffusion processes. It will be shown that any solution of a continuous SDE definedin Chap. 3 is infinitely H -differentiable. Further, if its generator A(t) is elliptic (orhypo-elliptic), its transition probability Ps,t (x, E) has a density ps,t (x, y), which isa rapidly decreasing C∞-function of x and y. We will show further that the weightedtransition function has also a rapidly decreasing C∞-function of x and y, and it is theheat kernel of the backward heat equation associated with the operator Ac(t), whichwere defined in Chap. 4. For the proof of these facts, we will apply the Malliavincalculus on the Wiener space discussed in Sects. 5.1, 5.2, and 5.3.

In Sects. 6.4, 6.5, and 6.6, we will study jump-diffusions. We will show that ifthe generator of the jump-diffusion is ‘pseudo-elliptic’, then its weighted transitionfunction has a smooth density. For the proof, we will apply the Malliavin calculuson the Wiener-Poisson space discussed in Sects. 5.9, 5.10, and 5.11.

In Sects. 6.7 and 6.8, we discuss short-time estimates of heat kernels, applying theMalliavin calculus again. These facts will be applied to two problems. In Sect. 6.9,we consider the solution of SDEs with big jumps, for which the Malliavin calculuscannot be applied. Instead, we take a method of perturbation and show that theperturbation preserves the smooth density. In Sect. 6.10, we show the existence ofthe smooth density of the laws of the killed elliptic diffusion or the killed pseudo-elliptic jump-diffusions. The density should be the heat kernel of the backward heatequation on the domain with the Dirichlet boundary condition.


245


https://doi.org/10.1007/978-981-13-3801-4_6

246 6 Smooth Densities and Heat Kernels

6.1 H -Derivatives of Solutions of Continuous SDE

Let us consider a continuous symmetric SDE on Rd defined on the Wiener space:

Xt = X0 +d ′∑

k=0

∫ t

t0

Vk(Xr, r) ◦ dWkr . (6.1)

Here W 0r = r and V0(Xr, r) ◦ dW 0

r = V0(Xr, r) dr . Further, (W 1t , . . . ,W

d ′t ), t ∈

[0,∞) is a d ′-dimensional Wiener process and ◦dWk(t), k = 1, . . . , d ′ denotesymmetric integrals. Coefficients Vk(x, t) = (V 1

k (x, t), . . . , Vdk (x, t)) are C

∞,1b -

functions on Rd×[0,∞). The solution defines a diffusion process with the generator

A(t) defined by (4.4). The operator is rewritten as (4.20). If the matrix A(x, t) :=(αij (x, t)) = V (x, t)V (x, t)T is positive definite for any x, t , the operator A(t)

is called elliptic. Further, if the matrix A(x, t) is uniformly positive definite, theoperator A(t) is called uniformly elliptic. The associated SDE (6.1) will be calledan elliptic SDE and a uniformly elliptic SDE, respectively.

Let {Φs,t } be the stochastic flow generated by the above equation. For a whilewe take s, t from the time interval T = [0, T ]. Then we can note that for any s < t

and x ∈ Rd , Φs,t (x) is a Wiener functional discussed in Sect. 5.1. It is an element

of (L∞−)d . We want to show (in Theorem 6.1.1) that Φs,t (x) is a d-dimensionalWiener functional belonging to the Sobolev space (D∞)d .

Let H be the Cameron–Martin space defined in Sect. 5.1. For a given h =(h1(r), . . . , hd ′(r)) ∈ H , consider an SDE with one-dimensional parameter λ inΛ = [−1, 1] on the time interval T:

Xλht = X0 +

d ′∑

k=1

∫ t

t0

Vk(Xλhr , r)(◦dWk

r +λhk(r) dr)+∫ t

t0

V0(Xλhr , r) dr. (6.2)

Let Xt be the solution of (6.1) starting from X0 at time t0 ∈ T and let Xλht be the

solution of (6.2). Then it holds that

Xλht (w) = Xt(w + λh), a.s. (6.3)

for any λ, where h(t) = ∫ t

0 h(r) dr . Indeed, we will approximate the solutions Xt

and Xλht as follows. Let N ∈ N. Let XN

t and Xλh,Nt be solutions of following

equations, respectively:

XNt = X0 +

d ′∑

k=1

∫ t

t0

Vk(XNφN(r), r) ◦ dWk

r +∫ t

t0

V0(XNφN(r), r) dr,

Xλh,Nt = X0 +

d ′∑

k=1

∫ t

t0

Vk(Xλh,NφN (r), r)(◦dWk

r + λhk(r)) dr

+∫ t

t0

V0(Xλh,NφN (r), r) dr,

6.1 H -Derivatives of Solutions of Continuous SDE 247

where φN(r) = m2N

, if m2N

≤ r < m+12N

. Then we have Xλh,Nt (w) = XN

t (w+ λh)

a.s. Let N tend to ∞. Then XNt converges to Xt in probability and X

λh,Nt converges

to Xλht in probability for any λ. Therefore we have Xλh

t (w) = Xt(w + λh) a.s.Let Xλh,x,s

t be the solution of the equation starting from x at time s. Then thefamily of the solutions {Xλh,x,s

t } has a modification which is infinitely differentiablewith respect to x (Theorem 3.4.2 in Chap. 3). We denote it by Φλh

s,t (x). It belongs to(L∞−)d for any s, t, x, λ, h. If λ = 0, it coincides with the stochastic flow Φs,t (x)

a.s.

Lemma 6.1.1 For a given h ∈ H and s, t (s < t), the family of solutions{Φλh

s,t (x), x ∈ Rd , λ ∈ Λ} has a modification which is continuously differen-

tiable in λ with respect to the metric of the F -space (L∞−)d . Further, we havesup|λ|≤1,x∈Rd E[| d

dλΦλh

s,t (x)|p] < ∞ for any p ≥ 2, s < t and h ∈ H .

Proof Using the Itô integral, Φλhs,t (x) satisfies

Φλhs,t (x) = x +

∫ t

s

α(Φλhs,r (x), r) dWr +

∫ t

s

βλ(Φλs,r (x), r) dr,

where α(x, r) = (V1(x, r), . . . , Vd ′(x, r)) and

βλ(x, r) = V0(x, r)+ 1

2

d ′∑

k=1

Vk(r)(Vk(r))(x)+ λ(

d ′∑

k=1

hk(r)Vk(x, r)).

It satisfies Conditions 1–3 of the master equation with parameter λ, x in Sect. 3.3.Indeed, αλ(x, r) := α(x, r) does not depend on λ and is Lipschitz continuous in x.βλ(x, r) is uniformly Lipschitz continuous with respect to x. With respect to λ, itsatisfies

∫

T

supx

|βλ(x, r)− βλ′(x, r)| dr ≤ c|λ− λ′|( ∫

T

|h(r)| dr)

and∫T|h(r)| dr < ∞. Therefore coefficients of the equation satisfy Conditions 1–

3 of the master equation. Further, βλ(x, r) is differentiable with respect to x andλ, and its derivatives satisfy Conditions 1–3 of the master equation. Then, in viewof Theorem 3.3.2, solutions {Φλh

s,t (x)} are included in the space (L∞−)d and arecontinuously differentiable with respect to λ in the space (L∞−)d . Further, we havesup|λ|<1,x∈Rd E[| d

dλΦλh

s,t (x)|p] < ∞ for any p ≥ 2. ��Lemma 6.1.2 Set d

dλΦλh

s,t (x)∣∣λ=0 = Ξh

s,t (x). Then it satisfies

Ξhs,t =

d ′∑

k=0

∫ t

s

∇Vk(Φs,r , r)Ξhs,r ◦ dWk

r +∫ t

s

( d ′∑

k=1

Vk(Φs,r , r)hk(r)

)dr (6.4)


for any h ∈ H , where ∇Vk(x, r) is the Jacobian matrix of the function Vk(x, r)

= (V 1k (x, r), . . . , V

dk (x, r)) given by (∂xj V

ik (x, r))i,j=1,...,d . Further, Ξh

s,t (x) isrepresented by

Ξhs,t (x) =

∫ t

s

∇Φs,t (x)∇Φs,r (x)−1

( d ′∑

k=1

Vk(Φs,r (x), r)hk(r)

)dr, (6.5)

where ∇Φs,t (x) is the Jacobian matrix of Φs,t (x) = (Φ1s,t (x), . . . , Φ

ds,t (x)).

Proof Φλhs,t (x) satisfies

Φλhs,t (x) = x+

d ′∑

k=0

∫ t

s

Vk(Φλhs,r (x), r) ◦ dWk

r + λ

∫ t

s

( d ′∑

k=1

Vk(Φλhs,r (x), r)h

k(r))dr.

Differentiate each term of the above with respect to λ and then set λ = 0. Then weget the equality (6.4) (See Theorem 3.3.2 and Proposition 2.4.3). On the other hand,the Jacobian matrix ∇Φs,t (x) satisfies the linear homogeneous equation

∇Φs,t (x) = I +d ′∑

k=0

∫ t

s

∇Vk(Φs,r (x), r)∇Φs,r (x) ◦ dWkr (6.6)

(see (3.56)). Now, let us compare the two equations (6.4) and (6.6). Then the solutionof the linear inhomogeneous equation (6.4) is written as (6.5), making use of thesolution ∇Φs,t (x) of the linear homogeneous equation (6.6). In fact, taking thedifferential of (6.5) with respect to t (Itô’s formula), we can prove directly thatΞs,t (x) of (6.5) satisfies SDE (6.4). ��Proposition 6.1.1 For any s < t and x, Φs,t (x) is H -differentiable. The derivative

DrΦs,t (x) = (D(1)r Φs,t (x), . . . , D

(d ′)r Φs,t (x)) is given a.e. dr dP by

D(k)r Φs,t (x) =

{∇Φs,t (x)∇Φs,r (x)−1Vk(Φs,r (x), r), if s < r < t,

0, if r < s, or r > t.

(6.7)It satisfies

sups<t,x

1√t − s

E[( ∫

T

|DrΦs,t (x)|2 dr) p

2] 1

p< ∞ (6.8)

for any p ≥ 2, where the supremum is taken for 0 ≤ s < t ≤ T , x ∈ Rd .

Proof Denote the right-hand side of (6.7) by F ′s,t,x . Its H -norm |F ′

s,t,x |H satisfiesE[|F ′

s,t,x |pH ] < ∞ for any p ≥ 2. Further, it holds that 〈F ′s,t,x, h〉H = Ξh

s,t (x) forany h ∈ H . Therefore we have for any p ≥ 2

6.1 H -Derivatives of Solutions of Continuous SDE 249

E[∣∣∣

Φs,t (x,w + λh)−Φs,t (x,w)

λ− 〈F ′

s,t,x(w), h〉∣∣∣p] → 0, as λ → 0,

by Theorem 3.3.2, 1. This shows that Φs,t (x) is H -differentiable and DΦs,t (x) =F ′s,t,x .

Since E[|∇Φs,t (x)|2p] and E[|∇Φs,r (x)−1|2p] together with Vk(Φs,r (x)) are all

bounded with respect to s, t, r, x, there is a positive constant c such that

E[( ∫

T

|DrΦs,t (x)|2 dr) p

2]≤ c(t − s)

p2 . (6.9)

Therefore we get (6.8). ��Since the flow {Φs,t (x)} satisfies Φs,t (x) = Φr,t (Φs,r (x)), we have ∇Φs,t (x) =

∇Φr,t (Φs,r (x))∇Φs,r (x). Then we get ∇Φs,t (x)∇Φs,r (x)−1 = ∇Φr,t (Φs,r (x)).

This expression is often used for (6.7).

Theorem 6.1.1 For any multi-index i and 0 ≤ s < t ≤ T , ∂ iΦs,t (x) is infinitelyH -differentiable. Further, for any m,p, ‖∂ iΦs,t (x)‖m,p is bounded with respect to0 ≤ s < t ≤ T , x ∈ R

d if |i| ≥ 1, and we have

sup0≤s<t≤T ,x

1√t − s

‖D∂ iΦs,t (x)‖m,p < ∞ (6.10)

for any |i| ≥ 0. Here ‖ ‖m,p are Sobolev norms for Wiener functionals defined inSect. 5.2.

Proof Since the pair (Φs,t (x),∇Φs,t (x)) is a solution of SDE (6.1) and (6.6),∇Φs,t (x) is H -differentiable by Proposition 6.1.1. This shows that ∂ iΦs,t (x) is H -differentiable if |i| = 1. Consider the triple (Φs,t ,∇Φs,t ,∇2Φs,t ), where ∇2Φs,t

is the Jacobian matrix of ∇Φs,t . It is again a solution of an SDE, and we find that∇2Φs,t (x) is also H -differentiable. Repeating this argument, we find that ∂ iΦs,t isH -differentiable for any i.

Φs,t (x) is twice H -differentiable. Indeed, since ∇Φs,t ,∇Φ−1s,r and Vk(Φs,r , r)

are H -differentiable, their product DrΦs,t ≡ ∇Φs,t∇Φ−1s,r Vk(Φs,r , r) is also H -

differentiable. This means that Φs,t is twice H -differentiable. In the same way,∂ iΦs,t (x) is also twice H -differentiable. Repeating this argument we find that itis m-times differentiable for any m ∈ N. We can verify (6.10) similarly to the proofof (6.9). ��Note Using Lemma 6.1.1, we showed that the flow Φs,t (x) belongs to (D∞)d forany 0 ≤ s < t ≤ T , x. In Watanabe [116] and Nualart [88], it is shown that Φs,t (x)

is an element of (D∞)d for any 0 ≤ s < t ≤ T and x ∈ Rd .


6.2 Nondegenerate Diffusions

Let R = RΦs,t (x) be the Malliavin covariance of the stochastic flow Φs,t (x) definedby (5.26). Note that DrΦs,t (x) satisfies (6.7). Since DrΦs,t (x) = 0 holds for0 < r < s and t < r < T , it is written as (omitting x),

RΦs,t (x) = ∇Φs,t

(∫ t

s

∇Φ−1s,r V (Φs,r , r)V (Φs,r , r)

T (∇Φ−1s,r )

T dr)∇ΦT

s,t ,

(6.11)where V (x, r) = (V i

k (x, r)) is a d × d ′-matrix and V (x, r)T is its transpose.The matrix V (x, r)V (x, r)T coincides with the matrix A(x, r) := (αij (x, r))

of coefficients of the second-order part of the differential operator A(r) givenby (4.20). We set R(θ) = (θ, Rθ) for θ ∈ Sd−1.

We saw in Sect. 4.1 that the stochastic process Xx,st := Φs,t (x) with parameter

x, s is a diffusion process with generator A(t). A system of diffusion processes{Xx,s

t = Φs,t (x), (x, s) ∈ T × Rd} is called nondegenerate, if the Malliavin

covariances RΦs,t (x) are invertible and the inverse belongs to L∞− for any s < t, x,and further, these inverses satisfy

sup|x|≤M

supθ∈Sd−1

E[RΦs,t (x)(θ)−p] < ∞ (6.12)

for any s < t,M > 0 and p > 2. Further, if

supx∈Rd

supθ∈Sd−1

E[RΦs,t (x)(θ)−p] < ∞

holds for any s < t and p > 2, the system of diffusion processes is called uniformlynondegenerate.

Proposition 6.2.1 If the operator A(t) of (4.4) is uniformly elliptic, the system ofduffusion processes with the generator A(t) is uniformly nondegenerate. Further,we have

sup0≤s<t≤T ,x

supθ∈Sd−1

(t − s)‖RΦs,t (x)(θ)−1‖m,p < ∞ (6.13)

for any m ∈ N and p ≥ 2.

Proof For fixed s, x, set ϕr = θ∇Φs,t∇Φ−1s,r . Then, RΦs,t (x)(θ) is written as

RΦs,t (x)(θ) =∫ t

s

ϕrA(Φs,r , r)ϕTr dr.

6.2 Nondegenerate Diffusions 251

Let λr be the minimal eigen-value of the matrix A(Φs,r , r). There exists c > 0 suchthat λr ≥ c for any r , since the matrices A(x, t) are uniformly positive definite withrespect to x, t . Then we have

RΦs,t (x)(θ) ≥ (t − s) · 1

t − s

∫ t

s

λr |ϕr |2 dr.

It holds that ϕr �= 0 a.s., since the matrix ∇Φs,t∇Φ−1s,r is nondegenerate for any

x, r, s, t . Therefore, we have

RΦs,t (x)(θ)−1 ≤ 1

(t − s)2

∫ t

s

1

λr |ϕr |2 dr, a.s. (6.14)

Then we get the inequality

E[RΦs,t (x)(θ)−p

] ≤ (c(t − s))−p supr

E[|ϕr |−2p].

We will show that the last term of the above is finite for any p. Let Vr,t (y)

be the inverse matrix of ∇Φr,t (y). Since ∇Φs,t (x)∇Φs,r (x)−1 = ∇Φr,t (Φs,r (x)),

Vr,t (Φs,r (x)) is the inverse of ∇Φs,t (x)∇Φs,r (x)−1. Then |ϕr |−1 ≤ |Vr,t (Φs,r )|

holds. We know by Lemma 3.7.1 that E[|Vr,t (y)|2p] is bounded with respect toy, r, t . Now since Vr,t (y) and Φs,r are independent, E[|Vr,t (Φs,r (x))|2p] is boundedwith respect to x, s, r, t . This proves supx,s,r,t E[|ϕr |−2p] < ∞. Consequently, weget

sups<t,x

supθ

(t − s)pE[RΦs,t (x)(θ)−p] < ∞.

This proves (6.13) for the case m = 0. Since ‖DΦs,t (x)‖m,p is bounded with respectto x, s, t by (6.10), we get (6.13) for any m,p in view of Proposition 5.3.1. ��

We will next consider the case where the operator A(t) is elliptic. Then theminimum eigen-values λr of A(Φs,r , r) may not be uniformly positive. We wantto localize the discussion. We need a lemma.

Lemma 6.2.1 Given x, s, consider the diffusion process Xx,st = Φs,t (x), t in

[s,∞). Let M > 0 and let τ = τ(x, s) be the first time t (> s) at which|Xx,s

t | ≥ M + 1 occurs. For any N > 2, there is cN > 0 such that the inequality

sup|x|≤M

P(τ(x, s) < t) ≤ cN(t − s)N (6.15)

holds for any s < t .


Proof By Chebyschev’s inequality, we have for any p ≥ 2,

sup|x|≤M

P(τ(x, s) < t) ≤ sup|x|≤M

P(

sups<r<t

|Xx,sr − x| ≥ 1

)

≤ sup|x|≤M

E[‖Xx,s − x‖2N

t

]

≤ cN(t − s)N .

The last inequality follows from Theorem 3.4.3. ��Proposition 6.2.2 If the differential operator A(t) of (4.4) is elliptic, the systemof diffusion processes with the generator A(t) is nondegenerate. Further, for anyM > 0 and m,p, we have

sup0≤s<t≤T ,|x|≤M

supθ∈Sd−1

(t − s)‖RΦs,t (x)(θ)−1‖m,p < ∞. (6.16)

Proof For given M > 0, x, s, let τ = τ(x, s) be the first time t at which|Xx,s

t | ≥ M + 1 occurs, where Xx,st = Φs,t (x). Let c is a positive constant such that

(u,A(x, t)u) ≥ c|u|2 holds if t ∈ T, |x| ≤ M + 1. Then the minimal eigen-valueλr of A(Φs,r (x), r) satisfies λr∧τ ≥ c. Instead of the inequality (6.14), we have theinequality

RΦs,t (x)(θ)−1 ≤ 1

(t ∧ τ − s)2

∫ t

s

1

λr∧τ |ϕr∧τ |2 dr ≤ 1

c(t ∧ τ − s)2

∫ t

s

1

|ϕr∧τ |2 dr,

almost surely. Therefore, we get the inequality

(t − s)pE[RΦs,t (x)(θ)−p

]

≤ c−p((t − s)4pE

[(t ∧ τ − s)−4p])

12

supr

E[|ϕr∧τ |−4p] 1

2 . (6.17)

For N > 4p, there is cN > 0 such that sup|x|≤M P(τ < t) ≤ cN(t − s)N holds forτ = τ(x, s) by Lemma 6.2.1. Then we have

(t − s)2pE[(t ∧ τ − s)−4p] 1

2≤((t − s)4p

∫ t

s

(r − s)−4pP (τ ∈ dr)+ P(τ ≥ t)) 1

2

≤ c′((t − s)4p

∫ t

s

(r − s)−4p+N−1 dr + 1) 1

2

for any 0 ≤ s < t < T, |x| ≤ M . The above is bounded with respect to these s, t, x

since we chose N > 4p.

6.3 Density and Fundamental Solution for Nondegenerate Diffusion 253

We can show sup|x|≤M,s,r,t E[|ϕr∧τ |−4p] < ∞ similarly to the proof ofProposition 6.2.1. Therefore (6.17) is bounded for 0 ≤ s < t < T and |x| ≤ M .Then we get (6.16) in view of Proposition 5.3.1. ��

6.3 Density and Fundamental Solution for NondegenerateDiffusion

Let ck(x, t), k = 0, . . . , d ′ be C∞,1b -functions. Let Gs,t (x) be an exponential

functional defined by


{ d ′∑

k=0

∫ t

s


}. (6.18)

We saw in Sect. 4.2 that the pair (Φs,t (x),Gs,t (x)) is a solution of an SDE on (d +1)-dimensional Euclidean space. We will apply Theorem 6.1.1 to the pair process.Then we find that ∂ iGs,t (x) is a smooth Wiener functional and its Sobolev norm‖∂ iGs,t (x)‖m,p is bounded with respect to s < t, x for any i and m,p.

We define the transition function of Φs,t (x) weighted by c = (c0, . . . , cd ′) by theformula

P cs,t (x, E) = E[1E(Φs,t (x))G

cs,t (x)].

Given a differential operator A(t) of (4.4) and the function c, we define an anotherdifferential operator Ac(t) by (4.11). Then P c

s,t (x, E) is the transition function ofthe semigroup generated by the operator Ac(t).

In this section, we will show the existence of the smooth density of theabove transition function, if the system of diffusions with the generator A(t) isnondegenerate. We first define its characteristic function by

ψcs,t,x(v) =

∫

Rd

ei(v,y)P cs,t (x, dy) = E[ei(v,Φs,t (x))Gs,t (x)]. (6.19)

For multi-index i = (i1, . . . , id ) of nonnegative integers i1, . . . , id , we set|i| = i1 + · · · id and ∂ i

x = ∂i1x1 · · · ∂idxd .

Lemma 6.3.1 Suppose that the system of diffusion processes with the generatorA(t) is nondegenerate. Then for any fixed s < t , ψc

s,t,x(v) is a C∞-function of xand is a rapidly decreasing C∞-function of v. Further, for any M > 0 and N ∈ N,there exists a positive constant c such that

∣∣∣∂ jv∂

ixψ

cs,t,x(v)

∣∣∣ ≤ c(1 + |x|)|j|

(1 + |v|)N−|i| (6.20)

holds for all |x| ≤ M and v ∈ Rd .


Furthermore, if the system of diffusions is uniformly nondegenerate, the aboveinequality holds for all x, v ∈ R

d .

Proof We will consider the differentiability of ψcs,t,x(v) by x. Differentiating

ei(v,Φs,t (x))Gs,t (x) with respect to x, we have ∂ ix(e

i(v,Φs,t (x))Gs,t (x)) =ei(v,Φs,t (x))Gi

s,t,x,v , where Gis,t,x,v is a finite sum of terms written as

∂ i0Gs,t (x) · i(v, ∂ i1Φs,t (x)) · · · i(v, ∂ ikΦs,t (x)), (6.21)

where |i0| + · · · + |ik| = |i|. We know ∂ i′Φs,t (x) and ∂ i′Gs,t (x) belong to D∞ andtheir norms ‖∂ i′Φs,t (x)‖m,p and ‖∂ i′Gs,t (x)‖m,p are bounded with respect to s, t, x,in view of Theorem 6.1.1. Then we can change the order of the expectation and thederivative operator and we find that ψc

s,t,x(v) is infinitely differentiable with respectto x and we get

∂ ixψ

cs,t,x(v) = E[ei(v,Φs,t (x))Gi

s,t,x,v]

for any i. Similarly, we can show the differentiability of the above function withrespect to v and we get the formula

∂ jv∂

ixψ

cs,t,x(v) = i|j|E[ei(v,Φs,t (x))Φs,t (x)

jGis,t,x,v]

for any j. Further, for any m,p there is a positive constant c such that the inequality

‖Φs,t (x)jGi

s,t,x,v‖m,p ≤ c|v||i|(1 + |x|)|j|

holds for any s < t and x, v ∈ Rd . Then in view of Theorem 5.3.1, for any N ∈ N

and M > 0, there exist c,m, p such that

|∂ jv∂

ixψ

cs,t,x(v)|

≤ c

|v|N(‖DΦs,t (x)‖m,p sup

θ∈Sd−1

‖RΦs,t (x)(θ)−1‖m,p

)N‖Φs,t (x)jGi

s,t,x,v‖m,p

≤ c′(1 + |x|)|j||v|N−|i|

holds for all |x| ≤ M and |v| ≥ 1. Further, the inequality holds for any x if Φs,t (x)

is uniformly nondegenerate. ��Theorem 6.3.1 Assume that the system of diffusion processes with the generatorA(t) is nondegenerate. Then the transition function P c

s,t (x, E) generated by Ac(t)

has a rapidly decreasing C∞-density pcs,t (x, y) for any s < t, x. Further, it satisfies

the following properties:


1. For any s < t , pcs,t (x, y) is a C∞-function of x, y and satisfies

∂ ix∂

jyp

cs,t (x, y) = (−i)|j|

( 1

2π

)d∫

Rd

e−i(v,y)vj∂ ixψ

cs,t,x(v) dv (6.22)

for any ∂ ix and ∂

jy , where ψc

s,t,x(v) is defined by (6.19).2. For any y, t , pc

s,t (x, y) is a C∞,1-function of x ∈ Rd and s (< t). It satisfies

∂

∂spcs,t (x, y) = −Ac(s)xp

cs,t (x, y). (6.23)

3. Let P c,∗s,t be the dual operator of P c

s,t . Then for any g ∈ C0(Rd), we have

Pc,∗s,t g(y) =

∫

Rd

pcs,t (x, y)g(x) dx, ∀y.

Further, for any x, s, pcs,t (x, y) is a C∞,1-function of y ∈ R

d and t (> s). Itsatisfies

∂

∂tpcs,t (x, y) = Ac(t)∗ypc

s,t (x, y), (6.24)

where Ac(t)∗ is the differential operator defined by (4.45).

Proof In view of Corollary 5.12.1, the transition function has a densitypcs,t (x, y), y ∈ R

d for any s, t, x and it is given by

pcs,t (x, y) = E[δy(Φs,t (x)) ·Gc

s,t (x)] =( 1

2π

)d∫

Rd

e−i(v,y)ψcs,t,x(v) dv.

(6.25)It is a continuous function of s < t, x, y and further, a rapidly decreasingC∞-function of y, since ψc

s,t,x(v) is a rapidly decreasing C∞-function of v, forany s < t, x.

We can change the order of the derivation with respect to x and the integralin equation (6.25), because of Lemma 6.3.1, We find that pc

s,t (x, y) is infinitelydifferentiable with respect to x and satisfies

( 1

2π

)d∫

e−i(v,y)∂ ixψ

cs,t,x(v) dv =

( 1

2π

)d

∂ ix

∫e−i(v,y)ψc

s,t,x(v) dv

= ∂ ixp

cs,t (x, y).

Next, differentiate both sides of the above equation by y. Then we get (6.22).We will prove 2. It holds that P c

s,t f−f = ∫ t

sAc(r)P c

r,t f dr for any C∞-functionf with compact supports. Therefore, the density function pc

s,t (x, y) satisfies


pcs,t (x, y)− δy =

∫ t

s

Ac(r)xpcr,t (x, y) dr.

Then pcs,t (x, y) is continuously differentiable with respect to s and the derivative

satisfies ∂∂spcs,t (x, y) = −Ac(s)xp

cs,t (x, y) for s < t . Consequently, for any t, y,

pcs,t (x, y) is a C∞,1-function of x and s.

We will next prove 3. We have Pc,∗s,t g(y) = ∫

pcs,t (x, y)g(x) dx a.e. y for any

g ∈ C0(Rd), since P

c,∗s,t is the dual of P c

s,t . The equality should holds for all y, sinceboth terms are continuous with respect to y. Further, the dual semigroup {P c,∗

s,t }satisfies P

c,∗s,t g = g + ∫ t

sAc(r)∗P c,∗

s,r g dr for any smooth g. Therefore pcs,t (x, y)

satisfies

pcs,t (x, y)− δx =

∫ t

s

Ac(r)∗ypcs,r (x, y) dr.

Then both sides are differentiable with respect to t and we get ∂∂tpcs,t (x, y) =

Ac(t)∗ypcs,t (x, y). ��

We shall consider the final value problem for a backward heat equationassociated with the operator Ac(t). Let p(x, s, ; y, t), x, y ∈ R

d , 0 < s < t < ∞be a continuous function of x, y, s, t and C2,1-function of x, s for any y, t . It iscalled the fundamental solution of the backward heat equation (4.18) or simply thebackward heat kernel associated with the operator Ac(s), if it satisfies

∂

∂sp(x, s; y, t) = −Ac(s)xp(x, s; y, t), x, y ∈ R

d , 0 < s < t < ∞, (6.26)

and the function

v(x, s) :=∫

Rd

p(x, s; y, t1)f1(y) dy, x ∈ Rd , 0 < s < t1 (6.27)

is a C2,1-function of (x, s) and is a solution of the final value problem of thebackward heat equation (4.18) for any slowly increasing continuous function f1on R

d .

Theorem 6.3.2 Assume that the system of diffusion processes with the generatorA(t) is nondegenerate. Let pc

s,t (x, y) be the density function of the transitionfunction P c

s,t (x, E) generated by Ac(t). Then p(x, s; y, t) := pcs,t (x, y) is the

fundamental solution of the backward heat equation associated with the operatorAc(t).

Proof The function p(x, s; y, t) ≡ pcs,t (x, y) satisfies (6.26) by the previous

theorem. We show that it is the fundamental solution for the backward heat equation.Since pc

s,t (x, y) is a rapidly decreasing C∞-function of y for any t, x, we can definev(x, s) by (6.27) for any slowly increasing continuous function f1.


We show that v(x, s) is a C∞-function of x and is differentiable with respectto s. For any M > 0 and positive integers n,N , there is a positive constantc such that for any |i| ≤ n, sup|x|≤M

∣∣∂ ixp

cs,t1

(x, y)∣∣ ≤ c

(1+|y|)N holds for any

y, since ∂ ixps,t1(x, y) are rapidly decreasing with respect to y. Now let p′ be a

positive integer such that f1(y)/(1+|y|)p′is a bounded function. Take N satisfying

N ≥ p + (d + 1). Then ∂ ixp

cs,t1

(x, y)f1(y)(1 + |y|)d+1 is bounded with respect to|x| < M and y ∈ R

d . Then, for the function v(x, s) = ∫pcs,t1

(x, y)f1(y) dy, wecan change the order of the derivative ∂ i and the integral by (1 + |v|)−(d+1) dv.It means that v(x, s) is n-times differentiable with respect to x and the equality∂ ixv(x, s) =

∫∂ ixp

cs,t1

(x, y)f1(y) dy holds.

Further, since ∂∂spcs,t1

(x, y) = −Ac(s)xpcs,t1

(x, y), for the function v(x, s), we

can also change the order of the derivative ∂∂s

and the integral. It means that v(x, s)is differentiable with respect to s and ∂

∂sv(x, s) = ∫

∂∂sps,t1(x, y)f1(y) dy holds.

Therefore, v(x, s) is a C∞,1-function and satisfies the backward heat equationassociated with the operator Ac(t).

Next, note the equality v(x, s) = E[f1(Φs,t1(x))Gs,t1(x)]. Given x ∈ Rd ,

the family of random variables {f1(Φs,t1(x))Gs,t1(x); 0 ≤ s < t1} are uniformlyintegrable and converges to f1(x) as s → t1. Then the function v(x, s) converges tof1(x) for any x as t → 0. We have thus shown that v(x, s) of (6.27) is a solution ofthe final value problem for the backward heat equation. ��

If the differential operator Ac(t) is nondegenerate, the associated semigroup{P c

s,t } of linear transformations can be extended to that of linear transformationsfrom the space of tempered distributions to the space of C∞-functions. The factwill be proved in Theorem 6.6.3 for the the semigroup associated with the integro-differential operator Ac,d

J (t).We will next consider the heat equation (4.21) associated with Ac(t). Let

p(x, t; y, s), x, y ∈ Rd , 0 < s < t < ∞ be a continuous function of x, y, s, t

and C2,1-function of x, t for any y, s. It is called the fundamental solution of theheat equation or simply heat kernel associated with Ac(t) if it satisfies the equation

∂

∂tp(x, t; y, s) = Ac(t)xp(x, t; y, s), x, y ∈ R

d , 0 < s < t < ∞, (6.28)

and the function

u(x, t) :=∫

Rd

p(x, t; y, t0)f0(y) dy, x ∈ Rd , t0 < t < ∞ (6.29)

is a solution of the initial value problem of the heat equation (4.21) for any slowlyincreasing continuous function f0 on R

d .In order to construct the fundamental solution for the heat equation associated

with the operator Ac(t), let us consider the backward SDE defined by (4.24), wherecoefficients Vk(x, r), k = 0, . . . , d ′ are C

∞,1b -functions. The solution generates a

backward flow {Φs,t }. It is a backward diffusion with the generator A(t) defined


by (4.4). Define P cs,t (x, E) = E[1E(Φs,t (x))G

cs,t (x)], where Gc

s,t (x) is givenby (4.27). It is a backward transition function of the backward semigroup generatedby the operator Ac(t).

Theorem 6.3.3 Assume that the system of backward diffusion processes with thegenerator A(t) is nondegenerate. Then the backward transition function generatedby Ac(t) has a rapidly decreasing C∞-density pc

s,t (x, y) for any t, s, x. It is a C∞,1b -

function of (x, t) for any y ∈ Rd and t (> s). Further, p(x, t; y, s) := pc

s,t (x, y) isthe fundamental solution of the heat equation associated with the operator Ac(t).

The proof is similar to that of Theorem 6.3.2. It is omitted.If coefficients Vk, k = 0, . . . , d ′ are time homogeneous, we can define the

fundamental solution of the heat equation associated with Ac, using the forwarddiffusion.

The smoothness of the transition density function pcs,t (x, y) with respect to x has

not been studied in the Malliavin calculus. We showed the property by using thesmoothness of the stochastic flow Φs,t (x) with respect to x.

If the operator A(t) is elliptic, both the forward diffusion and the backwarddiffusion generated by A(t) are nondegenerate. In this case, density functionspcs,t (x, y) and pc

s,t (x, y) are rapidly decreasing C∞-functions of x and y. Indeed,since the operator A(t) = 1

2

∑k≥1 Vk(t)

2 − V0(t) is elliptic, the dual transitionfunction P

c,∗s,t (y, E) := ∫

Epcs,t (x, y) dx should have a rapidly decreasing C∞

density by Theorem 6.3.1. This shows that pcs,t (x, y) is a rapidly decreasing C∞-

function of x.

Note We can relax the elliptic condition. For two vector fields V and W , we defineits Lie bracket by

[V,W ]f = VWf −WVf.

Then [V,W ] is again a vector field. Now let V0, V1, . . . , Vd ′ be the vector fieldsdefining the SDE. Set

Σ0 ={Vk(t); k = 1, . . . , d ′}, ΣM ={[Vk(t), V (t)]; k = 0, . . . , d ′, V (t) ∈ ΣM−1}.

If there exists N0 ∈ N such that the family of vector fields⋃N0

M=0 ΣM span Rd

for all x, t , then the operator A(t) is called hypo-elliptic or is said to satisfy theHörmander condition.

There are extensive studies for the existence of the smooth density for time-homogeneous hypo-elliptic operator A(t). It was shown by Malliavin [77],Kusuoka–Stroock [69] that if the operator A(t) satisfies the Hörmander condition,then the diffusion is nondegenerate. The proof is not simple. We refer it toKusuoka–Stroock [69], Norris [87], Nualart [88] and Komatsu–Takeuchi [58].Their discussions can be applied for time dependent hypo-elliptic operator.

6.4 Solutions of SDE on Wiener–Poisson Space 259

We can apply Theorems 6.3.1, 6.3.2, and 6.3.3 to hypo-elliptic diffusion. Notethat the operator A(t) is hypo-elliptic if A(t) is hypo-elliptic. Then its transitionfunction has a density ps,t (x, y), which is a rapidly decreasing C∞-function of x

and y. Further, the heat equation associated with the operator Ac(t) satisfying theHörmander condition has the fundamental solution.

Note The existence of the fundamental solution for a backward heat equation isknown in analysis. Let A(t) be an elliptic differential operator given by (4.20).We assume that coefficients αij , βi are bounded and are Hölder continuous. Thenthere exists a fundamental solution p(x, s; y, t) for the operator A(t), whichis continuous in s, t, x, y and is C2-class with respect to x. Results may befound in I’lin-Karashnikov-Oleinik [42], Friedman [28] and Dynkin [24]. Thefundamental solution p(x, s; y, t) for the operator A(t) is used for constructing adiffusion process with the generator A(t), without using SDEs. Define Ps,tf (x) =∫p(x, s; y, t)f (y) dy. Then {Ps,t } is a conservative semigroup. There exists a

Markov process Xt with transition probability Ps,t (x, E) = ∫Ep(x, s; y, t) dy. We

can show that Xt is a continuous process and has the strong Markov property. SeeDynkin [24], Chap. 5.

In this monograph, we started with SDE and proved the existence of thefundamental solution, using the Malliavin calculus. Our result is slightly strongerthan those, since p(x, s; y, t) is a C∞-function of x and y, although we assumedthat coefficients of the operator A(t) are of C∞

b -class.

6.4 Solutions of SDE on Wiener–Poisson Space

Let (Ω,B(Ω), P ) be the Wiener–Poisson space associated with the Lévy measureν discussed in Sect. 5.9. For the Lévy measure ν we assume that at the center it isnondegenerate and satisfies the order condition of exponent 0 < α < 2 with respectto a given family of star-shaped neighborhoods {A0(ρ)}. Further, we assume that νhas a weak drift.

Let us consider a symmetric SDE with jumps defined on Rd :

Xt = X0 +d ′∑

k=0

∫ t

t0


∫ t

t0

∫

|z|>0+{φr,z(Xr−)−Xr−}N(dr dz).

(6.30)Its diffusion part is the same as equation (6.1). For the jump part, we assume thatg(x, t, z) := φt,z(x)− x satisfies Conditions (J.1) and (J.2) in Sect. 3.2 in Chap. 3.

Let {Φs,t } be the stochastic flow generated by the above SDE. Let T = [0, T ] bethe time parameter of the Wiener–Poisson space. We will show that for any 0 ≤ s <

t ≤ T and x ∈ Rd , Φs,t (x) is a regular Wiener–Poisson functional belonging to the

space (D∞)d defined in Sect. 5.10. Let u = (r, z) ∈ U and ε+u be the transformationon the Poisson space defined in Sect. 5.4. Then the equality ε+u ω(E) = ω


(E)+ δu(E) holds for almost all ω. Therefore, for almost all u = (r, z) with respectto the measure n, Xt := Φs,t (x) ◦ ε+(r,z) satisfies the equation

Xt = x +d ′∑

k=0

∫ t

s

Vk(Xr ′ , r′) ◦ dWk

r ′ +∫ t

s

∫

|z|>0+{φr ′,z′(Xr ′−)−Xr ′−}N(dr ′ dz′)

+ {φr,z(Xr−)−Xr−}1(s,t](r), a.s.

If 0 ≤ r < s or t < r ≤ T , then we have Xt = Φs,t (x). If s ≤ r ≤ t , the solution Xt

is equal to Φr,t ◦ φr,z ◦ Φs,r−(x). Since Φs,r (x) = Φs,r−(x) holds for all x, almostsurely, we have

Φs,t (x) ◦ ε+(r,z) ={Φr,t ◦ φr,z ◦Φs,r (x) if s ≤ r ≤ t,

Φs,t (x), if r < s or r > t.(6.31)

Next, let j ≥ 2 and let u = {(ri, zi), i = 1, . . . , j} ∈ Uj . Let u′ be the subset of u

such that s ≤ ri ≤ t . It is written as {(r1, z1), . . . , (rj ′ , zj ′)}, where s < r1 < · · · <rj ′ < t and j ′ ≤ j . Then we have

Φs,t (x) ◦ ε+u = Φrj ′ ,t ◦ φrj ′ ,zj ′ ◦ · · · ◦Φr1,r2 ◦ φr1,z1 ◦Φs,r1(x), a.s. (6.32)

It is well defined and belongs to (L∞−)d for any s < t, x and u a.s. We will studythe H -differentiability of Dj

uΦs,t (x).

Lemma 6.4.1 For any x ∈ Rd , 0 ≤ s < t ≤ T and u ∈ U

j , DjuΦs,t (x) is H -

differentiable. Further, H -derivative DrDjuΦs,t (x) is given a.e. dr dP by

{D

ju(∇Φr,t (Φs,r (x))Vk(Φs,r (x), r)

), k = 1, . . . , d ′, s < r < t,

0, otherwise.(6.33)

Proof Given h = (h1(r), . . . , hd ′(r)) ∈ H , we set h(t) = ∫ t

0 h(r) dr . We considerΦλh

s,t (w) := Φs,t (x)(w + λh). Then,

Φλhs,t ◦ ε+u

= x +d ′∑

k=0

∫ t

s

Vk(Φλhs,r ◦ ε+u , r) ◦ dWk

r + λ

d ′∑

k=1

∫ t

s

Vk(Φλhs,r ◦ ε+u , r)hk(r) dr

+∫ t

s

∫

|z|>0+g(Φλh

s,r− ◦ ε+u , r, z)N(dr dz)+∑

i;s≤ri≤t

g(Φλhs,ri− ◦ ε+u , ri , zi),


if u = ((r1, z1), . . . , (rj , zj )) and r1 < · · · < rj . Then the solution Φλhs,t ◦ ε+u is

continuously differentiable with respect to λ in the F -space L∞−. See Remark afterTheorem 3.3.2. Let us denote its derivative at λ = 0 by Ξh

s,t . Then similarly toLemma 6.1.2, it satisfies an inhomogeneous linear SDE;

Ξhs,t =

d ′∑

k=0

∫ t

s

∇Vk(Φs,r ◦ ε+u , r)Ξhs,r ◦ dWk

r

+∫ t

s

∫

|z|>0+∇g(Φs,r− ◦ ε+u , r, z)Ξh

s,r−N(dr dz)

+∑

i;s≤ri≤t

∇g(Φs,ri− ◦ ε+u , ri , zi)Ξhs,ri− +

d ′∑

k=1

∫ t

s

Vk(Φs,r ◦ ε+u , r)hk(r) dr.

On the other hand, the Jacobian matrix ∇Φs,t (x)◦ε+u satisfies a homogeneous linearSDE:

∇Φs,t ◦ ε+u = I +d ′∑

k=0

∫ t

s

∇Vk(Φs,r ◦ ε+u , r)∇Φs,r ◦ ε+u ◦ dWkr

+∫ t

s

∫

|z|>0+∇g(Φs,r− ◦ ε+u , r, z)∇Φs,r− ◦ ε+u N(dr dz)

+∑

i;s≤ri≤t

∇g(Φs,rj− ◦ ε+u , ri , zi)∇Φs,ri ◦ ε+u .

Then the solution Ξhs,t of the inhomogeneous equation is written, using the solution

∇Φs,t ◦ ε+u of the homogeneous equations as

Ξhs,t =

∫ t

s

∇Φr,t (Φs,r ) ◦ ε+u( d ′∑

k=1

Vk(Φs,r ◦ ε+u , r)hk(r))dr.

This proves that Φs,t (x) ◦ ε+u is H -differentiable and DrΦs,t (x) ◦ ε+u is given by

{(∇Φr,t (Φs,r (x))Vk(Φs,r (x), r)) ◦ ε+u , k = 1, . . . , d ′, s < r < t,

0, otherwise.

Since DjuΦs,t (x) is written as a linear sum of Φs,t (x) ◦ ε+v with v ⊂ u, Dj

uΦs,t (x)

is also H -differentiable. Further, we have DrDjuΦs,t (x) = D

juDrΦs,t (x), since D

and D are commutative. Therefore we get (6.33) in view of Proposition 6.1.1. ��


For u = (r, z), we set γ (u) = |z| ∧ 1 and for u = ((r1, z1), . . . , (rj , zj )), we setγ (u) = (|z1| ∧ 1) · · · (|zj | ∧ 1). We set A(1) = {u = (r, z) ∈ U; |z| ≤ 1}.Lemma 6.4.2 For any 0 ≤ s < t ≤ T , j ∈ N, i and p ≥ 2, we have

sups<r<t,x

supu∈A(1)j

E[∣∣DrD

ju∂

iΦs,t (x)∣∣p]

γ (u)p< ∞, (6.34)

where the sup for s < r < t is the essential supremum with respect to the Lebesguemeasure and the sup for u means the essential supremum with respect to the productmeasure nj .

Proof We first consider the case i = 0 and j = 1. Let u = (r1, z1) ∈ U, wheres < r1 < t . If r1 > r , we have by Lemma 6.4.1,

D(k)r (Φr1,t ◦ φr1,z1 ◦Φs,r1) = ∇(Φr1,t ◦ φr1,z1 ◦Φr,r1)Vk(Φs,r , r).

Therefore, we have D(k)r Dr1,z1Φs,t = Dr1,z1∇Φr,t ·Vk(Φs,r , r). Similarly, if r1 < r ,

we have D(k)r Dr1,z1Φs,t (x) = ∇Φr,t · Dr1,z1Vk(Φs,r , r). We claim:

E[∣∣Dr1,z1∇Φr,t

∣∣p] ≤ c|z1|p, E[∣∣Dr1,z1Vk(Φs,r , r)

∣∣p] ≤ c|z1|p. (6.35)

We will show the second inequality. Since Φr1,r and Φs,r1 are independent, we havefor any p ≥ 2

E[∣∣Vk(Φr1,r ◦ φr1,z1 ◦Φs,r1 , r)− Vk(Φs,r , r)

∣∣p]

= E[E[∣∣Vk(Φr1,r ◦ φr1,z1(y), r)− Vk(Φr1,r (y), r)

∣∣p]∣∣y=Φs,r1

]. (6.36)

Further, there exists a positive constant cp such that

E[∣∣Vk(Φr1,r ◦ φr1,z1(y), r)− Vk(Φr1,r (y), r)

∣∣p] ≤ cp|φr1,z1(y)− y|p, ∀y, r

in view of (3.28). Therefore (6.36) is dominated by

cpE[|φr1,z1(y)− y|py=Φs,r1

]≤ cpE

[∣∣φr1,z1(Φs,r1)−Φs,r1

∣∣p]

≤ cp|z1|pE[∣∣∂zφr1,θz1(Φs,r1)

∣∣p],

where |θz1| ≤ |z1|. Then there exists a positive constant c′p such that

E[∣∣Dr1,z1Vk(Φs,r (x), r)

∣∣p] ≤ c′p|z1|p, ∀s < t, x, r, |z1| ≤ 1.

This proves the second inequality of (6.35). We can verify the first inequalityof (6.35) in the same way.


We will next consider the case j = 2. Let u = {(r1, z1), (r2, z2)} ∈ U2, where

r1 < r2. Then D(k)r D

juΦs,t is a sum of terms D

j ′u′∇Φr,t Du′′Vk(Φs,r , r), where

u′,u′′ ⊂ u. We shall consider the term Du′′Vk(Φs,r , r). If u′′ = u, it holds that

E[|D2

uVk(Φs,r (x), r)|p] =

∫I (Φr1,r2 ◦ φr1,z1 ◦Φs,r1 , Φs,r2) dP, (6.37)

where

I (y, y′) = E[∣∣∣{Vk(Φr2,r ◦ φr2,z2(y), r)− Vk(Φr2,r (y), r)} (6.38)

− {Vk(Φr2,r ◦ φr2,z2(y′), r)− Vk(Φr2,r (y

′), r)}∣∣∣p]

.

We show that there is a positive constant c1 such that

I (y, y′) ≤ c1|(φr2,z2(y)− y)− (φr2,z2(y′)− y′)|p. (6.39)

Set Xλr = Vk(Φr2,r (λ), r). Then the 2d-dimensional process (X

λ1r ,−X

λ2r ) satisfies

E[|(Xλ1r −Xλ2

r )− (Xλ′1r −X

λ′2r )|p] ≤ c1|(λ1 − λ2)− (λ′1 − λ′2)|p

with some positive constant c1, similarly to the inequality (3.28) in Lemma 3.3.3.Setting λ1 = φr2,z2(y), λ2 = y, λ′1 = φr2,z2(y

′) and λ′2 = y′. Then we get theinequality (6.39). Therefore (6.37) is dominated by

c1

∫ ∣∣∣(φr2,z2(y)− y)− (φr2,z2(y′)− y′)

∣∣∣p

y=Φr1,r2◦φr1,z1◦Φs,r1 ,y′=Φs,r2

dP

≤ c2|z2|pE[∣∣∂zφr2,θz2(Φr1,r2 ◦ φr1,z1 ◦Φs,r1)− ∂zφr2,θz2(Φs,r2)

∣∣p]

≤ c3|z2|pE[∣∣Φr1,r2 ◦ φr1,z1 ◦Φs,r1 −Φs,r2

∣∣p]

≤ c4|z2|p|z1|p.

A similar estimation is valid for Dj ′u′∇Φr,t . Therefore we get the inequality

sups<r<t,x

E[∣∣DrD

juΦs,t (x)

∣∣p] ≤ cj,p|z1|p · · · |zj |p (6.40)

in the case j = 2. We can verify the above inequality in the case j ≥ 3, similarly.Consequently, (6.34) holds in the case i = 0. Further, we have (6.34) for any i,since (Φs,t , ∂

i′x Φs,t , |i′| ≤ |i|) is a solution of an SDE on a certain high dimensional

Euclidean space. ��


We can show the existence of higher derivatives DirD

ju∂

iΦs,t (x), i = 2, . . . bythe arguments as in Lemma 6.4.2. Further, we have

supr∈[s,t]i ,x∈Rd

supu∈A(1)j

E[∣∣Di

rDju∂

iΦs,t (x)∣∣p]

γ (u)p< ∞ (6.41)

for any i, i, j, p. Therefore ∂ iΦs,t (x) is an element of (D∞)d for any i, s < t and x.We will check that the solution of the SDE is regular, i.e., it satisfies Condition

(R) in Sect. 5.7.

Lemma 6.4.3 Φs,t (x) satisfies Condition (R) for any s < t, x.

Proof We have Dr,zΦs,t (x) = Φr,t ◦ φr,z ◦Φs,r (x)−Φs,t (x), if s < r ≤ t . Further,Dr,zΦs,t (x) = 0 if r < s or r > t . Therefore, we have the equality

∂zk Dr,zΦs,t (x) ={∇Φr,t (φr,z(Φs,r (x))∂zkφr,z(Φs,r (x))), if s < r < t,

0, otherwise.(6.42)

In view of Lemma 6.4.1, E[|∇Φr,t (φr,z(Φs,r (x))) ◦ ε+u |p] is bounded with respectto x ∈ R

d , s < r < t, |z| ≤ 1 and u ∈ A(1)j . Since ∂zkφr,z(x) is a boundedfunction with respect to r, z, x, we find supx,s,r,t,z,u E[|∂zk Dr,zΦs,t (x)◦ε+u |p] < ∞.

Similarly, we can show that for any ∂ iz = ∂

i1z1 · · · ∂id′zd

′ , the inequality

supx,s,r,t,z,u

E[∣∣∂ i

zDr,zΦs,t (x) ◦ ε+u∣∣p] < ∞

holds. We will apply Morrey’s Sobolev inequality (Lemma 3.8.2). For any positiveinteger i ≥ 3, we have

E[ ∑

|i|≤2

sup|z|≤1

∣∣∣∂ izDr,zΦs,t (x) ◦ ε+u

∣∣∣p] ≤ CpE

[ ∑

|i|≤i

∫

|z|≤1

∣∣∣∂ izDr,zΦs,t (x) ◦ ε+u

∣∣∣p

dz].

Take the supremum with respect to x, s, r, t and u ∈ A(1)j , j = 1, . . . , n for theboth sides of the above. The supremum of the right-hand side is finite. Therefore thesupremum of the left-hand side is also finite. Therefore Condition (R) is satisfied.

��Norms ‖ ‖�m,n,p for Wiener–Poisson functionals were defined in Sect. 5.10. Using

these norms, we have the following theorem from (6.41).

Theorem 6.4.1 Let {Φs,t } be the stochastic flow generated by SDE (6.30). ThenΦs,t (x) is a regular Wiener–Poisson functional belonging to (D∞)d , for any s, t ∈ T

and x ∈ Rd . Further,

‖∂ ixΦs,t (x)‖�m,n,p, ‖DΦs,t (x)‖�m,n,p, ‖DΦs,t (x)‖�m,n,p

6.5 Nondegenerate Jump-Diffusions 265

are bounded with respect to s, t, x if |i| ≥ 1, where ‖ ‖�m,n.p are Sobolev norms forWiener-Poisson functionals defined in Sect. 5.10.

6.5 Nondegenerate Jump-Diffusions

The solution of SDE (6.30) is a jump-diffusion on Rd . We saw in Sect. 4.5 that it

generator AJ (t) is given by (4.52). Set V (x, t) = (V1(x, t), . . . , Vd ′(x, t)). If thematrix V (x, t)V (x, t)T is positive definite for any t, x, the operator AJ (t) is calledelliptic. Further, if the matrix V (x, t)V (x, t)T is uniformly positive definite withrespect to x, t , the operator AJ (t) is called uniformly elliptic. We shall considerthe possibly non-elliptic case. Let {Γρ} be the family of d ′ × d ′-matrices definedby (1.21) and let Γ0 be its limit as ρ → 0. For jump-maps φt,z, we will definetangent vector fields at z = 0 by

Vk(x, t) = ∂zkφt,z(x)

∣∣∣z=0

, k = 1, . . . , d ′.

Set V (x, t) = (V1(x, t), . . . , Vd ′(x, t)) and define d × d-matrix by

A(x, t) = V (x, t)V (x, t)T + V (x, t)Γ0V (x, t)T .

If A(x, t) is positive definite for any x, t , the operator (4.52) is called pseudo-elliptic. Further, if the matrix A(x, t) is uniformly positive definite, the operatorAJ (t) is called uniformly pseudo-elliptic.

Let {Φs,t } be the stochastic flow generated by SDE (6.30). The Malliavincovariance of Φs,t (x) at the center was defined by (5.155). If s < r ≤ t , we haveD

(k)r Φs,t (x) = ∇Φr,t (Φs,r (x))Vk(Φs,r (x)) and

∂zk Dr,zΦs,t (x)∣∣z=0 = ∂zkΦr,t ◦ φr,z ◦Φs,r (x)|z=0

= ∇Φr,t (Φs,r (x))∂zkφr,z(Φs,r (x))∣∣z=0 = ∇Φr,t (Φs,r (x))Vk(Φs,r (x), r).

If r > t or r < s, we have DrΦs,t (x) = 0 and Dr.zΦs,t (x) = 0. Therefore theMalliavin covariance at the center is written as

KΦs,t (x) =∫ t

s

∇Φr,t (Φs,r (x))V (Φs,r (x), r)V (Φs,r (x), r)T∇Φr,t (Φs,r (x))

T dr

+∫ t

s

∇Φr,t (Φs,r (x))V (Φs,r (x), r)Γ0V (Φs,r (x), r)T∇Φr,t (Φs,r (x))

T dr

=∫ t

s

∇Φr,t (Φs,r (x))A(Φs,r (x), r)∇Φr,t (Φs,r (x))T dr.


A system of jump-diffusions {Xx,st = Φs,t (x), (x, s) ∈ T × R

d} with thegenerator AJ (t) is called uniformly nondegenerate at the center, if KΦs,t (x) areinvertible and for any n, p > 2, supθ |KΦs,t (x)(θ)−1|�0,n,p are bounded with respect

to x ∈ Rd . Here | |�0,n,p are norms defined by (5.57). Further, if the above is bounded

with respect to x on any compact subset of Rd , the system of jump-diffusions iscalled nondegenerate at the center.

Theorem 6.5.1 If the intrgro-differential operator AJ (t) of (4.52) is uniformlypseudo-elliptic, the system of jump-diffusions with the generator AJ (t) is uniformlynondegenerate at the center. Further, the inequality

sups<t,x

supθ

(t − s)|KΦs,t (x)(θ)−1|�0,n,p < ∞ (6.43)

holds for any n ∈ N, p > 2.

For the proof, we need a lemma.

Lemma 6.5.1 Jacobian matrices ∇Φs,t (x) are invertible a.s. Let Vs,t (x) be theinverse matrix. Then for any p ≥ 2 and j ∈ N,

sups<t,x

supu∈A(1)j

E[|Vs,t (x) ◦ ε+u |p] < ∞. (6.44)

Proof The stochastic flow Φs,t = Φs,t (x) satisfies

Φs,t = x +d ′∑

k=0

∫ t

s

Vk(Φs,r , r) ◦ dWkr +

∫ t

s

∫

|z|>0+g(Φs,r−, r, z)N(dr dz).

Differentiating each term of the above with respect to x, we get an SDE for theJacobian matrix Jt = Js,t = ∇Φs,t (x). It is written as

Jt = I +d ′∑

k=0

∫ t

s

∇Vk(r)Jr ◦ dWk(r)+∫ t

s

∫

|z|>0+∇g(r, z)JrN(dr dz),

where ∇Vk(r) := ∇Vk(Φs,r , r) and ∇g(r, z) := ∇g(Φs,r−, r, z). Now consider anSDE for an unknown d × d-matrix-valued process Vt = Vs,t :

Vt = I −d ′∑

k=0

∫ t

s

Vr∇Vk(r) ◦ dWkr −

∫ t

0

∫

|z|>0+Vr−∇h(r, z)N(dr dz), (6.45)

where ∇h(r, z) := ∇h(φr,z(Φs,r−), r, z) and h(x, r, z) is given by h(x, r, z) :=x − φ−1

r,z (x). We shall prove that Vt is the inverse matrix of Jt . Apply the rule of


the differential calculus for jump processes (Theorem 3.10.1) to the product VtJt .Using symmetric integrals, it holds that

VtJt = VsJs −∑

k

∫ t

s

Vr∇Vk(r)Jr ◦ dWjr +

∑

k

∫ t

0Vr∇Vk(r)Jr ◦ dW

jr

+∫ t

s

∫

|z|>0+

{Vr−(I−∇h(r, z))(I+∇g(r, z))Jr−−Vr−Jr−

}dN.

We have

(I −∇h(r, z))(I +∇g(r, z)) = ∇φ−1r,z (φr,z(Φs,r−))∇φr,z(Φs,r−) = I.

Consequently, we get VtJt = I , proving that Vs,t is the inverse matrix of Js,t .Now, equation (6.45) is a linear equation for Vt , where coefficients ∇Vj (· · · ) and

∇h(· · · ) are bounded uniformly with respect to parameter x. Rewrite the equationusing Itô integrals. Then we can apply Lemma 3.3.2 and show the inequalitysups<t,x E[|Vs,t (x)|p] < ∞ for any p > 2.

Further, the above discussion is valid if we transform Js,t and Vs,t by Js,t ◦ ε+uand Vs,t ◦ε+u , respectively; Vs,t ◦ε+u is the inverse of Js,t ◦ε+u and the former satisfiesthe inequality of the lemma. ��Proof of Theorem 6.5.1 Our discussion is close to that of Proposition 6.2.1. Setϕr = ϕ

s,t,xr = θ∇Φr,t (Φs,r (x)). Then KΦs,t (x)(θ) is written as

KΦs,t (x)(θ) =∫ t

s

ϕrA(Φs,r (x), r)ϕTr dr. (6.46)

For any s, t, x, it holds that ϕr = ϕ(s,t,x)r �= 0 a.s., since the Jacobian matrix

∇Φr,t (x) is invertible for any x, r, t . Let λr be the minimal eigen-value of the matrixA(Φs,r (x), r). Then we have

KΦs,t (x)(θ)−1 ≤ 1

(t − s)2

∫ t

s

1

λr |ϕr |2 dr

similarly to the proof of Proposition 6.2.1. The inequality is valid for the transfor-mation ε+u . Since λr ≥ c > 0 holds for any r and ω, we get the inequality

E[(KΦs,t (x)(θ) ◦ ε+u )−p] ≤ (c(t − s))−p supr

E[|ϕr ◦ ε+u |−2p].

We will show that the last term of the above is finite for any p. Let Vr,t (x) be theinverse matrix of ∇Φr,t (x). Since ∇Φr,t (Φs,r (x))Vr,t (Φs,r (x)) = I , we have

|ϕr ◦ ε+u |−1 ≤ |Vr,t (Φs,r (x)) ◦ ε+u |. (6.47)


The above is L2p-bounded for any 2p > 2 in view of Lemma 6.5.1. Then we getsups,t,x,r,u E[|ϕr ◦ ε+u |−2p] < ∞. Consequently, the inequality (6.43) holds. Thenthe jump-diffusion is uniformly nondegenerate at the center. ��

We shall next consider the pseudo-elliptic case. We want to show that the jump-diffusion is nondegenerate at the center. In the case of elliptic diffusions, we provedthis by introducing localizing stopping times τ(x, s) of Lemma 6.2.1. Loosely, thelemma shows that the process Φs,t (x) moves slowly, since P(τ(x, s) − s < δ) =O(δN) holds for any N > 2. However, the fact does not hold for jump-diffusions.Jump-diffusions could move rapidly. Even so, we will show that a hitting timeτ(x, s) of Φs,t (x) to a set V satisfies P(τ(x, s) − s < δ) = O(δN), if it jumpsat least N times before hitting the set V .

We will discuss it precisely. Let {φt,z} be jump-maps of the jump-diffusion. Fora subset U of Rd , we set

φ(U) =⋃

φt,z(U), φi(U) = φ(φi−1(U)), i = 2, 3, . . . ,

where ∪ is taken for all t ∈ T,z ∈ Supp(ν). It holds that U ⊂ φi (U) ⊂ φi+1(U) forany set U . Similarly, we define

φ−1(V ) =⋃

φ−1t,z (V ), φ−i (V ) = φ−1(φ−i−1(V )), i = 2, 3, . . .

For two subsets U,V of Rd , we set d(U, V ) = infx∈U,y∈V d(x, y), where d(x, y)

is the metric of two points x, y.In the following argument, we often set Φs,∞ = ∞ conventionally and we use

the notation ∞−∞ = 0.

Lemma 6.5.2 Let N be a positive integer satisfying d(φN(U), V ) > 0. For givenx, s, let τ = τ(x, s) be the hitting time of the jump-diffusion process X

x,st =

Φs,t (x), t > s to the set V . Then there exists c > 0 such that

supx∈U

P (τ(x, s) < t) ≤ c(t − s)N (6.48)


Proof For ε > 0 we define the ε-neighborhood of the set U ⊂ Rd by Uε =

{x; d(x,U) ≤ ε}. We set

ψε(U) = φ(Uε), ψnε (U) = ψε(ψ

n−1ε (U)), n = 2, . . . , N

inductively. Then if d(φN (U), V ) > 0 holds, d(ψNε (U), V ) > 0 holds for

sufficiently small ε > 0. We have U ⊂ ψε(U) ⊂ · · · ⊂ ψNε (U) ⊂ V.


Let x ∈ U . Let σ1 = σ1(x, s) be the first leaving time of Xt = Xx,st = Φs,t (x)

from the ball Bε(x) = {y ∈ Rd; |y − x| < ε}. Then Xσ1 ∈ ψ1

ε (U), if σ1 < T a.s.

We define stopping times σ2 = σ2(x, s), . . . , σN = σN(x, s) inductively as

σn = σn(x, s) = inf{r > σn−1; |Xr −Xσn−1 | ≥ ε},= ∞, if {· · · } is empty.

Then we have Xσn ∈ ψnε (U) if σn < ∞ a.s. for any n = 2, . . . , N . Consequently,

we have XσN /∈ V if σN < T a.s. This means τ ≥ σN a.s.Now, we have τ ≥ σN ≥ ∑N

n=1(σn − σn−1), where σ0 = s. Therefore,

{τ < t} ⊂N⋂

n=1

{|Xσn −Xσn−1 |1(σn−σn−1)<t−s ≥ ε}.

We shall consider the conditional probability of the last term: Since Φs,u(x) =Φs,t (Φt,u(x)) holds for any s < t < u and x a.s., σN(x, s) coincides with the firstleaving time of X(N−1)

r := ΦσN−1,r (XσN−1) from the ball Bε(XσN−1). Therefore wehave

σN(x, s) = σ1(XσN−1 , σN−1), a.s.

By Chebyschev’s inequality and the strong Markov property, we have

P(|XσN −XσN−1 |1σN−σN−1<t−s ≥ ε|FσN−1)

≤ ε−2E[|XσN −XσN−1 |21σN−σN−1<t−s |FσN−1 ]≤ ε−2E[|Xσ1(y,r) − y|21σ1(y,r)<t−s]

∣∣y=XσN−1 ,r=σN−1

≤ ε−2E[|Xσ1(y,r)∧t − y|2]∣∣y=XσN−1 ,r=σN−1

≤ cε−2(t − s), a.s.

for any x ∈ U . Here we used inequality supy E[|Φs,σ1(y,r)∧t (y) − y|2] ≤ c(t − s).Therefore,

P({τ < t}) ≤ cε−2(t − s)P (

N−1⋂

n=1

{|Xσn −Xσn−1 |1σn−σn−1<t−s > ε})

for any x ∈ U . Repeating this argument, P(τ < t) ≤ (cε−2)N(t − s)N holds forany x ∈ U . ��


Lemma 6.5.3 Let K be a positive constant satisfying |g(x, z)| ≤ K for any x, z.For given M > 0, ε > 0 and a positive integer N , set MN = M + N(K + ε). Letτ = τ(x, s) be the hitting time of the process X

x,st = Φs,t (x) to the set {y; |y| >

MN }. Then there is a positive constant c such that

sup|x|≤M

P(τ(x, s) < t) ≤ c(t − s)N (6.49)


Proof Set U = {y; |y| ≤ M}. Then φN(U) ⊂ {y; |y| ≤ M + NK}. Henceif V = {y; |y| > MN }, we have d(ψN(U), V ) > 0. Therefore (6.49) holds byLemma 6.5.2. ��Theorem 6.5.2 If the operator AJ (t) of (4.52) is pseudo-elliptic, the system ofjump-diffusions with the generator AJ (t) is nondegenerate at the center. Further,the inequality

sups,t∈T,s<t,|x|≤M

supθ∈Sd−1

(t − s)|KΦs,t (x)(θ)−1|�0,n,p < ∞ (6.50)

holds for any n ∈ N, p > 2 and M > 0.

Proof Consider again the formula (6.46). Let λr be the minimal eigen-value ofA(Φs,r (x), r). It may not be uniformly positive with respect to r, ω. So we needa localization. For a given p > 2, we will fix N ∈ N such that N > 2p.Let τ = τ(x, s) be the stopping time satisfying (6.49). Since A(x, t) is positivedefinite, we can take c > 0 such that θA(x, t)θT ≥ c holds for all θ ∈ Sd−1,|x| ≤ M + N(K + ε). Then we have KΦs,t (x)(θ) ≥ ∫ t∧τ

sλr |ϕr |2 dr for |x| ≤ M .

Then KΦs,t (x)(θ) are invertible a.s. and inverses satisfy

KΦs,t (x)(θ)−1 ≤ c(t ∧ τ − s)−2∫ t∧τ

s

1

λr |ϕr |2 dr.

Note λr ≥ c for r < τ . Then we get

(t − s)pE[KΦs,t (x)(θ)−p] ≤ c−p((t − s)4pE[(t ∧ τ − s)−4p]) 12 sup

rE[|ϕr∧τ |−4p] 1

2 .

We can verify sups<t,|x|≤M(t − s)4pE[(t ∧ τ − s)−4p] < ∞, similarly to the proofof Proposition 6.2.2. Further, we can show supx,r E[|ϕr∧τ |−4p] < ∞, similarly tothe proof of Proposition 6.2.1. Consequently, we get

sups<t,|x|≤M

supθ

(t − s)pE[KΦs,t (x)(θ)−p] < ∞.

Then we get the inequality (6.50) in the case n = 0 (ε+u is identity).


In the case where n ≥ 1, let τ (x, s) be the hitting time of the process Xx,st =

Φs,t (x) to the set {|y| > M + (N + n)(K + ε)}. Then we have the inequalityτ (x, s) ◦ ε+u ≥ τ(x, s) a.s. if u ∈ A(1)k , where k ≤ n. Therefore,

sup|x|≤M

supu∈A(1)k

P (τ (x, s) ◦ ε+u < t) ≤ c(t − s)N , s < t.

Using this stopping time, we have

sups<t,|x|≤M

supu∈A(1)k

supθ

(t − s)pE[(KΦs,t (x)(θ) ◦ ε+u )−p] < ∞.

Then we get (6.50) for the case n ≥ 1. ��Now, we will discuss the relation with δ-nondegenerate property. Let δ be any

constant satisfying 1 < δ < 2/α. Define

QΦs,t (x)ρ (θ) =

∫

T

(θ,DrΦs,t (x))2 dr

+ 1

|v|2ϕ(ρ)∫

A(ρ)

|ei(v,DuΦs,t (x)) − 1|2n(du)∣∣∣v=ρ

− 1δ θ

.

In the next proposition, we extend a result for diffusion processes stated inPropositions 6.2.1 and 6.2.2 to that for jump-diffusion processes.

Proposition 6.5.1 Let {Φs,t } be the stochastic flow associated with the operatorAJ (t).

1. If AJ (t) is uniformly pseudo-elliptic, QΦs,t (x)ρ (θ) are invertible and satisfy the

inequality

sups,t∈T,s<t,x

sup(ρ,θ)

(t − s)‖QΦs,t (x)ρ (θ)−1‖�m,n,p < ∞ (6.51)

for any m, n, p.

2. If AJ (t) is pseudo-elliptic, QΦs,t (x)ρ (θ) are invertible and for any M > 0 satisfy

the inequality

sups<t,|x|≤M

sup(ρ,θ)

(t − s)‖QΦs,t (x)ρ (θ)−1‖�m,n,p < ∞ (6.52)

for any m, n, p.

Proof We shall consider the uniformly pseudo-elliptic case only. We will applyTheorem 5.11.2 for KΦs,t (x). Note that it satisfies (6.43) and that Cδ′

n,p(Φs,t (x))

given by (5.99) is bounded with respect to t, x. Then we have


sups<t,x

sup(ρ,θ)

(t − s)|QΦs,t (x)ρ (θ)−1|�0,n,p < ∞

by Theorem 5.11.2. Next, we will prove (6.51) by applying Theorem 5.10.2. Notethat there is q > 1 such that the inequality

(t − s)‖QΦs,t (x)

ρ′ (θ)−1‖�m,n,p

≤ cm,n,p

(1 + 1

t − s‖QΦs,t (x)

ρ′ (θ)‖�m,n,2pq

)q(1 + |(t − s)Q

Φs,t (x)

ρ′ (θ)−1|�0,n,2pq)q

holds by Theorem 5.10.2. We shall consider the right hand side. Since

1

t − sQ

Φs,t (x)

ρ′ (v) = 1

t − s

∫ t

0

(v,DrΦs,t (x))2

|v|2 dr

+ 1

ϕ(ρ′)1

t − s

∫ t

s

∫

A0(ρ′)

|ei(v,DuΦs,t (x)) − 1|2|v|2 n(du)

holds, we can show that its ‖ ‖�m,n,p-norm is dominated by

c(‖DΦs,t (x)‖�m,n,2p)2 + c′(1 + ‖DΦs,t (x)‖�m,n,2(n+1)p)

2(n+1),

similarly to the proof of Proposition 5.11.1. Consequently, there is a positiveconstant c0 such that

sup(ρ,θ)

(1 + 1

t − s‖QΦs,t (x)

ρ′ (θ)‖�m,n,2pq

)q ≤ c0.

Then we have (6.51). ��Note If the generator A(t) is not pseudo-elliptic, we define a sequence of a familyof vector fields ΣM ;M = 0, 1, 2, . . . by

Σ0 = {V1, . . . , Vd ′ , V1, . . . , Vd ′ },ΣM = {[V0, V ], [Vk, V ], [Vk, V ]; k = 1, . . . , d ′, V ∈ ΣM−1}, M = 1, 2, . . .

If⋃

M ΣM span Rd for any x, the generator A is said to satisfy the Hörmander

condition or called hypo pseudo-elliptic. The existence of the smooth density forhypo pseudo-elliptic jump-diffusion was studied by Kunita [64]. The discussion iscomplicated.

6.6 Density and Fundamental Solution for Nondegenerate Jump-Diffusion 273

6.6 Density and Fundamental Solution for NondegenerateJump-Diffusion

Let Φs,t (x) be the jump-diffusion (stochastic flow) with generator (4.52). For agiven x and s < t , we consider an exponential functional:

Gs,t = exp{ d ′∑

k=0

∫ t

s

ck(Φs,r , r) ◦ dWkr +

∫ t

s

∫

|z|>0+log dr,z(Φs,r−)N(dr dz)

},

(6.53)where ck(x, t), k = 0, . . . , d ′ are C


d × T, and dt,z(x) is a

positive C∞,1,2b -function on R

d × T × Rd ′ such that dt,0(x) = 1 holds for any

x, t . The pair (Φs,t (x),Gs,t (x)y) is a solution of an Rd+1-dimensional SDE and

we may apply Theorem 6.4.1. Then we find that ∂ iGs,t (x) are in D∞ and norms‖∂ iGs,t (x)‖�m,n,p are bounded with respect to s < t, x for any i and m, n, p. Weset c = (c0(x, t), . . . , cd ′(x, t)) and d = (dt,z(x)) and denote the above Gs,t

by Gc,ds,t . We define the law of Φs,t (x) weighted by G

c,ds,t (x) by P

c,ds,t (x, E) =

E[1E(Φs,t (x))Gc,ds,t (x)].

Given an integro-differential operator AJ (t) of (4.52) and functions c,d, wedefine another operator Ac,d

J (t) by (4.58). Then it is the generator of the semigroup

{P c,ds,t } (Theorem 4.5.2).

Lemma 6.6.1 Assume that the system of jump-diffusions with generator AJ (t)

is nondegenerate at the center. Then the characteristic function ψc,ds,t,x(v) of the

transition function Pc,ds,t (x, E) has properties similar to those of ψc

s,t,x(v) inLemma 6.3.1.

Proof The characteristic function of P c,ds,t (x, E) is written as

ψc,ds,t,x(v) = E[ei(v,Φs,t (x))G

c,ds,t (x)].

It is a C∞-function of v, because Φs,t (x) has finite moments of any order. We

will show that it is a rapidly decreasing. Since QΦs,t (x)ρ (θ) satisfies (6.52) by

Proposition 6.5.1, for any N ∈ N and M > 0, there are m, n, p and a positiveconstant c such that

∣∣∣ψc,ds,t,x(v)

∣∣∣≤ C

|v|Nγ0

({‖DΦs,t (x)‖�m,n,2p +

(1 + ‖DΦs,t (x)‖�m,n,2(n+1)p

)n+1}

× sup(ρ,θ)

‖QΦs,t (x)ρ (θ)−1‖�m,n,p

)N‖Gc,ds,t (x)‖�m,n,p (6.54)


for all |v| ≥ 1, 0 ≤ s < t ≤ T and |x| ≤ M in view of Theorem 5.11.1. Thenwe can show that ψc,d

s,t,x(v) is infinitely differentiable with respect to x and v andsatisfies the inequality similar to (6.20), replacing N by Nγ0. ��Theorem 6.6.1 Assume that the system of jump-diffusions with the generator AJ (t)

is nondegenerate at the center. Then for any 0 ≤ s < t < ∞ and x ∈ Rd , the

transition function Pc,ds,t (x, E) generated by A

c,dJ (t) has a rapidly decreasing C∞-

density pc,ds,t (x, y) with respect to the Lebesgue measure. Further, it satisfies the

following properties.

1. For any s < t , it is a C∞-function of x, y and satisfies, for any ∂ ix and ∂k

y ,

∂ ix∂

jyp

c,ds,t (x, y) = (−i)|j|

( 1

2π

)d∫

Rd


c,ds,t,x(v) dv. (6.55)

2. For any y, t , it is a C∞,1-function of x ∈ Rd and s (< t). It satisfies

∂

∂sp

c,ds,t (x, y) = −A

c,dJ (s)xp

c,ds,t (x, y). (6.56)

3. For any x, s, it is a C∞,1-function of y ∈ Rd and t (> s). It satisfies

∂

∂tp

c,ds,t (x, y) = A

c,dJ (t)∗yp

c,ds,t (x, y), (6.57)

where Ac,dJ (t)∗ is the operator given by (4.77).

The above theorem can be verified similarly to Theorem 6.3.1 (diffusion case),making use of Lemma 6.6.1.

We consider the final value problem of the backward heat equation associatedwith the operator A

c,dJ (t). Its fundamental solution is defined in the same way as

the fundamental solution of the backward heat equation associated with Ac(t) inSect. 6.3.

Theorem 6.6.2 Assume that the system of jump-diffusions with generator AJ (t) isnondegenerate at the center. Let pc,d

s,t (x, y) be the smooth density of the transition

function Pc,ds,t (x, E) generated by A

c,dJ (t). Then p(x, s; y, t) := p

c,ds,t (x, y) is the

fundamental solution of the backward heat equation associated with the operatorA

c,dJ (t).

We can extend the function f1(x) of the final condition for the backward heatequation to any tempered distribution f1. Schwartz’s distribution is discussed brieflyin Sect. 5.12. We may assume that f1 belongs to S−N for sufficiently large N .Assume that the operator A

c,dJ (t) is uniformly nondegenerate at the center. Then

Theorem 5.12.1 and its proof tells us that for any N ∈ N and fixed s < t , there arem, n, p and a positive constant CN such that

6.6 Density and Fundamental Solution for Nondegenerate Jump-Diffusion 275

‖f (Φs,t (x))‖∗m,n,p ≤ CN |f |−N, f ∈ S−N (6.58)

holds for any x, where the norm | |−N is defined by (5.170). Indeed, in Theo-rem 5.12.1, the constant CN in (5.176) is given by sup‖G‖�m,n,p≤1 |ψx

G|N , where

ψxG(v) = E[ei(v,Φs,t (x))G]. The above function is a rapidly decreasing C∞-function

of v. Further, for any positive integer m and multi-index i, (1 + |v|2)m∂ ivψ

xG(v) is

uniformly bounded with respect to x, v and G with ‖G‖�m,n,p ≤ 1, since ψxG satisfies

the inequality (6.54) and

‖DΦs,t (x)‖�m,n,p, ‖DΦs,t (x)‖�m,n,p, sup(ρ,θ)

‖QΦs,t (x)ρ (θ)−1‖�m,n,p

are bounded with respect to x. Then sup‖G‖�m,n,p≤1 |ψxG|N is bounded with respect

to x.

Theorem 6.6.3 Assume that the system of jump-diffusions with the generator AJ (t)

is uniformly nondegenerate at the center. Then the semigroup of linear operatorsP

c,ds,t is extended to that of linear operators from the space of tempered distributions

to the space of C∞-functions.For a tempered distribution f1, set v(x, s) = E[f1(Φs,t1(x)) ·Gc,d

s,t1(x)]. Then it

is a C∞,1-function of (x, s). Further, it is differentiable with respect to s (< t1) andsatisfies the backward integro-differential equation

∂

∂sv(x, s) = −A

c,dJ (s)v(x, s) (6.59)

and the final condition lims→t1 v(x, s) = f1.

Proof We will approximate f1 by a sequence of smooth functions fm,m = 2, . . .in S with respect to the norm | |−N . Set vm(x, s) = E[fm(Φs,t (x))Gs,t (x)], whereGs,t = G

c,ds,t Then vm(x, t) converges to v(x, s) uniformly in x, in view of the

inequality (6.58). Therefore v(x, s) is a continuous function. Further, we have

∂ ivm(x, t) = ∂ iE[fm(Φs,t (x))Gs,t (x)]=

∑E[∂ i1fm(Φs,t (x)) · ∂ i2Φs,t (x), ∂

i3Gs,t (x)],

where the sum is taken for i1, i2 and i3 such that |i1|, |i2|, |i3| ≤ |i|. The right-handside of the above converges again uniformly in x in view of (6.58). Since this isvalid for any derivative operator ∂ i, the limit function v(x, t) is a C∞-function of x.

Now each function vm(x, s) satisfies ∂∂svm(x, s) = −A

c,dJ (s)vm(x, s) for s < t1.

Then the limit function v(x, s) should be differentiable with respect to s < t1 andsatisfy the same equation.


Let ϕ ∈ S∞. By the change of variables, we have

∫v(x, s)ϕ(x)dx =

∫E[f1(Φs,t1(x))Gs,t1(x)ϕ(x)] dx

=∫

E[f1(y)Gs,t1(Ψs,t1(y))ϕ(Ψs,t1(y))| det∇Ψs,t1(y)|] dy.

Let s tend to t1. Then the last term converges to 〈f1, ϕ〉. Therefore v(x, s) convergesto f1 as s → t1. ��

Finally, we will discuss the fundamental solution for the heat equation associatedwith the nondgenerate operator A

c,dJ (t). Consider the backward SDE defined with

characteristics (Vj (x, t), j = 0, . . . , d ′, g(x, t, z), ν). Let Φs,t (x) be the backward

flow of diffeomorphisms generated by the backward SDE and let Gc,ds,t be the expo-

nential function defined in Sect. 4.5. Then Pc,ds,t (x, E) = E[1E(Φs,t (x))G

c,ds,t (x)] is

the backward transition function with the generator Ac,dJ (t).

Theorem 6.6.4 Assume that the system of backward jump-diffusions generatedby AJ (t) is nondegenerate at the center. Then its backward transition functionP

c,ds,t (x, E) has a rapidly decreasing C∞-density p

c,ds,t (x, y) for any 0 ≤ s < t < ∞

and x ∈ Rd . Further, it is a C∞,1-function of (x, t) ∈ R

d × (s,∞) for any y, s

and p(x, t; y, s) := pc,ds,t (x, y) is the fundamental solution of the heat equation

associated with the operator Ac,dJ (t).

Remark We shall consider a jump-diffusion where its Lévy measure may not satisfythe order condition. In this case Theorem 6.6.1 may not hold. We will consider anelliptic jump-diffusion. Suppose that the generator is elliptic and jump-map φt,z

satisfies Condition (J.2). Then the Malliavin covariance RΦs,t (x) of Φs,t (x) definedby (6.11) is invertible and the inverse satisfies

supθ∈Sd−1,|x|≤M

(t − s)E[RΦs,t (x)(θ)−p] 1p < ∞, ∀p ≥ 2 (6.60)

for any 0 < M < ∞. Indeed, discussions of Sect. 6.3 are also valid for jump-diffusions, if we use Lemma 6.5.2 instead of Lemma 6.11. Then the transitionfunction P

c,ds,t (x, E) has a rapidly decreasing C∞-density. Further, the density

function pc,ds,t (x, y) is a rapidly decreasing C∞-function of x. See Remark in

Sect. 5.11.If jump-maps φt,z do not satisfy Condition (J.2) (φt,z are not diffeomorphic),

the smoothness of the density is not known even if AJ (t) is elliptic. Indeed, theflow Φs,t may not be diffeomorphic and hence its Jacobian matrix ∇Φs,t may notinvertible. Then the invertibility of the Malliavin covariance is not clear.

6.7 Short-Time Estimates of Densities 277

6.7 Short-Time Estimates of Densities

In the remainder of this chapter (Sects. 6.7, 6.8, 6.9, and 6.10) we will study ellipticdiffusion and pseudo-elliptic jump-diffusions. We will study short-time asymptoticsof the smooth densities pc

s,t (x, y) studied in Sect. 6.3 and pc,ds,t (x, y) studied in

Sect. 6.6 as t → s. We will be able to obtain the same short-time estimates for thefundamental solutions for heat equations associated with the elliptic operator Ac(t)

and pseudo-elliptic operator Ac,dJ (t). However, we will not repeat these arguments.

We will first state the assertion for a diffusion process.

Theorem 6.7.1 Assume that the partial differential operator Ac(t) is elliptic. Letpcs,t (x, y) be the smooth density of the transition function P c

s,t (x, E) generated

by Ac(t). Let ∂ ix and ∂

jy be any given differential operators with indexes i and j,

respectively.

1. For any T > 0 and M > 0, there exists a positive constant c such that

∣∣∣∂ ix∂

jyp

cs,t (x, y)

∣∣∣ ≤ c

(t − s)|i|+|j|+d

2

, ∀0 ≤ s < t ≤ T (6.61)

holds for all |x| ≤ M,y ∈ Rd .

2. If Ac(t) is uniformly elliptic, for any T > 0, there exists c > 0 such that theabove inequality holds for all x, y ∈ R

d .

Proof We will first consider the case where the diffusion is elliptic. For a givenT > 0, we consider the Wiener space of time parameter T = [0, T ] and apply theMalliavin calculus on the Wiener space. Let 0 ≤ s < t ≤ T and set ψc

s,t,x(v) =E[ei(v,Φs,t (x))Gs,t (x)], where Gs,t (x) = G

c,ds,t (x). Then we have from (6.22)

∂ ix∂

jyp

cs,t (x, y) = (−i)|j|

( 1

2π

)d∫

Rd


cs,t,x(v) dv. (6.62)

It holds that

∂ ixψ

cs,t,x(v) = E

[ei(v,Φs,t (x))Gi

s,t,x,v

],

where Gis,t,x,v is written as a linear sum of terms (6.21). We can rewrite it as

Gis,t,x,v =

∑|v||i|−|i0|Gi0,...,ij

s,t,x,θ ,

where

Gi0,...,ijs,t,x,θ = i|i|−|i0|∂ i0Gs,t (x)(θ, ∂

i1Φs,t (x)) · · · (θ, ∂ ijΦs,t (x)),


and the summation is taken for multi-indexes il such that il ≤ i and |i0|+· · ·+|ij| =|i|. Setting ψ

i0,...,ijs,t,x,θ (v) = E[ei(v,Φs,t (x))G

i0,...,ijs,t,x,θ ], we have

∣∣∣vj∂ ixψ

cs,t,x(v)

∣∣∣ ≤∑

|v||i|+|j|−|i0||ψ i0,...,ijs,t,x,θ (v)|

≤ (1 ∨ |v|)|i|+|j| ∑ |ψ i0,...,ijs,t,x,θ (v)|. (6.63)

Therefore we get

supy∈Rd

∣∣∣∂ ix∂

jyp

cs,t (x, y)

∣∣∣ ≤( 1

2π

)d∫

Rd

(1 ∨ |v|)|i|+|j| ∑ |ψ i0,...,ijs,t,x,θ (v)| dv.

Set w = √t − sv and change the variable. Since 1 ∧ |v| ≤

√T√t−s

(1 + |w|) and

dv = (t − s)−d/2 dw, the above is dominated by

c

(t − s)|i|+|j|+d

2

∫

Rd

(1 ∨ |w|)|i|+|j||ψ(w)| dw,

where

ψ(w) = E[ei(w,Hs,t,x )Gi0,...,ijs,t,x,θ ], Hs,t,x = Φs,t (x)√

t − s.

Take any M > 0 and consider

‖DHs,t,x‖m,p, supθ∈Sd−1

‖Gi0,...,ijs,t,x,θ‖m,p, sup

θ∈Sd−1

‖RHs,t,x (θ)−1‖m,p. (6.64)

The first and the second term are bounded with respect to s < t, |x| ≤ M

by Theorem 6.1.1. Further, the last term is bounded at the same region byProposition 6.2.2. Then, for any N there exists a positive constant cN such thatinequalities |ψ(w)| ≤ cN (if |w| ≤ 1) and ≤ cN

|w|N (if |w| ≥ 1) hold for any s < t

and |x| ≤ M in view of Theorem 5.3.1. Consequently, we obtain from (6.62)

supy∈Rd

∣∣∣∂ ix∂

jyp

cs,t (x, y)

∣∣∣ ≤ cN

(t − s)|i|+|j|+d

2

(1 +

∫

|w|≥1|w||i|+|j|−N dw

)

≤ c′N(t − s)

|i|+|j|+d2

(6.65)

for all s < t, |x| ≤ M , if N > d + |i| + |j|. Therefore we get the inequality (6.61)for all |x| ≤ M,y ∈ R

d , proving the first assertion of the theorem.


Finally, if the process is uniformly elliptic, all terms in (6.64) are bounded for alls < t, x in view of Theorem 6.1.1 and Proposition 6.2.1. Then (6.65) holds for alls < t, x. This proves the second assertion of the theorem. ��Remark The short-time estimate for fundamental solution for the elliptic oper-ator (4.20) is known in analysis. Assuming its coefficients are bounded andHolder continuous, inequality (6.61) is shown in the case |i| ≤ 2 and j = 0(Il’in–Kalashnikov–Oleinik [42], Dynkin [24]). We got a stronger result using theMalliavin calculus, assuming that coefficients of the operator A(t) is of C∞

b -class.The short-time estimate of the fundamental solution of a non-elliptic but hypo-elliptic operator seems to be open.

We will next consider jump-diffusions. Suppose first that their generators areelliptic or uniformly elliptic. Then discussions in the proof of Theorem 6.7.1 arevalid for these jump-diffusions. Therefore the assertions of Theorem 6.7.1 are validfor elliptic and uniformly elliptic jump-diffusions. We will consider the density forpseudo-elliptic jump-diffusions.

Theorem 6.7.2 Assume that the operator AJ (t) is pseudo-elliptic. Let pc,ds,t (x, y)

be the smooth density of transition function Pc,ds,t (x, E) generated by A

c,dJ (t). Let α0

be an arbitrary constant such that α < α0 < 2, where α is the exponent of the Lévymeasure ν. Let ∂ i

x and ∂jy be any given differential operators with indexes i and j.

1. For any T > 0 and M > 0, there exists a positive constant c such that∣∣∣∂ i

x∂jyp

c,ds,t (x, y)

∣∣∣ ≤ c

(t − s)|i|+|j|+d

2−α0

, ∀0 ≤ s < t ≤ T (6.66)

holds for all |x| ≤ M,y ∈ Rd .

2. If the operator AJ (t) is uniformly pseudo-elliptic, for any T > 0 there existsc > 0 such that the above inequality holds for all x, y ∈ R

d .

A similar result is shown in Ishikawa–Kunita–Tsuchiya [46] in the case of jumpprocesses. An interesting point in the inequality (6.66) is that the term t−β ′

, whereβ ′ = |i|+|j|+d

2−α0, is common for all jump-diffusions and all Lévy measures with the

same exponent 0 < α < 2 of the order condition, but the constant c should dependon coefficients Vk(x, t), k = 1, . . . , d ′, Vk(x, t), k = 1, . . . , d ′. It seems that α0 =α should be the critical value for the estimate (6.66), but we do not know whetherthe estimate is valid for α0 = α.

It may be an interesting problem to find the short-time asymptotics for non-elliptic or non-pseudo-elliptic, but nondegenerate SDEs such as hypoelliptic SDEs.In such a case the exponent β ′ in t−β ′

in the inequality (6.66) should be changed. Itseems that systematic results are not known.

The proof of Theorem 6.7.2 is not so simple as the proof of Theorem 6.7.1. Weneed four lemmas. Given α < α0 < 2, let us choose 1 < δ <

α0α

. For a given T > 0,we set T = [0, T ] and consider the Wiener–Poisson space of time parameter T.With this constant δ, we consider δ-nondegenerate condition with respect to QF (ρ)

defined by (5.146).


Lemma 6.7.1 For any positive integer N0 > 2, there exist m, n ∈ N, p ≥ 2 and apositive constant c such that for |v| ≥ 1 the inequality


]∣∣∣ ≤ c

|v|(1− αδ2 )κ

({‖DF‖�m,n,p + ‖DF‖�m,n,2(n+1)p

}(6.67)

×{

1 + |v|− αδ2 ‖DF‖�m,n,2(n+1)p

}n

sup(ρ,θ)

‖QFρ (θ)

−1‖�m,n,p

)κ‖G‖�m,n,p

holds for all κ ∈ [1, N0] and G ∈ D∞.

Proof We modify the proof of Theorem 5.11.1. Let us show that for any positiveinteger N , there exist m, n, p such that the above inequality holds with κ = N .Instead of (5.153), we use the inequality

‖e−i(v,DF ) − 1‖m,n,p;A(ρ)

≤ c|v|ϕ(ρ) 12 ‖DF‖�m,n,(n+1)p

{1 + |v|− αδ

2 ‖DF‖�m,n,(n+1)p

}n

which follows from (5.129), replacing the term |v|ρϕ(ρ) 12 by the bigger term

|v|− αδ2 . Then the discussion in the proof of Theorem 5.11.1 leads to (6.67) with

κ = N .Since norms ‖ ‖�m,n,p etc. are nondecreasing with respect to m, n, p, we can

choose m, n, p, c such that the inequality of the lemma holds for any N =2, 3, . . . , N0. Then by the interpolation, there exists another constant c > 0 suchthat the inequality of the lemma holds for all κ ∈ [0, N0]. ��

Let 0 ≤ s < t ≤ T and Φs,t (x) be the solution of a nondegenerate SDE (6.30)starting from x at time s. Let γ be a positive constant. We consider a random variable

Hs,t,x = 1

(t − s)γΦs,t (x).

We are interested in the decay property of characteristic functions of Hs,t,x , ast → s. In the following three lemmas, the operator AJ (t) is assumed to be pseudo-elliptic.

Lemma 6.7.2 For any M > 0 and κ0 > 2, there exist m, n, p and a positiveconstant c such that for any |x| ≤ M and 1 < κ < κ0 the inequality

∣∣∣E[ei(v,Hs,t,x )G

]∣∣∣ ≤ c( (t − s)γ−1

|v|(1− αδ2 )

)κ‖G‖�m,n,p (6.68)

holds for all 0 ≤ s < t ≤ T and |v| ≥ (t − s)−2γαδ .


Further, If AJ (t) is uniformly pseudo-elliptic, for any κ0 > 2 there existc,m, n, p such that for any x ∈ R

d and 1 < κ < κ0, the inequality (6.68) holds for

all 0 ≤ s < t ≤ T and |v| ≥ (t − s)−2γαδ .

Proof We will apply Lemma 6.7.1, setting F = Hs,t,x . Consider the right-hand sideof (6.67). Since

c1 := sups<t,|x|≤M

{‖DΦs,t (x)‖�m,n,p + ‖DΦs,t (x)‖�m,n,2(n+1)p

}(6.69)

is finite by Theorem 6.4.1, we have{‖DHs,t,x‖�m,n,p + ‖DHs,t,x‖�m,n,2(n+1)p

}{1 + |v|− αδ

2 ‖DHs,t,x‖�m,n,2(n+1)p

}n

≤ c1

(t − s)γ,

for any |v| ≥ (t − s)−2γαδ and |x| ≤ M . Note that

QHs,t,xρ (θ) = 1

(t − s)2γ QΦs,t (x)

ρ′ (θ), where ρ′ = (t − s)δγ ρ. (6.70)

Then we have

‖QHs,t,xρ (θ)−1‖�m,n,p = (t − s)2γ ‖QΦs,t (x)

ρ′ (θ)−1‖�m,n,p. (6.71)

We know by Proposition 6.5.1

sup|x|≤M,s<t

sup(ρ′,θ)

(t − s)‖QΦs,t (x)

ρ′ (θ)−1‖�m,n,p ≤ c2. (6.72)

Therefore we have

sup|x|≤M

sup(ρ,θ)

‖QHs,t,xρ (θ)−1‖�m,n,p ≤ c2(t − s)2γ−1. (6.73)

Then we get (6.68) from Lemma 6.7.1. ��Lemma 6.7.3 For any M > 0 and N0 > 2, there exist m, n, p and a positiveconstant c such that

|E[ei(v,Φs,t (x))G]| (6.74)

≤⎧⎨

⎩

‖G‖�m,n,p if v ∈ Rd ,

c

(t − s)(1−αδγ

2 )κ |v|(1− αδ2 )κ

‖G‖�m,n,p, if |v| ≥ (t − s)−c0γ

holds for all 1 < κ < N0, 0 ≤ s < t ≤ T and |x| ≤ M , where c0 = 2αδ

+ 1.


Further, if AJ (t) is uniformly pseudo-elliptic, for any N0 > 2, there exist m, n, p

and a positive constant c such that (6.74) holds for all 1 < κ < N0, 0 ≤ s < t ≤ T

and x ∈ Rd .

Proof Since the equality E[ei(v,Φs,t (x))G] = E[ei((t−s)γ v,Hs,t,x )G] holds, we getfrom Lemma 6.7.2 the inequality

|E[ei(v,Φs,t (x))G]| ≤ c( (t − s)γ−1

((t − s)γ |v|)(1− αδ2 )

)κ‖G‖�m,n,p

≤ c

(t − s)(1−αδγ

2 )κ |v|(1− αδ2 )κ

‖G‖�m,n,p, (6.75)

if |v| ≥ (t − s)−( 2αδ+1)γ , for any |x| ≤ M , s < t and 1 ≤ κ ≤ N0. Further, if the

SDE is uniformly nondegenerate at the center, the above holds for all x ∈ Rd , s < t

and 1 ≤ κ ≤ N0. ��Lemma 6.7.4 Let ψc,d

s,t,x(v) be the characteristic function of P c,ds,t (x, E). For any

M > 0 and N0 > 2, there exist positive constants c3, c4 such that

∣∣∣∂ ixψ

c,ds,t,x(v)

∣∣∣ ≤⎧⎨

⎩

c3(1 ∨ |v|)|i|, if v ∈ Rd ,

c4

(t − s)(1−αδγ

2 )κ|v||i|−(1− αδ

2 )κ , if |v| ≥ (t − s)−c0γ (6.76)

holds for all 1 ≤ κ ≤ N0, 0 ≤ s < t ≤ T and |x| ≤ M .If the SDE is uniformly nondegenerate at the center, there exist k, p, c > 0 such

that the above holds for all 0 ≤ s < t ≤ T , x ∈ Rd and 1 ≤ κ ≤ N0.

Proof The function ψc,ds,t,x(v) is infinitely differentiable with respect to x. It holds

that

∂ ixψ

c,ds,t,x(v) = E

[ei(v,Φs,t (x))Gi

s,t,x,v

], (6.77)

where Gis,t,x,v is written as a linear sum of terms written as (6.21). Then the

inequality (6.63) holds for ψc,ds,t,x . Apply (6.75) for each G = G

i0,...,ijs,t,x,θ and sum

up for i0, . . . , ij. Then we get (6.76) for any |x| ≤ M , s < t and 1 ≤ κ ≤ N0. Thelatter assertion will be obvious. ��Proof of Theorem 6.7.2 It is sufficient to prove (6.66) in the case 0 < t− s < 1. Wewant to prove the inequality by applying the following formula:

∂ ix∂

jyp

c,ds,t (x, y) = (−i)|j|

( 1

2π

)d∫

Rd


c,ds,t,x(v) dv.

Note that (t − s)−c0γ > 1. From Lemma 6.7.4, for any M > 0 there exist positiveconstants c1, c2 such that the inequality


∣∣∣vj∂ ixψ

c,ds,t,x(v)

∣∣∣ ≤

⎧⎪⎪⎨

⎪⎪⎩

c0, if |v| ≤ 1,c1|v||i|+j|, if 1 < |v| ≤ (t − s)−c0γ ,

c2

(t − s)(1−αδγ

2 )κ|v||i|+|j|−(1− αδ

2 )κ , if |v| > (t − s)−c0γ

(6.78)holds for all s < t, |x| < M . Note that the right-hand side is an integrable functionof v, if and only if κ satisfies |i|+|j|−(1− αδ

2 )κ < −d or equivalently κ >|i|+|j|+d

1− αδ2

holds.We will take γ = γ0 satisfying γ0 ≥ (2 − αδ) ∨ 1

αδ. Next, set κ0 = |i|+|j|+d

1− α02

.

Then the right-hand side of (6.78) is an integrable function of v. In fact, |i| + |j| −(1− αδ

2 )κ0 < −d holds, because αδ < α0. Further, if |x| < M , we have from (6.78)

∣∣∣∫

Rd


c,ds,t,x(v) dv

∣∣∣ ≤ c0

∫

|v|≤11 dv (6.79)

+ c1

∫

|v|≤(t−s)−c0γ0|v||i|+|j| dv + c2

(t − s)(1−αδγ0

2 )κ0

∫

|v|≥1|v||i|+|j|−(1− αδ

2 )κ0 dv

≤ c′0 +c′1

(t − s)|i|+|j|+dc0γ0

+ c′2(t − s)

(2−αδγ0)|i|+|j|+d

2−α0

,

since∫|v|≥1|v||i|+|j|−(1− αδ

2 )κ0 dv is finite. We have |i|+|j|+dc0γ0

≤ |i|+|j|+d2−α0

because1

c0γ0≤ 1

2−α0. Furthermore, 2 − αδγ0 ≤ 1 holds because γ0 ≥ 1

αδ. Therefore we

have

1

(t − s)|i|+|j|+d

γ0

≤ 1

(t − s)|i|+|j|+d

2−αδ

,1

(t − s)(2−αδγ0)|i|+|j|+d

2−αδ

≤ 1

(t − s)|i|+|j|+d

2−αδ

.

Consequently, we get the inequality

supy∈Rd

|∂ ix∂

jyp

c,ds,t (x, y)| =

( 1

2π

)d∫

Rd

|vj∂ ixψ

c,ds,t,x(v)| dv

≤ c3

(t − s)|i|+|j|+d

2−αδ

, (6.80)

for all 0 ≤ s < t ≤ T , if |x| < M .If AJ (t) is uniformly pseudo-elliptic, we can take constants c1, c2 in (6.78)

such that the inequality holds for all 0 ≤ s < t ≤ T and x ∈ Rd . Then the

inequality (6.80) holds for all x, y ∈ Rd and 0 ≤ s < t ≤ T . Then the second

assertion of the theorem follows. ��


Remark If the generator of a jump-diffusion satisfying Condition (J.2) is elliptic,the same short-time estimate of (6.61) is valid for its transition function. Indeed,the proof of Theorem 6.7.1 can be applied to elliptic jump-diffusions satisfyingCondition (J.2).

6.8 Off-Diagonal Short-Time Estimates of Density Functions

We shall next consider the off-diagonal short-time asymptotics of density functionsfor elliptic diffusions and pseudo-elliptic jump-diffusions. These properties arequite different between diffusions and jump-diffusions. We first consider a diffusionprocess.

Theorem 6.8.1 Let pcs,t (x, y) be the smooth density of transition function

P cs,t (x, E), where its generator Ac(t) is elliptic. Let ∂

jy be any given differential

operator of index j. Suppose that U,V are bounded disjoint open subsets of Rd

such that d(U , V ) > 0. Then for any T > 0 and N ∈ N, there exists a positiveconstant c = cT ,N,j,U,V such that

supx∈U,y∈V

∣∣∣∂ jyp

cs,t (x, y)

∣∣∣ ≤ c(t − s)N , ∀0 ≤ s < t ≤ T . (6.81)

For the proof of the theorem, we will consider a diffusion Xx,st = Φs,t (x) pinned

at the point y ∈ Rd at time t , or a diffusion conditioned by {Xx,s

t = y}. Later wewill consider a similar problem for a jump-diffusion. So we will define the pinnedprocess for a jump-diffusion.

We defined in Sect. 5.12 the composite of a tempered distribution and a non-degenerate smooth Wiener–Poisson functional. Let Φs,t (x) be the jump-diffusiondetermined by a pseudo-elliptic SDE (6.30) and let Ψs,t (y) = Φ−1

s,t (y). Then

both Φs,t (x) and Ψs,t (y) are nondegenerate Wiener–Poisson functionals for anys < t, x, y. Let δx and δy be the delta functions of points x and y, respectively.Then the composite δy(Φs,t (x)) and δx(Ψs,t (y)) are well defined as generalized

Wiener–Poisson functionals. It holds that pc,ds,t (x, y) = E[δy(Φs,t (x))Gs,t (x)],

where Gs,t (x) is given by (6.53) (see Corollary 5.12.1).

Lemma 6.8.1 We have

E[ n∏

i=1

fi(Φs,ti (x))δy(Φs,t (x))]= E

[ n∏

i=1

fi(Ψti ,t (y))δx(Ψs,t (y))| det∇Ψs,t (y)|]

(6.82)for any s < t1 < · · · tn < t , x, y ∈ R

d and C∞0 -functions fi, i = 1, . . . , n.

6.8 Off-Diagonal Short-Time Estimates of Density Functions 285

Proof Applying the change-of-variable formula and then taking expectations(Sect. 4.6), we get the dual formula

∫

Rd

ψ(x)E[∏

i

fi(Φs,ti (x))ϕ(Φs,t (x))]dx

=∫

Rd

E[∏

i

fi(Ψti ,t (y))ψ(Ψs,t (y))| det∇Ψs,t (y)|]ϕ(y) dy

for C∞0 -functions fi, i = 1, . . . , n and ϕ,ψ . Let ϕ and ψ tend to δy and δx ,

respectively. Then we get the formula of the lemma. ��We need an another lemma. Lemma 6.2.1 is modified as follows, whose proof

can be done similarly to the proof of Lemma 6.2.1.

Lemma 6.8.2 Let U,V be bounded open subsets of Rd such that d(U , V ) > 0. Letτ(x, s) be the hitting time of the diffusion process Xt = X

x,st := Φs,t (x) to the set

V . Then for any N ≥ 2, there exists a positive constant c such that

supx∈U

P (τ(x, s) < t) ≤ c(t − s)N , ∀s < t. (6.83)

Proof of Theorem 6.8.1 We will consider the case Gs,t (x) = 1 only, since discus-sions for the case Gs,t (x) > 0 are similar. We first consider the case j = 0. Let U,V

be given sets in the assertion of Theorem 6.8.1. Take a bounded open set W whichsatisfies U ⊂ W ⊂ V c, d(U ,Wc) > 0 and d(W , V ) > 0. Let x ∈ U and y ∈ V . Letτ = τ(x, s) be the first leaving time of the diffusion process Xt = X

x,st = Φs,t (x)

from the set W . Then it holds that P(τ ≤ t) ≤ c(t − s)N by Lemma 6.8.2. Sinced(W , y) > 0, we have

ps,t (x, y) = E[δy(Φs,t (x)); s < τ ≤ t

](6.84)

= E[δy(Φs,t (x)); s < τ ≤ (t + s)/2

]+ E[δy(Φs,t (x)); (t + s)/2 < τ ≤ t

].

Using the strong Markov property, the first term of the right-hand side is estimatedas

E[δy(Φs,t (x)); s < τ ≤ (t + s)/2

] = E[Pτ,t δy(Φs,τ (x)); s < τ ≤ (t + s)/2

]

= E[pτ,t (Φs,τ (x), y); s < τ ≤ (t + s)/2

].

Note that Φs,τ ∈ W . Then we have

pτ,t (Φs,τ (x), y) ≤ c(t − τ)−d0 ≤ c( t − s

2

)−d0, (6.85)


where d0 = d2 , in view of Theorem 6.7.1. Therefore, we get

E[δy(Φs,t (x)); s < τ ≤ (t + s)/2

] ≤ c1

( t − s

2

)−d0P(s < τ ≤ (t + s)/2)

≤ c2(t − s)N−d0 . (6.86)

We shall next consider the second term of (6.84). It holds by Lemma 6.8.1 that

E[δy(Φs,t (x)); (t + s)/2 < τ ≤ t] (6.87)

= E[| det∇Ψs,t (y)|δx(Ψs,t (y)); (t + s)/2 < τ ≤ t].If ps,t (x, y) > 0, the conditional probability measure given {Φs,t (x) = y} isdefined by

P(A|Ψs,t (y) = x) = E[1Aδx(Ψs,t (y))]ps,t (x, y)

.

Under this measure, it holds that Φs,r (x) = Ψr,t (Φs,t (x)) = Ψr,t (y) a.s. Then τ

may be regarded as the last exit time of the backward process Yr := Ψr,t (y) (t, yare fixed) from W . Let σ∗ be the first exit time of the same process Yr from W c. Itis defined by σ∗ = sup{r; Yr ∈ W c}. Then it holds that t − τ ≥ t − σ∗. We willapply Lemma 6.8.2 to the backward process Yr . Then, similarly to (6.86), we have

E[δy(Φs,t (x)); (t + s)/2 < τ ≤ t] (6.88)

≤ E[| det∇Ψs,t (y)|δx(Ψs,t (y)); (t + s)/2 < σ∗ ≤ t] ≤ c3(t − s)N−d0 .

Substituting (6.86) and (6.88) to (6.84), we get the inequality (6.81) of the theoremif we rewrite N − d0 as N .

Finally, if j �= 0, we consider the composite of the generalized function(−1)j∂

jxδy and Φs,t (x). Then we get the assertion (6.81) similarly. ��

In the case of a jump-diffusion, the problem is more complicated, sinceLemma 6.8.2 does not hold. In order that the inequality (6.83) is valid, the distanceof two disjoint sets U and V should be big so that the point x ∈ U reaches the pointy ∈ V after sufficiently many jumps. It is explained below. Let {φr,z} be jump-mapsof the jump-diffusion. Let U be an open subset of Rd . In Sect. 6.5, we defined theset φ(U) = ∪φr,z(U) etc.

The following is a consequence of Lemma 6.5.2.

Lemma 6.8.3 Let N be a positive integer satisfying d(φN (U), φ−N(V )) > 0. Letτ = τ(x, s) be the hitting time of the process X

x,st := Φs,t (x), t ∈ [s,∞) to the set

V . Then there exists c > 0 such that

supx∈U

P (τ(x, s) < t) ≤ c(t − s)N , ∀s < t. (6.89)

6.8 Off-Diagonal Short-Time Estimates of Density Functions 287

Theorem 6.8.2 Let pc,ds,t (x, y) be the smooth density of transition function

Pc,ds,t (x, E), where its generator A

c,dJ (t) is pseudo-elliptic. Let α0 be an arbitrary

positive constant such that α < α0 < 2, where α is the exponent of theLévy measure. Suppose that U,V are bounded open sets of R

d such thatd(φN (U), φ−N(V )) > 0 holds for some positive integer N . Then for any T > 0there exists a positive constant c = cT ,j,U,V such that

supx∈U,y∈V

∣∣∣∂ jyp

c,ds,t (x, y)

∣∣∣ ≤ c(t − s)N− |j|+d

2−α0 , ∀0 ≤ s < t ≤ T . (6.90)

Proof We will consider the case Gs,t (x) = 1, i.e., c = 0,d = 1, since discussionsfor the case Gs,t (x) �= 1 are similar. We denote p

c,ds,t (x, y) by ps,t (x, y). We first

consider the case j = 0. Let x ∈ U and y ∈ V . Let τ be the first leaving time of theprocess Xt = Φs,t (x) from φN (U). Then it holds that P(τ ≤ r) ≤ c(r − s)N byLemma 6.8.3. Further, we have

ps,t (x, y) = E[δy(Φs,t (x)); s < τ ≤ t

](6.91)

= E[pτ,t (Φs,τ (x), y) : s < τ ≤ (t + s)/2

]

+ E[| det∇Ψs,t (y)|δx(Ψs,t (y)); (t + s)/2 < τ < t],as in the proof of the previous theorem. Note that Φs,τ (x) ∈ φN (U) and φN (U) isa bounded set, because g(x, t, z) is a bounded function. Then we have

pτ,t (Φs,τ (x), y) ≤ c(t − τ)−d0 ≤ c( t − s

2

)−d0,

where d0 = d2−α0

, in view of Theorem 6.7.2. Therefore, we get

E[pτ,t (Φs,τ (x), y) : s < τ ≤ (t + s)/2

] ≤ c1

( t − s

2

)−d0P(s < τ ≤ (t + s)/2)

≤ c2(t − s)N−d0 .

We shall next consider the last term of (6.91). Under the conditional measureP(A|Ψs,t (y) = x), it holds that Φs,r (x) = Ψr,t (Φs,t (x)) = Ψs,t (y) a.s. Then τ

may be regarded as the last exit time of the left continuous backward process Ys =Ψs−,t (y) (t is fixed) from φ−N(Uc). Let σ∗ be the first exit time of the same processYs from φ−N(Uc). It is defined by σ∗ = sup{s; Ys ∈ φ−N(V )}. Then it holds thatt − τ ≥ t − σ∗, since d(φN (U), φ−N(V )) > 0. We will apply Lemma 6.8.1 to thebackward process Ψs,t (y) (t being fixed). Then, similarly to (6.86), we have

E[| det∇Ψs,t (y)|δx(Ψs,t (y)); (t + s)/2 < σ∗ < t] ≤ c3(t − s)N−d0 .

Substituting these two inequalities into (6.91), we get the inequality (6.90) of thetheorem in the case |j| = 0.


Finally, if j �= 0, we consider the composite (−1)j∂jxδy and Φs,t (x). Then we get

the assertion (6.90) similarly. ��Note The short-time estimate and the off-diagonal short-time estimate for diffusionprocesses seems to be known more or less. These estimates for jump processes arestudied by Picard [93] and Ishikawa [43], not using the Malliavin calculus. Ishikawaet al. [46] obtained the short-time estimates for jump processes using the Malliavincalculus. Their results are close to our Theorem 6.7.1. The present off-diagonalshort-time estimate using the Malliavin calculus might be new.

6.9 Densities for Processes with Big Jumps

Let us return to SDE (6.30). For drift and diffusion coefficients V0(x, t), V1(x, t), . . .,Vd ′(x, t), we assumed that they are of C

∞,1b -class; for jump coefficients φt,z, we

assumed that they satisfy Conditions (J.1) and (J.2) stated in Sect. 3.2. In this sectionwe will assume the same condition for drift and diffusion coefficients. But for jumpcoefficients, we will assume Conditions (J.1)K and (J.2)K stated in Sect. 3.9.

For the Lévy measure ν, we assume again that it has a weak drift and that atthe center it satisfies the order condition of exponent 0 < α < 2 with respect to afamily of star-shaped neighborhoods. We assume further that its generator is pseudo-elliptic. We should remark that the pseudo-elliptic property is concerned with thediffusion part and small jumps only, since vector fields Vk(t), k = 1, . . . , d ′ aredetermined by jump-maps φr,z; |z| < γ for arbitrary small γ > 0.

We are again interested in the existence of the smooth density of the law of thesolution Φs,t (x). However, since equation (6.30) may admit big jumps, the solutionmay not belong to L∞−. So we cannot apply the Malliavin calculus to the solutiondirectly. To avoid this difficulty, we will first consider an SDE where big jumpsare truncated, for which the smooth density of the law exists. Then we will adjoinbig jumps as a perturbation and show that such a perturbation should preserve thesmoothness of the density.

In this section, we will restrict our attention to the law of the solution Φs,t (x)

and will avoid some complicated discussions and notations related the weightedlaws. Our discussions will be extended to the weighted law of Φs,t (x) as inSects. 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, and 6.8.

Suppose we are given an SDE (6.30), where coefficients of the equation satisfyConditions (J.1)K and (J.2)K . We will consider an SDE truncating jumps whichare bigger than δ > 0. The truncated equation is given by

Xt = X0 +d ′∑

k=0

∫ t

t0

Vk(Xr, r) ◦ dWkr

+∫ t

t0

∫

δ≥|z|>0+{φr,z(Xr−)−Xr−}N(dr dz). (6.92)

6.9 Densities for Processes with Big Jumps 289

The solutions of the above equation define a stochastic flow of diffeomorphisms{Φs,t } by Theorem 3.9.1, since the truncated function g(x, t, z)1|z|≤δ satisfiesConditions (J.1) and (J.2).

For a given T > 0, we set T = [0, T ]. We will fix 0 ≤ s < t ≤ T and a pointx. We shall construct the solution of equation (6.30) starting from x at time s, byadding big jumps to Φs,t (x). Let U = T×R

d ′0 and u = ((t1, z1), . . . , (tn, zn)) be an

element of Un (n ≥ 1) such that |zi | > δ, i = 1, . . . , n and s < t1 < · · · < tn ≤ t .We consider Φu

s,t (x) := Φs,t (x) ◦ ε+u . It is represented by

Φus,t (x) = Φtn,t ◦ φtn,zn ◦ · · · ◦ Φt1,t2 ◦ φt1,z1 ◦ Φs,t1(x), (6.93)

as in (6.32). We set Φφs,t (x) = Φs,t (x) if u = ∅ ∈ U

0 (empty set). Now, let q(ω) be a

Poisson point process on U′ = [s, T ]×R

d ′0 with intensity measure dtνδ(dz), where

νδ(dz) = 1(δ,∞)(|z|)ν(dz). Let N(dr dz) be the associated Poisson random measure(Sect. 1.4). Let Dq be the domain of the point process q. Since λδ ≡ ν({|z| > δ})is finite, the set Dq ∩ (s, t] can be written, if it is non-empty, as {s = τ0 < τ1 <

· · · < τn}, where τm are stopping times defined inductively by τ0 = s and byτm = inf{t > τm−1; Nt − Nτm−1 ≥ 1} (= ∞ if the set {· · · } is empty). HereNt := N((s, t] × {|z| > δ}) is a Poisson process with intensity λδ . We set u(q) = ∅if τ1 > t and u(q) = ((τ1, q(τ1)), . . . , (τn, q(τn))) if τn ≤ t < τn+1. Then wedefine Φs,t (x) := Φ

u(q)s,t (x). It is written as

{Φs,t (x), if t < τ1,

Φτn,t ◦ φτn,q(τn) ◦ · · · ◦ Φτ1,τ2 ◦ φτ1,q(τ1) ◦ Φs,τ1(x), if τn ≤ t < τn+1.

Therefore Φs,t (x) is a solution of the original equation (6.30) starting from x at times (see Sect. 3.2 in Chap. 3).

Now, we assume that the generator of the process Φs,t (x) is pseudo-elliptic. Thenthe law of Φs,t (x); Ps,t (x, E) = P(Φs,t (x) ∈ E) has a rapidly decreasing C∞-density by Theorem 6.6.1. We denote it by ps,t (x, y). Further, the stochastic processXt = Φu

s,t (x) is a Markov process with fixed discontinuity, jumping from Φus,ti− to

φti ,zi (Φus,ti−) at fixed times ti; i = 1, . . . , n. Its transition probability has a C∞-

density given by

pus,t (x, y) =

∫· · ·

∫

(Rd )nps,t1(x, x1)pt1,t2(φt1,z1(x1), x2)× · · ·

× ptn,t (φtn,zn(xn), y) dx1 · · · dxn. (6.94)

Further, setting pφs,t (x, y) = ps,t (x, y), we have

P(Φs,t (x) ∈ E) = P(Φu(q)s,t ∈ E) =

∫

E

E[pu(q)s,t (x, y)] dy.


Therefore, the function defined by

ps,t (x, y) ≡ E[pu(q)s,t (x, y)] (6.95)

is the density functions of transition probability Ps,t (x, E) of Φs,t (x).We want to prove the smoothness of the above ps,t (x, y) under additional

conditions.

Theorem 6.9.1 Consider a system of uniformly pseudo-elliptic jump-diffusions,which satisfies Conditions (J.1)K and (J.2)K .

1. Assume that {φt,z} satisfies

supt,x

∫

|z|>δ

∣∣∣ det∇φ−1t,z (x)

∣∣∣ν(dz) < ∞ for some δ > 0, (6.96)

where det∇φ−1t,z (x) is the Jacobian determinant of the inverse map φ−1

t,z . Thenthe function ps,t (x, y) given by (6.95) is a continuous function of x, y ∈ R

d forany 0 ≤ s < t ≤ T . Further, for any α < α0 < 2, there exists c0 > 0 such thatthe following holds for all 0 ≤ s < t ≤ T and x, y ∈ R

d :

ps,t (x, y) ≤ c0

(t − s)d

2−α0

. (6.97)

2. Let m0 be a positive integer. Assume further that

supt,x

∫

|z|>δ

∣∣∣∂ ixφt,z(x)

∣∣∣p

ν(dz) < ∞ for some δ > 0 (6.98)

holds for ∂ ix with 1 ≤ |i| ≤ m0 and p > 1. Then ps,t (x, y) is a C

m0b -function of x.

Further, there exists cm0 > 0 such that the following holds for all 0 ≤ s < t ≤ T

and x, y ∈ Rd if |i| ≤ m0:

∣∣∣∂ ixps,t (x, y)

∣∣∣ ≤ cm0

(t − s)|i|+d2−α0

. (6.99)

Conditions (6.96) and (6.98) indicate that derivatives of jump-maps φ−1r,z and φr,z

should not be big but moderate, though φr,z can be big. Loosely, the image of a ballby these maps should not be too distorted.

We discuss two lemmas. Let u = {(t1, z1), . . . , (tn, zn)}, where s ≤ t1 <

· · · < tn ≤ t . We write it as u = (t, z), where t = (t1, . . . , tn) andz = (z1, . . . , zn) ∈ (Rd ′

0 )n. We define a probability measure on (Rd ′0 )n by

ν(n)δ (dz) = νδ(dz1) · · · νδ(dzn), where νδ(dz) = 1|z|>δ

λδν(dz) and λδ = ν({|z| > δ}).

We fix δ > 0 stated in the theorem and we set for t ∈ T and x, y ∈ Rd


pts,t (x, y) :=

∫

(Rd′0 )n

p(t,z)s,t (x, y)ν

(n)δ (dz),

where p(t,z)s,t (x, y) is defined by (6.94). Then we get the formula

ps,t (x, y) =∞∑

n=0

E[p

u(q)s,t (x, y)1τn≤t<τn+1

](6.100)

≤ e−λδ(t−s)ps,t (x, y)+∞∑

n=1

(λδ(t − s))n

n! e−λδ(t−s) supt∈(s,t]n+

pts,t (x, y),

where (s, t]n+ is the set of all t = (t1, . . . , tn) such that s < t1 < · · · < tn ≤ t . Wewill show that the above infinite sum is uniformly convergent with respect to x, y

under additional conditions for jump coefficients φt,z(x).We set

P ts,t f (x) =

∫

Rd

pts,t (x, y)f (y) dy.

We define the supremum norm by |f |∞ = supx |f (x)|.Lemma 6.9.1 Assume condition (6.98). For any given nonnegative integer m0,there exists a positive constant C such that for any f ∈ C

m0b , inequalities

sups<t

supt∈(s,t]n+

∣∣∣∂ iP ts,t f

∣∣∣∞ ≤ Cn+1∑

i′⊂i

∣∣∂ i′f∣∣∞, n = 0, 1, 2, . . . (6.101)

hold for any i with |i| ≤ m0.

Proof If i = 0, inequality (6.101) is obvious. We will show the inequality in thecase |i| = 1. We first consider the case n = 0. Since Ps,t f (x) = E[f (Φs,t (x))], wehave

∂Ps,t f (x) = E[∂(f ◦ Φs,t (x))] = E[∂f ◦ Φs,t (x)∇Φs,t (x)].

We know supx,s<t E[|∇Φs,t (x)|p] < ∞ for any p > 1. Therefore we get theinequality supt∈T |∂Ps,t f |∞ ≤ c|∂f |∞.

We next show the inequality for n ≥ 1. We will rewrite P ts,t f as P t

s,t f =∫R

d′0Ps,t1(P

t′t1,t

f ◦ φt1,z1)νδ(dz1), where t′ = (t2, . . . , tn). Then we get from the

above inequality


supx

∣∣∣∂P ts,t f (x)

∣∣∣ ≤ c supx

∫

Rd′0

|∂(P t′t1,t

f ◦ φt1,z1)(x)|νδ(dz1)

≤ c supx

∫

Rd′0

|∂(P t′t1,t

f ) ◦ φt1,z1(x)||∂φt1,z1(x)|νδ(dz1)

≤ c(

supx

∫

Rd′0

|∂φt1,z1(x)|νδ(dz1))

supx

|∂P t′t1,t

f (x)|

≤ c1|∂P t′t1,t

f |∞.

Therefore we obtain |∂P t1,...,tns,t f |∞ ≤ C1|∂P t2,...,tn

t1,tf |∞. Repeating this argument,

we get the inequality of the lemma. ��In the following discussion we will use the short-time estimate of Theorem 6.7.2.

Lemma 6.9.2 If conditions (6.96) and (6.98) are satisfied, for any given nonneg-ative integer m0, there exists a positive constant C such that for any differentialoperator ∂ i

x with |i| ≤ m0, inequalities

supx,y∈Rd ,t∈(s,t]n+

∣∣∣∂ ixp

ts,t (x, y)

∣∣∣ ≤ Cn+1(n+1)βi(t−s)−βi , n = 1, 2, . . . (6.102)

hold for any s < t , where βi = |i|+d2−α0

.

Proof We will consider the case where i ≥ 1. The case i = 0 will be provedsimilarly. We set s = t0 and t = tn+1, conventionally. There exists 0 ≤ k ≤ n suchthat tk+1 − tk ≥ 1

n+1 (t − s). We consider the case 1 ≤ k ≤ n. The case k = 0 willbe treated similarly. We rewrite pt

s,t as

pts,t (x, y) =

∫

Rd

∫

Rd′0

pt1s,tk

(x, xk)pt2tk,t

(φtk,zk (xk), y) dxkνδ(dzk),

where t = (t1, . . . , tn), t1 = (t1, . . . , tk−1) and t2 = (tk+1, . . . , tn). We set f (xk) =∫p

t2tk,t

(φtk,zk (xk), y)νδ(dzk) (y being fixed) and apply the previous lemma. Then wehave

supx

∣∣∣∂ ixp

ts,t (x, y)

∣∣∣ ≤ Ck∑

|i′|=|i|supxk

∣∣∣∫

Rd′0

∂ i′xkp

t2tk,t

(φtk,zk (xk), y)νδ(dzk)

∣∣∣ (6.103)

for any s < t and t ∈ (s, t]n+. Set t3 = (tk+2, . . . , tn). We apply Theorem 6.7.2 forps,t (x, y) in place of pc

s,t (x, y). Setting k + 1 = l, we have


∣∣∣∂ i′xkp

t2tk,t

(φtk,zk (xk), y)

∣∣∣

≤∫

Rd

∫

Rd′0

∣∣∂ i′xkptk,tl (φtk,zk (xk), xl)

∣∣pt3tl ,t

(φtl ,zl (xl), y) dxlνδ(dzl)

≤ C1(tl − tk)−βi

∫

Rd

∫

Rd′0

pt3tl ,t

(φtl ,zl (xl), y) dxlνδ(dzl)


∫

Rd

∫

Rd′0

pt3tl ,t

(x′l , y)| det∇φ−1tl ,zl

(x′l )| dx′l νδ(dzl)


(supx

∫

Rd′0

| det∇φ−1tl ,zl

(x)|νδ(dzl)) ∫

Rd

pt3tl ,t

(x′l , y) dx′l .

In view of condition (6.96), we have suptl ,x

∫ | det∇φ−1tl ,zl

(x)|νδ(dzl) < ∞. Since

suptl ,y

∫p

t3tl ,t

(x′l , y) dx′l < ∞, the above is dominated by C2(tl − tk)−βi . Note (tl −

tk) ≥ 1n+1 (t − s). Then from (6.103), we can conclude that there exists a positive

constant C3 such that

supx,y

|∂ ixp

ts,t (x, y)| ≤ Cn+1

3 (n+ 1)βi(t − s)−βi , n = 1, 2, . . .

holds for any s < t and t ∈ (s, t]n+. ��Next, we give a simple lemma, which will be used in the proof of Theorem 6.9.1.

Lemma 6.9.3 Let D be a domain of Rd and let {fn} be a sequence of functions inCm0(D). Assume that the sequences {fn} and {∂ ifn}, |i| ≤ m0 converge uniformlyto f and f i, respectively. Then f belongs to Cm0(D) and the equality ∂ if = f i

holds for any |i| ≤ m0.

The proof of the lemma is straightforward. It is omitted.

Proof of Theorem 6.9.1 We will prove assertion 2, assuming that the generator isuniformly pseudo-elliptic. Consider

pis,t (x, y) := E[∂ i

xpu(q)s,t (x, y)]

=∞∑

n=0

E[∂ ixp


]. (6.104)

If |i| ≤ m0, it holds by Lemma 6.9.2 that

∣∣E[∂ ixp


]∣∣ (6.105)

≤ supt∈Tn

∣∣∣∂ ixp

ts,t (x, y)

∣∣∣ · (λδ(t − s))n

n! e−λδ(t−s)


≤ (Cλδ(t − s))n(n+ 1)βi

n! e−λδ(t−s)(t − s)−βi .

The infinite sum of the last term of (6.105) with respect to n ∈ N is convergent. Thenthe infinite sum (6.104) is uniformly convergent with respect to x, y. Therefore,the limit function pi

s,t (x, y) is bounded and continuous in x, y for any |i| ≤ m0.Consequently, ps,t (x, y) := p0

s,t (x, y) is the continuous density of Ps,t (x, E).Further, ps,t (x, y) is m0-times continuously differentiable with respect to x

and the equality ∂ ixps,t (x, y) = pi

s,t (x, y) holds for any |i| ≤ m0 in view ofLemma 6.94. Since (6.105) holds, the infinite sum pi

s,t (x, y) satisfies the inequality|pi

s,t (x, y)| ≤ cit−βi .

Other assertions will be shown similarly. ��Theorem 6.9.2 In Theorem 6.9.1, assume further that for some δ > 0, inequality

supt,y

∫

|z|>δ

∣∣∂ jyφ

−1t,z (y)

∣∣pν(dz) < ∞ (6.106)

holds for any 1 ≤ |j| ≤ n0 and p > 1. Then ps,t (x, y) is a Cn0b -function of y.

Further, there exists cn0 > 0 such that if |j| ≤ n0, the following inequality holds forall 0 ≤ s < t ≤ T and x, y ∈ R

d :

∣∣∣∂ jyps,t (x, y)

∣∣∣ ≤ cn0

(t − s)|j|+d2−α0

. (6.107)

Proof We need the dual property of the jump-diffusion with respect to L2(dx) onR

d . We will define the dual of P ts,t f by

Pt,∗s,t g(y) =

∫

Rd

pts,t (x, y)g(x) dx. (6.108)

Then it holds that |P t,∗s,t g|∞ ≤ c∗|g|∞. Further, similarly to Lemma 6.9.1, there

exists a positive constant C such that

supt∈(s,t]n+

∣∣∣∂ jPt,∗s,t g

∣∣∣∞ ≤ Cn+1∑

j′⊂j

∣∣∂ j′g∣∣∞, n = 1, 2, . . . (6.109)

Then we can verify that ps,t (x, y) is a Cn0b -function of y and satisfies the short-time

estimate (6.107) by a similar argument. ��In Theorem 6.6.1, we showed that the density function ps,t (x, y) is a rapidly

decreasing C∞-function of x and y, under Conditions (J.1) and (J.2). Hence jumpsare assumed to be bounded. However, in Theorem 6.9.2, the density functionps,t (x, y) may not be rapidly decreasing, because of big jumps. For example, any

6.10 Density and Fundamental Solution on Subdomain 295

stable process satisfies conditions of the above theorem, and its law has a C∞-density function, but the density function is not rapidly decreasing if it is notGaussian.

Assertions of Theorem 6.9.1 can be extended to the smooth density pc,ds,t (x, y) of

a weighted law Pc,ds,t (x, E) = E[1E(Φs,t (x))Gs,t (x)], if Gs,t (x) is an exponential

functional given by (6.53), where dr,z(x) is assumed to be a bounded positivefunction. For details, see the proof of Theorem 7.5.1 in Sect. 7.5,

Note The existence of the measurable density can be shown without Condition(J.2)K . It means that maps {φt,z, |z| > δ} may not be diffeomorphic for some > 0.In fact, ps,t (x, y) of (6.95) is well defined and is the density function of the transitionprobability, without condition (J.2)K .

However, it seems to be a hard question whether we can remove Condition(J.2)K for the existence of the smooth density. The continuity and the differentia-bility are not clear without the diffeomorphic property of maps φr,z. In Theorem 3.2in Kunita [61], the existence of the smooth density is asserted without (J.2)K , buthis discussion contains gaps.

6.10 Density and Fundamental Solution on Subdomain

In Sect. 4.8, we considered the weighted transition function of a jump-diffusionkilled outside of a subdomain D of Rd . The weighted transition function Q

c,ds,t (x, E)

is equal to

Qc,ds,t (x, E) = E[1E(Φs,t (x))G

c,ds,t (x)1τ(x,s)>t ],

where τ(x, s) is the first leaving time of the process Xt = Xx,st := Φs,t (x), t in

[s,∞) from the set D. We saw in Sects. 6.3 and 6.6 that, if Xt is nondegenerate, theweighted transition function P

c,ds,t (x, E) = E[1E(Φs,t (x))G

c,ds,t (x)] has a smooth

density pc,ds,t (x, y). Since Q

c,ds,t (x, E) ≤ P

c,ds,t (x, E) holds for any s, x, t , the measure

Qc,ds,t (x, E) has always a density q

c,ds,t (x, y) such that qc,d

s,t (x, y) ≤ pc,ds,t (x, y) holds

a.e. y. We want to show its smoothness. Suppose that the jump-diffusion is pseudo-elliptic. If there are no jumps entering Dc from D a.s., it should have a C∞-density.However, if the process has jumps from Dc into D, the result is not simple. Thesmoothness of the density is related to the off-diagonal short-time estimate studiedin Sect. 6.8.

We first consider a diffusion process. Since d = 1, we denote Qc,ds,t (x, E) and

qc,ds,t (x, y) by Qc

s,t (x, E) and qcs,t (x, y), respectively.

Lemma 6.10.1 Suppose that Xx,st := Φs,t (x) is an elliptic diffusion process of

initial state (x, s). Then for any x ∈ D and s < t , Qcs,t (x, E);E ⊂ D has a C∞-

density qcs,t (x, y). It is continuously differentiable with respect to t and satisfies


∂

∂tqcs,t (x, y) = Ac(t)∗yqc

s,t (x, y). (6.110)

Further, the density has the following two properties:

1. Let U be an open subset of Rd such that U ⊂ D. Then for any multi-index j andT > 0, there exists a positive constant c such that the inequality

∣∣∣∂ jyq

cs,t (x, y)

∣∣∣ ≤ c

(t − s)|j|+d

2

, ∀0 ≤ s < t ≤ T (6.111)

holds for any x ∈ D,y ∈ U .2. Let V be an open subset of D satisfying V ⊂ U ⊂ U ⊂ D. Then for any j and

T > 0 there exists a positive constant c′ such that the inequality

|∂ jyq

cs,t (x, y)| ≤ c′, ∀0 ≤ s < t ≤ T (6.112)

holds for all x ∈ Uc ∩D, y ∈ V .

Proof For a given T > 0, we take 0 ≤ s < t ≤ T . Let τ = τ(x, s) be the firstleaving time of the process Xt = X

x,st = Φs,t (x) from the set D. We have by the

strong Markov property

Qcs,t f (x) = P c

s,t f (x)− E[P cτ,t f (Φs,τ (x))G

cs,τ (x)1s<τ<t

]. (6.113)

Set for x ∈ Rd and y ∈ V ,

qcs,t (x, y) := pc

s,t (x, y)− E[pcτ,t (Φs,τ (x), y)G

cs,τ (x)1s<τ<t

]. (6.114)

Then it is the density function of Qcs,t (x, E). Further, for a multi-index j set

qjs,t (x, y) := ∂ j

ypcs,t (x, y)− E

[∂ jyp

cτ,t (Φs,τ (x), y)G

cs,τ (x)1s<τ<t

]. (6.115)

The last integral is well defined and is continuous with respect to t, y. Indeed, sinceΦs,τ (x) ∈ ∂D if τ < t , for any nonnegative integer N , there exists a positiveconstant c such that

|∂ jyp

cτ,t (Φs,τ (x), y)| ≤ c(t − τ)N ≤ c′ < ∞,

in view of Theorem 6.8.1. Then qjs,t (x, y) is well defined and is continuous with

respect to y ∈ U . Since this is valid for any j, qcs,t (x, y) is infinitely continuously

differentiable with respect to y and the equality ∂jyq

cs,t (x, y) = q

js,t (x, y) holds for

any j.


Now, since Qcs,t (x, E ∩ V ) has the smooth density qc

s,t (x, y), y ∈ V for any V

such that V ⊂ D, Qcs,t (x, E ∩ D) has also a smooth density qc

s,t (x, y), y ∈ D forany 0 ≤ s < t ≤ T and x ∈ D. The function qc

s,t (x, y) satisfies

∂

∂t

∫qcs,t (x, y)f (y) dy =

∫qcs,t (x, y)A

c(t)f (y) dy =∫

Ac(t)∗yqcs,t (x, y)f (y) dy

for any f ∈ C0(D) in view of Proposition 4.8.1. Therefore qcs,t (x, y) is differen-

tiable with respect to t and should satisfy the equation (6.110).Note that the last term of (6.115) is bounded. Then ∂

jyq

cs,t (x, y) has the same

short-time estimate as ∂jyp

cs,t (x, y), which proves (6.111) for qc

s,t (x, y).

We will show the second assertion of the lemma. We know that |∂ jyp

cs,t (x, y)|

is bounded with respect to x ∈ Uc ∩ D, y ∈ V and 0 ≤ s < t ≤ T , in viewof Theorem 6.8.1. Further, the last term of equation (6.115) is also bounded withrespect to these x, y, s, t . Consequently, qj

s,t (x, y) is also bounded with respect tothese x, y, s, t . ��

We are also interested in the regularity of the above qcs,t (x, y) with respect to x.

Lemma 6.10.2 Suppose that Xx,st = Φs,t (x) is an elliptic diffusion process of

initial state (x, s). Then the density function qcs,t (x, y) of Qc

s,t (x, E) is a C∞,1-function of x ∈ D and s(< t) for any y, t , and satisfies

∂

∂sqcs,t (x, y) = −Ac(s)xq

cs,t (x, y). (6.116)

Further, the density has the following two properties:

1. Let U be an open subset of Rd such that U ⊂ D. Then for any multi-index i andT > 0, there exists a positive constant c such that the inequality

∣∣∣∂ ixq

cs,t (x, y)

∣∣∣ ≤ c

(t − s)|i|+d

2

, ∀0 ≤ s < t ≤ T (6.117)

holds for any x ∈ U ,y ∈ D.2. Let V be an open subset of D satisfying V ⊂ U ⊂ U ⊂ D. Then for any i and

T > 0 there exists a positive constant c′ such that the inequality

|∂ ixq

cs,t (x, y)| ≤ c′, ∀0 ≤ s < t ≤ T (6.118)


Proof The smoothness of qcs,t (x, y) with respect to x is not clear from the proof

of Lemma 6.10.1, since in the expression (6.114), the stopping time τ = τ(x, s)

depends on x. Instead, we will use the dual property. The dual of P cs,t (x, E) is

written as Pc,∗s,t (y, E) = E[1E(Ψs,t (y))Gc

s,t (y)| det∇Ψs,t (y)|]. Since the generator


of the backward diffusion Ψs,t (y) is A(t) = 12

∑k≥1 Vk(t)

2 − V0(t), it is elliptic.

Then we can apply the same argument to the backward process Ψs,t (y). Wediscussed the dual of the semigroup Qc

s,t in Sect. 4.8. It is given by Qc,∗s,t of (4.86).

Then its transition function Qc,∗s,t (y, E) should have a smooth density q

c,∗s,t (y, x),

x ∈ D.We want to show that qc,∗

s,t (y, x) = qcs,t (x, y) holds for any s < t and x, y ∈ D.

We set Q′s,t f (x) = ∫

qc,∗s,t (y, x)f (y) dy. Then the equality

∫

D

g(x)Q′s,t f (x) dy =

∫

D

f (y)Qc,∗s,t g(y) dy =

∫

D

Qcs,t f (x)g(x) dx

holds for any f, g. Therefore we have Q′s,t f (x) = Qc

s,t f (x) almost everywhere.We show that the equality holds for all x ∈ D. For δ > 0, we have

Qcs−δ,t f (x) = Qc

s−δ,sQcs,t f (x) = Qc

s−δ,sQ′s,t f (x), ∀x ∈ D,

since kernels Qcs−δ,s(x, ·) are absolutely continuous. Let δ tend to 0. Since Q′

s,t f (x)

is a continuous function, we get the equality Qcs,t f (x) = Q′

s,t f (x) for all x ∈ D.Consequently, for any x ∈ D q

c,∗s,t (y, x) = qc

s,t (x, y) holds for almost all y ∈ D.Then, for any x and y we have

qc,∗s,t+δ(y, x) =

∫Q

c,∗t,t+δ(y, dz)q

∗s,t (z, x) dz =

∫Q

c,∗t,t+δ(y, dz)q

cs,t (x, z) dz.

Let δ → 0. Then we obtain qc,∗s,t (y, x) = qc

s,t (x, y) for any s < t and x, y ∈ D.Now, the density function q

c,∗s,t (y, x) of the dual transition function satisfies

assertions of Lemma 6.10.1, interchanging x and y. Then qcs,t (x, y) should satisfy

assertions of the present lemma. ��We will summarize results for the killed diffusion process.

Theorem 6.10.1 Consider a system of elliptic diffusions on Rd with the generator

A(t) of (4.4). Let Qcs,t (x, ·) be the weighted transition function of the diffusion

killed outside of a bounded domain D. For any s < t and x ∈ D, it has a C∞-density qc

s,t (x, y), y ∈ D. The density function is a C∞,1-function of x, s for anyy, t . It satisfies short-time estimates (6.111), (6.112), (6.117), and (6.118). Further,it satisfies the backward equation (6.116).

Corollary 6.10.1 For any bounded continuous function f and t , Qcs,t f (x) is a

C∞,1-function of x, s. Further, it satisfies the backward heat equation

∂

∂sQc

s,t f (x) = −Ac(s)Qcs,t f (x).

As an application of Corollary 6.10.1, we will consider the final value problemof the backward heat equation on a bounded domain D of an Euclidean space.


⎧⎨

⎩

∂

∂sv(x, s) = −Ac(s)v(x, s), x ∈ D, 0 < s < t1

lims→t1 v(x, s) = f1(x), ∀ ∈ D,(6.119)

where Ac(s) is the differential operator defined by (4.11) and f1(x) is a givenbounded continuous function. Let v(x, s), (x, s) ∈ D × (0, t1) be a function ofC

2,1b -class of (x, s). It is called a solution of the final value problem if v(x, s)

satisfies (6.119). For the uniqueness of the solution, we need a stochastic Dirichletboundary condition. A continuous function v(x, t) defined on D × (0, t1) is said tosatisfy the stochastic Dirichlet boundary condition, if for any x ∈ D and t0 ≤ s

∃ limt↑τ(x,s) v(Φs,t (x), t) = 0, if τ(x, s) < t, a.s. P . (6.120)

Let p(x, s; y, t), x, y ∈ D, 0 < s < t < ∞ be a continuous function of x, y, s, t .It is called the fundamental solution of equation (6.119) if it is of C2,1-class withrespect to x, s for any y, t and satisfies ∂

∂sp(x, s; y, t) = −Ac(s)xp(x, s; y, t) and

further, v(x, s) := ∫p(x, s; y, t1)f1(y) dy, x ∈ D, 0 < s < t1 is a solution

of (6.119) satisfying the stochastic Dirichlet boundary condition for any t1 and abounded continuous function f1 on D.

Theorem 6.10.2 Suppose the operator Ac(t) is elliptic. For any bounded contin-uous function f1 on D, the solution of the final value problem of the backwardheat equation associated with the operator Ac(t) on the domain D with stochasticDirichlet boundary condition exists uniquely. It is given by

v(x, s) = E[f1(Φs,t1(x))Gcs,t1

(x)1τ(x,s)>t1]. (6.121)

Let qcs,t (x, y) be the density of Theorem 6.10.1. Then p(x, s; y, t) := qc

s,t (x, y) isits fundamental solution.

Proof We saw in Corollary 6.10.1 that the function (6.121) is a solution of the finalvalue problem. We will check that the above v(x, s) satisfies the stochastic Dirichletboundary condition. By the Markov property, we have for s < t < t1

v(Φs,t (x), t) = E[f1(Φs,t1(x))G

cs,t1

(x)1τ(x,s)≥t1

∣∣∣Ft

].

Let t tend to τ(x, s). Then the above converges to 0 on the set {τ(x, s) < t1}.Therefore, the Dirichlet boundary condition is satisfied.

We want to prove the uniqueness of the solution. Let v′(x, s) be a solution of thefinal value problem (6.119) satisfying the stochastic Dirichlet boundary condition.Apply Itô’s formula to Xt := v′(Φs,t , t)G

cs,t . Then

Xt = v′(x, s)+∫ t

s

Gcs,r

(Ac(r)+ ∂

∂s

)v′(Φs,r , r) dr +Mt,


where Mt is a martingale with mean 0. Since v′(x, s) satisfies equation (6.119), theterm

∫ t

s· · · dr in the above expression is 0. Therefore we find that Xt is actually a

martingale. Consequently, E[Xt1 ] is equal to v′(x, s), i.e., we get

v′(x, s) = E[f1(Φs,t1(x))Gcs,t1

(x)1τ(x,s)≥t1 ].

The uniqueness follows.Finally, qc

s,t (x, y) is the fundamental solution, since it satisfies (6.116). ��Next, the initial value problem for the heat equation associated with the

operator Ac(t) on the domain D is defined in the same way. For this problem,we consider a backward stochastic flow {Φs,t } generated by a backward symmetricSDE with coefficients Vj (x, t), j = 0, . . . , d ′. Let τ (x, t) be the exit time of thebackward process X

x,ts = Φs,t (x) from the domain D. Then Qc

s,t (x, E) defined by

E[1E(Φs,t (x))Gs,t (x)1τ (t,x)<s] has a density qcs,t (x, y), which is a C∞-function of

x and y. The fundamental solution of the heat equation on D with the stochasticDirichlet boundary condition is defined similarly.

The following can be verified similarly to the previous theorem.

Theorem 6.10.3 Suppose that operator Ac(t) is elliptic. For any bounded continu-ous function f0, the solution of the heat equation associated with the operator Ac(t)

on the domain D with stochastic Dirichlet boundary condition exists uniquely. Itis given by u(x, t) = E[f0(Φt0,t (x))G

ct0,t

(x)1τ (t,x)<t0 ]. Further, p(x, t; y, s) :=qcs,t (x, y) is its fundamental solution.

We next consider a pseudo-elliptic jump-diffusion process. We consider again theweighted transition function Q

c,ds,t (x, E). Notations φN (U) etc. in the next lemma

are introduced in Sect. 6.5. Let α0 be any positive constant satisfying α < α0 < 2,where α is the exponent of the Lévy measure.

Lemma 6.10.3 For a given n0 ∈ N, let N be a positive integer such that N >n0+d2−α0

.Suppose that U is a bounded open subset of D such that

d(φN (U), φ−N(Dc)) > 0. (6.122)

Then the measure Qc,ds,t (x, ·) on the set U has a Cn0-density q

c,ds,t (x, y) for any s < t

and x ∈ D. Further, the density has the following two properties:

1. For any j with |j| ≤ n0 and T > 0, there exists c > 0 such that the inequality

∣∣∣∂ jyq

c,ds,t (x, y)

∣∣∣ ≤ c

(t − s)|j|+d2−α0

, ∀0 ≤ s < t ≤ T (6.123)

holds for all x ∈ D, y ∈ U .


2. Let V be an open subset of U such that d(φN (V ), φ−N(Uc)) > 0. Then for anyj and T > 0 there exists c′ > 0 such that the inequality

|∂ jyq

c,ds,t (x, y)| ≤ c′, ∀0 ≤ s < t ≤ T (6.124)


Proof We will repeat an argument similar to the proof of Lemma 6.10.1. Let U bean open subset of D satisfying properties stated in the lemma. Define the functionq

c,ds,t (x, y) by (6.114) replacing pc

s,t (x, y) by pc,ds,t (x, y). It is the density function

of Qc,ds,t (x, E) for any x ∈ R

d and y ∈ U . We can define the function qjs,t (x, y)

by (6.115) replacing pcs,t (x, y) by p

c,ds,t (x, y). Indeed, since Φs,τ (x) ∈ Dc if

τ = τ(x, s) < t , we have by Theorem 6.8.2,

|∂ jyp

c,dτ,t (Φs,τ (x), y)| ≤ c(t − τ)

N− |j|+d2−α0 ≤ c < ∞.

Then qjs,t (x, y) is well defined and is continuous with respect to y ∈ U for any

j with |j| ≤ n0. Therefore, qc,ds,t (x, y) is n0-times continuously differentiable with

respect to y ∈ V and the equality ∂jyq

c,ds,t (x, y) = q

js,t (x, y) holds for |j| ≤ n0.

Since the last term of (6.115) is bounded, ∂ jyq

c,ds,t (x, y) has the same short-time

estimate as ∂jyp

c,ds,t (x, y), which proves (6.123) for qc,d

s,t (x, y).

We show the second assertion of the lemma. We know that |∂ jyp

c,ds,t (x, y)| is

bounded with respect to x ∈ Uc ∩D, y ∈ V and s < t , in view of Theorem 6.8.2.Further, the last term of equation (6.115) is also bounded with respect to thesex, y, t . Consequently, qj

s,t (x, y) is also bounded with respect to these x, y, t . ��Now if jump-maps φt,z map D onto D, diffeomorphically, both φt,z and φ−1

t,z

should map Dc into itself. Therefore we have d(φN (U), φ−N(Dc)) > 0 for any N

if U is a compact subset of D. Therefore Qc,ds,t (x, E) has a C∞-density if x ∈ U .

Next, we will apply the same argument to the dual process. Then we find thatq

c,ds,t (x, y) is a C∞-function of x, similarly to the case of diffusion. Therefore, we

have the following theorem.

Theorem 6.10.4 Consider a system of pseudo-elliptic jump-diffusions on Rd with

the generator AJ (t) of (4.52). Assume that jump-coefficients φt,z map a boundeddomain D ⊂ R

d onto itself for all t, z. Then the process killed outside of D isa jump-diffusion. Its weighted transition function Q

c,ds,t (x, E) has a C∞-density

qc,ds,t (x, y). The density function is a C∞,1-function of x ∈ D, s (< t) for any

y, t . It satisfies the short-time estimates (6.123), (6.124). Further, p(x, s; y, t) =q

c,ds,t (x, y) is the fundamental solution of the backward heat equation with the

stochastic Dirichlet boundary condition on D, associated with the operator Ac,dJ (t).


By the similar argument, we can obtain the fundamental solution for the heatequation on D associated with the integro-differential operator Ac,d(t). Details areomitted.

Note The smoothness of the density function qc,ds,t (x, y) with respect to y in the case

c = 0,d = 1, was shown by Picard–Savona [95] for jump processes.The existence of fundamental solutions for various type of heat equations are

known in analysis and mathematical physics. A well known case is that the operatorA(t) is a Laplacian and domain D of the heat equation is an interval, a squares ora ball. Results for arbitrary operators Ac(t), A

c,dJ (t) and arbitrary domains D might

be new.The stochastic Dirichlet boundary condition should be transformed to the usual

Dirichlet boundary condition if the boundary ∂D is regular, say if it is of C1-class.For the proof, further discussions will be needed.

So far, we showed the existence of the smooth density qcs,t (x, y) for elliptic

diffusions (or pseudo-elliptic jump-diffusions) on a subdomain of an Euclideanspace. We can extend the result for a nondegenerate diffusion (or a nondegeneratejump-diffusion) if its transition density pc

s,t (x, y) satisfies the short time estimatelike (6.81) (or (6.90)). It is expected that a hypo-elliptic diffusion (and pseudo-hypo-elliptic jump diffusions) admits such an estimate, but for the proof furtherdiscussions will be needed.

Chapter 7Stochastic Flows and Their Densitieson Manifolds

Abstract In this chapter, we will study stochastic flows and jump-diffusions onmanifolds determined by SDEs. If the manifold is not compact, SDEs may not becomplete; solutions may explode in finite time. Then solutions could not generatestochastic flow of diffeomorphisms; instead they should define a stochastic flow oflocal diffeomorphisms. These facts will be discussed in Sect. 7.1. In Sect. 7.2, it willbe shown that the stochastic flow defines a jump-diffusion on the manifold. Then,the dual process with respect to a volume element will be discussed. Further, inSect. 7.3, the Lévy process on a Lie group and its dual with respect to the Haarmeasure will be discussed.

In Sect. 7.4, we consider an elliptic diffusion process on a connected manifold.We show the existence of the smooth density with respect to a volume element, bypiecing together smooth densities on local charts which were obtained in Sect. 6.10.The result of the section can be applied to diffusion processes on Euclidean spacewith unbounded coefficients, where the explosion may occur. For a pseudo-ellipticjump-diffusion on a connected manifold, we need additional arguments, sincesample paths may jump from a local chart to other local charts. It will be discussedin Sect. 7.5.

Finally, in Appendix, we collect some basic facts on manifolds and Lie groupsthat are used in this chapter.

7.1 SDE and Stochastic Flow on Manifold

Let M be a C∞-manifold of dimension d. Related notations and terminologies formanifolds are collected in the Appendix at the end of the chapter. If M is notcompact, we consider its one point compactification M ∪ {∞}. If M is compact,∞ is adjoined to M as an isolated point. Given a function f on M , we setf (∞) = 0, conventionally. A vector field V (t) with parameter t ∈ T = [0,∞)

is called a C∞,1-vector field, if V (t)f (x) is a C∞,1-function of (x, t) for any C∞-function f on M . With a local coordinate (x1, . . . , xd), V (t)f (x) is represented by∑

i Vi(x, t)

∂f∂xi

(x), where V 1(x, t), . . . , V d(x, t) are C∞,1-functions.


303


https://doi.org/10.1007/978-981-13-3801-4_7

304 7 Stochastic Flows and Their Densities on Manifolds

Let V0(t), . . . , Vd ′(t) be C∞,1-vector fields on M . Let {φt,z, (t, z) ∈ T×Rd ′ } be

a family of C∞-maps from M into itself, satisfying:

Condition (J.1)’. (x, t, z) → φt,z(x) is a C∞,1,2-map from M × T × Rd ′ to M .

Further, it satisfies φt,0(x) = x for any x, t .Condition (J.2)’. φt,z;M → M are diffeomorphic maps for all t, z. Let φ−1

t,z be

the inverse map. Then (x, t, z) → φ−1t,z (x) is a C∞,1,2-map.

Let z = (z1, . . . , zd′) ∈ R

d ′ . We set

Vk(t)f (x) = ∂zkf (φt,z(x))

∣∣∣z=0

, k = 1, . . . , d ′. (7.1)

Then Vk(t), k = 1, . . . , d ′ are C∞,1-vector fields, called the tangent vector fields ofmaps {φt,z} at z = 0.

We shall define a symmetric stochastic differential equation on manifold M

associated with the above Vk(t) and φt,z. Let {Fs,t , 0 ≤ s < t < ∞} bethe two-sided filtration generated by a d ′-dimensional Wiener process Wt =(W 1

t , . . . ,Wd ′t ), t ∈ T and a Poisson random measure N(dr dz) on T×R

d ′0 with the

Lévy measure ν, which is independent of Wt . We set Ft = F0,t . Let 0 ≤ t0 < ∞.Let τ∞ be a stopping time with values in [t0,∞) ∪ {∞}. Let Xt, t0 ≤ t < τ∞be a cadlag M-valued {Ft }-adapted process such that limt↑τ∞ Xt = ∞ wheneverτ∞ < ∞. The stopping time τ∞ is called the terminal time or the explosion time. LetC∞

0 (M) be the set of all C∞-functions of compact supports. Xt is called a solutionof a symmetric SDE with characteristics (Vk(t), k = 0, 1, . . . , d ′, φt,z, ν), startingfrom X0 at time t0, if f (Xt ), t < τ∞ is a local semi-martingale for any f ∈ C∞

0 (M)

and satisfies

f (Xt )= f (X0 +d ′∑

k=0

∫ t

t0

Vk(r)f (Xr) ◦ dWkr (7.2)

+ limε→0

{∫ t

t0

∫

|z|≥ε

{f (φr,z(Xr−))−f (Xr−)

}N(drdz)−

d ′∑

k=1

bkε

∫ t

t0

Vk(r)f (Xr−)dr}

for t0 < t < τ∞. Here ◦dWkr , k = 1, . . . , d ′ mean symmetric integrals by Wiener

processes Wkr , k = 1, . . . , d ′ and ◦dW 0

r means the usual integral dr . Further,bkε = ∫

ε≤|z|≤1 zkν(dz). The terminal time τ∞ depends on the initial condition of

the process Xt . We denote it by τ∞(X0, t0).Let us consider the relation with SDE on Euclidean space Rd defined in Chap. 3.

If Xt is a solution of equation (3.10) on Rd , it satisfies (7.2) for any smooth function

f by setting φt,z(z) = g(x, t, z) + x, as it was shown in (3.69). Conversely, ifa process Xt on R

d satisfies (7.2) for any smooth function f , Xt satisfies (3.10).However, in this chapter, we do not assume the boundedness of coefficientsVk(x, t), g(x, t, z) := φt,z(x)− x and their derivatives in equation (7.2). Hence the

7.1 SDE and Stochastic Flow on Manifold 305

explosion may occur for the solution in the Euclidean space. Hence our SDE (7.2)may be regarded as a generalization of SDE (3.10).

Theorem 7.1.1 For the symmetric SDE (7.2) on the manifold M , we assume thatVk(t), k = 0, . . . , d ′ are C∞,1-vector fields and {φt,z} satisfies Condition (J.1)’.Then, given x ∈ M and 0 ≤ s < ∞, the equation (7.2) has a unique solutionX

x,st (ω), t < τ∞(x, s, ω), starting from x at time s. Further, for any s < t ,

the solution has a modification Φs,t (x, ω), t < τ∞(x, s, ω) such that there isa measurable subset Ωs,t of Ω with P(Ωs,t ) = 1 and for any ω ∈ Ωs,t thefollowing properties hold. The terminal time τ∞(x, s, ω) is lower semi-continuouswith respect to x for any s, so that the set Ds,t (ω) := {x; τ∞(x, s, ω) > t} are opensubsets of M . Further, maps Φs,t (ω);Ds,t (ω) → M are C∞.

The family of pairs {Φs,t ,Ds,t } satisfying properties of the above theorem iscalled a stochastic flow of C∞-maps on the manifold M .

Proof We will construct the solution of equation (7.2) by piecing together solutionsdefined on local charts of the manifold M . Let us first choose a family of local chartsUα ⊂ Dα, α = 1, 2, . . . satisfying the following:

(i)⋃

α Uα = M .(ii) The number of α such that x ∈ Uα is at most finite for any x.

(iii) For any α, Dα is compact and satisfies Uα ⊂ Dα .

We will consider SDE on Dα . Let ψα be a diffeomorphism from Dα to an open set inR

d (local coordinate). Then the SDE is transformed to an SDE on ψα(Dα) (⊂ Rd).

Set Vk(t) = dψα(Vk(t)), Vk(t) = dψα(Vk(t)) and g(x, t, z) = ψα(φt,z(x)) − x ifφt,z(x) ∈ Dα , where x = ψα(x). The functions {Vk(x, t), k = 0, . . . , d ′, g(x, t, z)}are defined on the set ψα(Dα). Since the set is relatively compact in R

d , thesefunctions are extended to R

d as C∞,1b -functions and C

∞,1,2b -functions, respectively.

Consider an SDE on Rd :

Xt = x +d ′∑

k=0

∫ t

s

Vk(Xr , r) ◦ dWkr (7.3)

+ limε→0

{ ∫ t

s

∫

|z|≥ε


k=1

bkε

∫ t

s

Vk(Xr−, r) dr}.

It generates a stochastic flow {Φs,t } of C∞-maps on Rd (Theorem 3.4.4).

Given x ∈ M and 0 ≤ s < ∞, we will define an M-valued process as follows. Ifx ∈ Uα , we set

Φ(α)s,t (x) = ψ−1

α (Φs,t (x)), if t < τDα (x, s) := inf{t > s; Φs,t (x) /∈ ψα(Dα)}.


Then, by the pathwise uniqueness of the solution, it holds that Φ(α)s,t (x) = Φ

(β)s,t (x)

if x ∈ Uα ∩ Uβ and t < τDα (x, s) ∧ τDβ (x, s). Now set

τ1(x, s) := min{α;x∈Uα}

τDα (x, s). (7.4)

Then τ1(x, s) − s is strictly positive for any x, s a.s. We can define the solutionof (7.2) up to time τ1(x, s) as follows. For t < τ1(x, s), take α such that τUα (x, s) >

t . Then we define Φs,t (x) by Φ(α)s,t (x). Then the process Φs,t (x), t < τ1(x, s) is

indeed a solution of SDE (7.2) on the manifold M .We will prolong the solution for t ≥ τ1 = τ1(x, s). Define an R

d ′0 -valued random

variable by S1=∫ ∫

{(τ1,z);z∈Rd′0 } zN(dr dz). Set Φs,τ1 = φτ1,S1 ◦Φs,τ1− and define

Φs,t (x) = Φτ1,t (Φs,τ1(x)), if τ1(x, s) ≤ t < τ2(x, s) := τ1(Φs,τ1(x), τ1).

Then it satisfies SDE (7.2) for s < t < τ2(x, s). We will continue this procedure.Then we can define Φs,t (x) for s < t < τ∞(x, s) := limk→∞ τk(x, s), whereτk(x, s) = τ1(Φs,τk−1(x), τk−1). It satisfies SDE (7.2) for s < t < τ∞(x, s).

The above τ∞(x, s) is a terminal time. Indeed, suppose τ∞(x, s) < ∞. Thenthe sequence of M-valued random variables {Φs,τn−(x), n = 1, 2, . . .} shouldconverges to a point y ∈ M or diverges to ∞. But the former case cannot occur,because of the definition of stopping times τn(x, s). Therefore, the latter case occursonly, proving that τ∞(x, s) is the terminal time.

The functional Φs,t (x), s < t < τ∞(x, s) constructed above has the flowproperty: There exists Ω1 ⊂ Ω with P(Ω1) = 1 such that for any fixed ω ∈ Ω1,Φs,t (x, ω) satisfies

Φr,t (Φs,r (x, ω), ω) = Φs,t (x, ω), s < ∀r < t < τ∞(x, s, ω).

Set Ds,t (ω) = {x ∈ M; τ∞(x, s, ω) > t}. Then Φs,t (ω) is a map from Ds,t (ω) to M .We will prove that it is a C∞-map for almost all ω. Take any x0 ∈ Ds,t and considerthe sample path Φs,t (x0, ω). There is Ω2 ⊂ Ω with P(Ω2) = 1 such that for fixedω ∈ Ω2, we may choose s = t0 < t1 < · · · < tn = t and local charts Uαi

, i =1, . . . , n from the family {Uα} such that the trajectory {Φs,r (x0, ω); r ∈ [ti−1, ti)}is included in Uαi

for any i = 1, . . . , n. Then there is an open neighborhood U0 ofx0 (depending on ω) such that

⋃

x∈U0

{Φs,r (x, ω); r ∈ [ti−1, ti)} ⊂ Dαi, i = 1, . . . , n.

If ω ∈ Ω1 ∩ Ω2, Φs,t (ω) is a C∞-map, since the flow property

Φs,t (ω) = Φtn−1,t (ω) ◦Φtn−1,tn−2(ω) ◦ · · · ◦Φs,t1(ω)


holds and the right hand side is composites of C∞-maps. Therefore Φs,t (ω) is aC∞-map on U0 if ω ∈ Ω1 ∩ Ω2. Further, U0 ⊂ Ds,t (ω) and hence Ds,t (ω) is anopen set. We have thus shown the assertion of the theorem. ��

Now, for the symmetric SDE (7.2), we will assume further that jump-maps φt,z

satisfy Condition (J.2)’. We shall study the diffeomorphic property of the flow. Weneed the backward symmetric SDE on the manifold M . Let 0 < t1 < ∞. An {Fs,t1}-adapted M-valued backward caglad process Xs, t1 ≥ s > τ∞ is called a solutionof a backward symmetric SDE with characteristics (−Vk(t), k = 0, . . . , d ′, φ−1

t,z , ν)

starting from x at time t1 if f (Xs), s ∈ (τ∞, t1] is a backward local semi-martingalefor any f ∈ C∞

0 (M) and satisfies

f (Xs) = f (x)−d ′∑

k=0

∫ t1

s

Vk(r)f (Xr ) ◦ dWkr (7.5)

+ limε→0

{∫ t1

s

∫

|z|≥ε

{f (φ−1r,z (Xr+))−f (Xr+)}N(dr dz)+

d ′∑

k=1

bkε

∫ t1

s

Vk(r)f (Xr ) dr},

for t1 > s > τ∞. Here ◦dWkr , k = 1, . . . , d ′ mean backward symmetric integrals by

processes W kt = Wk

t −WkT , t ∈ T, where Wt = (W 1

t , . . . ,Wd ′t ), t ∈ T is a Wiener

process and ◦dW 0r means the usual integral dr . Further, Vk(t), k = 1, . . . , d ′ are

tangent vector fields of {φt,z} at z = 0. τ∞ is the (backward) terminal time ofthe backward process Xs ; it means that it is a backward stopping time with valuesin [0, t1] ∪ {−∞} such that limt↓τ∞ Xt = ∞ holds if τ∞ > 0. The equation

has a unique backward solution, which we denote by Xx,t1s (ω), s > τ∞. The

backward terminal time is denoted by τ∞(x, t1, ω). Then for any 0 < s < t , thebackward solution has a modification Ψs,t (x, ω), s > τ∞(x, t, ω) such that for anyω ∈ Ωs,t ⊂ Ω with P(Ωs,t ) = 1, the set

Ds,t (ω) = {x ∈ M; τ∞(x, t, ω) < s} (7.6)

is open and maps Ψs,t (ω); Ds,t (ω) → M are C∞-maps. {Ψs,t (x, ω), Ds,t (ω)}is called the backward stochastic flow of C∞-maps generated by the backwardsymmetric SDE (7.5).

We first consider a continuous SDE.

Lemma 7.1.1 Let {Φs,t ,Ds,t } be the continuous stochastic flow of C∞-maps on themanifold M defined by the continuous symmetric SDE with coefficients Vk(x, t), k =0, . . . , d ′. Let {Ψs,t , Ds,t } be the continuous backward stochastic flow of C∞-mapsdefined by a continuous symmetric backward SDE with coefficients −Vk(x, t), k =0, . . . , d ′. Then we have the following:


1. For any 0 < s < t , Φs,t are C∞-maps from Ds,t to Ds,t a.s. Further, these areone to one and onto and inverse maps are C∞ a.s.

2. For any 0 < s < t < T , it holds a.s. that Ψs,t (Φs,t (x)) = x if x ∈ Ds,t andΦs,t (Ψs,t (x)) = x if x ∈ Ds,t .

Proof Let Vk(x, t) be vector fields on the Euclidean space Rd defined in the proof

of the previous theorem. Let {Φs,t } be the forward flow on Rd generated by SDE

with coefficients Vk(x, t), k = 0, . . . , d ′ and let {Ψs,t } be the backward flow on Rd

generated by the backward SDE with coefficients −Vk(x, t), k = 0, . . . , d ′. Sincethese coefficients are of C

∞,1b -class, the explosion time of Φs,t (x) is ∞ for any

x, s a.s. and the explosion time of the backward flow Ψs,t (x) is −∞ for any t, x

a.s. as we have seen in Chap. 3. Further, we saw in Sect. 3.7 that Φs,t (Ψs,t (x)) =Ψs,t (Φs,t (x)) = x holds for all 0 < s < t and x ∈ R

d , a.s.Let Uα be a local chart of M . We transform Φs,t and Ψs,t by the inverse map ψ−1

α .Set Φs,t = ψ−1

α (Φs,t ) and Ψs,t = ψ−1α (Ψs,t ). Then if x ∈ Uα and τUα (x, s) > t ,

we have τ∞(Φs,t (x), t) < s and Ψs,t (Φs,t (x)) = x holds. The same fact holds ifx ∈ M satisfies τ∞(x, s) > t . Repeating this argument inductively, we find that

τ∞(x, s) > t &⇒ τ∞(Φs,t (x), t) < s and Ψs,t (Φs,t (x)) = x.

Consequently, Φs,t maps Ds,t into Ds,t , a.s.Next, apply a similar argument to Ψs,t instead of Φs,t . Then we find that Ψs,t

maps Ds,t into Ds,t and the equality Φs,t (Ψs,t (x)) = x holds for x ∈ Ds,t .Consequently, the map Φs,t : Ds,t → Ds,t is one to one and onto, and further,the inverse is a C∞-map. See the discussion at the beginning of Sect. 3.7. Thereforewe get the assertion of the lemma. ��

We next consider the case with jumps. Let Vk(t) be tangent vector fields of {φ−1t,z }

at z = 0. Then it holds that Vk(t)f = −Vk(t)f, k = 1, . . . , d ′, where Vk(t)f aretangent vector fields of {φt,z} at z = 0.

Lemma 7.1.2 Let {Φs,t ,Ds,t } be the stochastic flow of C∞-maps on the manifoldM defined by symmetric SDE (7.2). Let {Ψs,t , Ds,t } be the backward stochastic flowof C∞-maps on the manifold M defined by the backward symmetric SDE (7.5). Thenassertions 1 and 2 of Lemma 7.1.1 are valid.

Before the proof, we prepare another lemma.

Lemma 7.1.3 Let (Uα,ψα) be a local chart of M . For x ∈ Uα , set x = ψα(x),which is an element of Rd . For maps φt,z(x) satisfying Condition (J.1)’, there existδ0 > 0 and g(x, t, z) such that φt,z(x) := g(x, t, z)+ x are diffeomorphisms on R

d

for |z| < δ0, t ∈ T and ψα(φt,z(x)) = φt,z(x) holds for x ∈ Uα .


Proof Choose a sufficiently small δ0 > 0 and a function g(x, t, z), x ∈ Rd , |z| < δ0

such that for any z the equality g(x, t, z) = ψα(φt,z(x)) − x holds if x ∈ ψα(Uα)

and |∂x g(x, t, z)| < 1/2 holds for any x ∈ Rd . Then the map φt,z;Rd → R

d isone to one for any t, z, since |∂x g(x, t, z)| ≤ 1/2 for any x, t, z. We will show thatmaps φt,z are onto for any z. Let Rd be the one point compactification of Rd . Itis homeomorphic to the d-dimensional unit sphere. We set φt,z(∞) = ∞. Sincelimx→∞ φt,z(x) = ∞ holds, the extended map φt,z(x); {|z| ≤ δ0} × R

d → Rd is

continuous. Further, φt,0 is the identity. Then the map φt,z is an onto map for anyt, z by the homotopy theory. This indicates that the map φt,z;Rd → R

d is also anonto map for any z. We have thus seen that φt,z are diffeomorphisms for any t and|z| < δ0. ��Proof of Lemma 7.1.2 Let δ0 be the positive number of Lemma 7.1.3 and let0 < δ < δ0. We consider a forward SDE with jumps on the manifold M withcharacteristics (Vk(x, t), k = 0, . . . , d ′, φt,z(x), νδ), where the associated Poissonrandom measure is Nδ(dr dz) = 1(0,δ](z)N(dr dz) and its Lévy measure νδ is givenby νδ(dz) = 1(0,δ](|z|)ν(dz). The forward stochastic flow generated by the SDE isdenoted by {Φδ

s,t ,Dδs,t }. Further, we consider a backward SDE with jumps with

characteristics (−Vk(x, t), k = 0, . . . , d ′, φ−1t,z , νδ). The backward stochastic flow

generated by the backward SDE is denoted by {Ψ δs,t , Dδ

s,t }.The forward equation on M is transformed to a forward equation on R

d withcharacteristics (Vk(x, t), k = 0, . . . , d ′, g(x, t, z), νδ), where g(x, t, z) is the func-tion defined in Lemma 7.1.3. The backward SDE on M is transformed to a backwardequation on R

d with characteristics (−Vk(x, t), k = 0, . . . , d ′,−h(x, t, z), νδ),where h(x, t, z) = x− φ−1

t,z (x). Let Φδs,t be the flow on R

d generated by the forward

SDE and let Ψ δs,t be the backward flow on R

d generated by the backward SDE.

Since φt,z(x) = g(x, t, z)+ x are diffeomorphic and φ−1t,z (x) = h(x, t, z)+ x, these

flows satisfy Φδs,t (Ψ

δs,t (x)) = Ψ δ

s,t (Φδs,t (x)) = x on the Euclidean space. Further,

it holds that Φδs,t (x) = ψ−1

α (Φδs,t (x)) for t < τδ

Dα(x, s) (the first leaving time of

Xt = Xx,s,δt = Φδ

s,t (x) from Dα) and Ψ δs,t (x) = ψ−1

α (Ψ δs,t (x)) for s < σδ

Dα(x, t)

(the backward first leaving time of Xx,t,δs = Ψ δ

s,t (x) from Dα). Therefore we have

Φδs,t (Ψ

δs,t (x)) = Ψ δ

s,t (Φδs,t (x)) = x for x ∈ Dα if t < τδ

Dα(x, s). Then we can

prolong the equality Φδs,t (Ψ

δs,t (x)) = Ψ δ

s,t (Φδs,t (x)) = x for x ∈ M and t < τ1(x, s).

Now the stochastic flow Φs,t (x) on the manifold M is decomposed as

Φs,t (x) ={Φδ

s,t (x), if t < τ1

Φδτn,t

◦ φτn,Sn · · ·φτ1,S1 ◦Φδs,τ1

(x), if τn ≤ t < τn+1,(7.7)

where τn are jumping times of the Poisson process Nδt = N((s, t] × {|z| > δ})

and S1, . . . , Sn are random variables given by Si =∫ ∫

{(τi ,z);|z|>δ} zN(dr dz), i =1, . . . , n. Further the backward flow Ψs,t (x) is represented by


Ψs,t (x) ={Ψ δs,t (x), if t < τ1,

Ψ δs,τ1

◦ φ−1τ1,S1

· · ·φ−1τn,Sn

◦ Ψ δτn,t

(x), if τn ≤ t < τn+1.(7.8)

Therefore we get Φs,t (Ψs,t (x)) = Ψs,t (Φs,t (x)) = x if t < τDα (x, s). We canprolong the equality Φs,t (Ψs,t (x)) = Ψs,t (Φs,t (x)) = x for any x and t < τ1(x, s).Repeating this argument, the equality holds for any x ∈ M and t < τ∞(x, s).Finally, the equality implies that maps Φs,t ;Ds,t → Ds,t are one to one and ontoa.s. and further, the inverse maps are C∞. Hence we established the lemma. ��

From the above lemmas, we get the following assertion.

Theorem 7.1.2 Assume that Vk(t), k = 0, . . . , d ′ are C∞,1-vector fields and {φt,z}satisfy Conditions (J.1)’ and (J.2)’ for the symmetric SDE (7.2) on the manifold M .Then maps Φs,t ;Ds,t → M of Theorem 7.1.1 are diffeomorphic a.s. for any s < t .

Let {Ψs,t , Ds,t } be the backward flow defined by SDE (7.5) on the manifold M

with characteristics (−Vk(t), k = 0, . . . , d ′, φ−1t,z , ν). Then Φs,t are maps from Ds,t

onto Ds,t . Further, it holds that Ψs,t (x) = Φ−1s,t (x) if x ∈ Ds,t .

The family of pairs {Φs,t ,Ds,t } satisfying properties of the above theorem iscalled a stochastic flow of local diffeomorphisms on the manifold M . SDE (7.2)is called complete if τ∞(x, s) = ∞ holds a.s. for any x, s. Further, it is calledstrongly complete if infy∈M τ∞(x, s) = ∞ holds a.s for any s. It is equivalent tothat Ds,t = M holds a.s. for any s < t .

Corollary 7.1.1 Assume that SDE (7.2) on the manifold M is strongly complete.Then Ds,t = M holds a.s. and {Φs,t } satisfies the same properties stated inTheorem 3.9.1, replacing the state space R

d by the manifold M .

Proposition 7.1.1 If the manifold M is compact and SDE (7.2) is time homoge-neous, the SDE is strongly complete.

Proof Let us consider the stopping time τ1(x, s) defined by (7.4). We define astopping time by σ1(s) = infx∈M τ1(x, s). Then σ1(s) > s a.s. since M is compact.Further, set σn(s) = σ1(σn−1(s)) for n ≥ 2. Then it holds that σn(s) ≤ τn(x, s) forall x. Set σ∞(s) = limn→∞ σn(s). We want to prove σ∞(s) = ∞ for any s a.s. Thisshould imply that the SDE is strongly complete.

Consider a sequence of random variables σn+1(s) − σn(s), n = 1, 2, . . .. Notethat σn+1(s)− σn(s) = σ1(σn(s))− σn(s). Since σ1(t)− t is Ft,∞-measurable and1σn(s)≤t is F0,t -measurable, these two random variables are independent. Therefore,for any bounded continuous function f we have

E[f (σn+1(s)− σn(s))|Fσn] = E[f (σ1(t)− t)]|t=σn(s) = E[f (σ1(0))],

because laws of σ1(t) − t are common for all t . This proves that σn+1(s) − σn(s)

and F0,σn are independent.

7.2 Diffusion, Jump-Diffusion and Their Duals on Manifold 311

The above discussion shows that σn+1(s)− σn(s), n = 1, 2, . . . are independentand identically distributed. Since

σN(s)− s =N−1∑

n=0

{σn+1(s)− σn(s)}, (σ0(s) = s)

(σN(s) − s)/N converges to a positive constant, by law of the large numbers. Thisproves σ∞(s) = ∞ a.s for all s > 0. Finally, σN(s) is right continuous in s for anyN . Then σ∞(s) = ∞ holds for all s a.s. ��Remark Let N be a connected component of the manifold M . Then jump-mapsφt,z maps N onto itself diffeomorphically. Indeed, since φt,0 are identity maps andφt,z(x) are continuous with respect to t, z, x, we have φt,z(N) ⊂ N . Then thestochastic flow Φs,t should also map N onto itself diffeomorphically a.s. Hence it issufficient that we consider SDE and stochastic flow on each connected components.

We will relax Conditions (J.1)’ and (J.2)’ to the following:

Condition (J.1)′K . (i) The map (x, t, z) → φt,z(x) is of C∞,1,2-class on the spaceM×T×{|z| ≤ c} for some c > 0. (ii) It is of C∞-class on M for any t ∈ T, |z| >c and is piecewise continuous in (t, z) ∈ T× {|z| > c}.

Condition (J.2)′K . For any t, z, the map φr,z;M → M is a diffeomorphism.Further, the inverse maps φ−1

t,z satisfy Condition (J.1)′K .

Then the assertion of Theorem 7.1.2 is valid. In this case, the stochastic flow Φs,t

may jump from a connected component N to other connected components.

Note The continuous SDE on manifolds was studied by Ikeda–Watanabe [41]. Ourdefinition of the SDE on manifold is close to their definition.

We saw in Chap. 3 that the SDE on a Euclidean space is strongly complete, ifcoefficients Vk(x, t) are of C∞,1

b -class and g(x, t, z) are C∞,2,1b -class. It is expected

to get criteria for vector fields Vk(t) and jump-maps φt,z under which an SDE isstrongly complete. For continuous SDEs or diffusion processes, we refer to Xue-Mei Li [76].

The SDE with jumps on manifolds was studied by Fujiwara [29] and Kunita[60]. The present definition of the SDE is taken from [60]. For the constructionof stochastic flows on a compact manifold, Fujiwara [29] applied the Whitneyembedding theorem of a manifold into a higher-dimensional Euclidean space.Assumptions required in the paper are different from ours.

7.2 Diffusion, Jump-Diffusion and Their Duals on Manifold

In Sect. 7.1, we studied a symmetric SDE on a C∞-manifold M of dimension d. Inthe sequel, we assume that the Lévy measure has a weak drift. Then the last term(jump part) of (7.2) is split into the difference of two terms:


∫ t

t0

∫

|z|>0+

{f (φr,z(Xr−))−f (Xr−)

}N(dr dz)−

∫ t

t0

d ′∑

k=1

bk0Vk(r)f (Xr) dr,

where the first term is the improper integral by the Poisson random measure N andbk0 = limε→0 b

kε . Then rewriting V0(t) − ∑

k bk0Vk(t) as V0(t), SDE (7.2) can be

rewritten simply as

f (Xt )=f (X0)+d ′∑

k=0

∫ t

t0

Vk(r)f (Xr) ◦ dWkr

+∫ t

t0

∫

|z|>0+{f (φr,z(Xr−))−f (Xr−)

}N(dr dz). (7.9)

We define an integro-differential operator AJ (t) by

AJ (t)f = 1

2

d ′∑

k=1

Vk(t)2f + V0(t)f +

∫

|z|>0+{f ◦ φt,z−f }ν(dz). (7.10)

Then, using the Itô integral, Eq. (7.9) is rewritten as

f (Xt )=f (X0)+∫ t

t0

AJ (r)f (Xr) dr +d ′∑

k=1

∫ t

t0

Vk(r)f (Xr) dWkr

+∫ t

t0

∫

|z|>0+{f (φr,z(Xr−))−f (Xr−)

}N(dr dz). (7.11)

It is called a symmetric SDE with characteristics (Vk(t), k = 0, . . . , d ′, φt,z, ν).We will define stochastic flow generated by the above SDE slightly different from

that defined in Sect. 7.1. Let M ′ = M ∪{∞}, where ∞ is a cemetery adjoined to themanifold M as a one-point compactification if M is noncompact, and as an isolatedpoint if M is compact. For a given function f on M , we extend it to a function onM ∪ {∞} by setting f (∞) = 0.

Let Xx,st , t < τ∞(x, s) be the solution starting from x at time s, where τ∞(x, s)

is the terminal time. For t ≥ τ∞(x, s), we set Xx,st = ∞. Then X

x,st , t ∈ [s,∞)

is an M ∪ {∞}-valued process and satisfies the above equation (7.11) for all t > s.Theorems 7.1.1 and 7.1.2 tell us that Xx,s

t , t > s has a modification {Φs,t (x), t > s}satisfying the following properties:

1. The terminal time τ∞(x, s) is lower semi-continuous with respect to x for any s.2. Set Ds,t = {x; τ∞(x, s) > t}. These are open subsets of M a.s., for any s < t .

Further, maps Φs,t ;Ds,t → M are C∞ a.s.3. If jump-maps φt,z satisfy Condition (J.2)’ or identity maps, then maps Φs,t :

D → M are into diffeomorphisms a.s.


We set Φs,t (∞) = ∞ for any t ≥ s. Then Φs,t are maps from M ∪ {∞}into itself and satisfy Φt,u(Φs,t (x)) = Φs,u(x) for any x ∈ M ∪ {∞} ands < t < u. The family of maps {Φs,t , s < t} is called again the stochasticflow of local diffeomorphisms generated by a forward SDE with characteristics(Vk(t), k = 0, . . . , d ′, φt,z, ν).

Now, since Φs,t (x) and Φt,u(x) are independent for any s < t < u, the stochasticprocess X

x,st = Φs,t (x) is a Markov process of initial state (x, s) with transition

function

Ps,t (x, E) = P(Φs,t (x) ∈ E) = P(Φs,t (x) ∈ E, τ∞(x, s) > t)

for x ∈ M and E ⊂ M . The Markov process Xx,st is called a jump-diffusion

process on the manifold M with characteristics (Vk(t), k = 0, . . . , d ′, φt,z, ν).For f ∈ Cb(M), we set Ps,tf (x) := ∫

MPs,t (x, dy)f (y). It is written as

Ps,tf (x) = E[f (Φs,t (x))] = E[f (Φs,t (x))1τ∞(x,s)>t ]. Then it is a boundedcontinuous function of x. Hence {Ps,t } is a semigroup of linear transformations ofCb(M).

The above Markov process Xx,st = Φs,t (x), t ∈ [s,∞) satisfies equation (7.11)

with the initial condition X0 = x and t0 = s. Take expectations for each termof (7.11). Expectations of the third and the fourth terms of the right-hand side are 0,since these are martingales. Consequently, we get

E[f (Φs,t (x))] = f (x)+ E[ ∫ t

s

AJ (r)f (Φs,r (x)) dr].

Therefore we have Ps,tf (x) = f (x)+ ∫ t

sPs,rAJ (r)f (x) dr for any f ∈ C∞

0 (M).This shows that the semigroup {Ps,t } satisfies Kolmogorov’s forward equation withrespect to the operator AJ (t).

If φt,z are identity maps for any t, z in equation (7.9), the Markov processXt is called a diffusion process on the manifold M with coefficients (Vk(t), k =0, . . . , d ′). In this case, we define a differential operator A(t) by

A(t)f (x) = 1

2

d ′∑

k=1

Vk(t)2f (x)+ V0(t)f (x). (7.12)

Then the semigroup satisfies Ps,tf (x) = f (x) + ∫ t

sPs,rA(r)f (x) dr for any f ∈

C∞0 (M). Hence the semigroup {Ps,t } satisfies Kolmogorov’s forward equation with

respect to the operator A(t).It is expected that if f is in C∞

0 (M), Ps,tf (x) should be smooth with respect tox and should satisfy Kolmogorov’s backward equation. If the associated stochasticflow is strongly complete, these fact is true, but if it is not strongly complete, thesefacts are not obvious, since it is not evident whether Ps,tf (x) is smooth or not withrespect to x. Then it is not clear whether the (backward) heat equation associated


the operator A(t) or AJ (t) has a solution. In Sect. 7.4, we will show these factsfor elliptic diffusion, by constructing the fundamental solution for the differentialoperator A(t).

Let us consider the dual processes, assuming that the manifold M is orientable.We will restrict our attention to jump-diffusions. A similar result for diffusions willbe obtained with the obvious change. From now, we assume further that jump-mapsφt,z satisfy Condition (J.2)’ in Sect. 7.1. Hence these maps are diffeomorphic. Letdx be a given volume element of the manifold. We will study the dual of the abovejump-diffusion with respect to dx. An operator AJ (t)

∗;C∞0 (M) → C∞(M) is

called a dual of the operator AJ (t) with respect to dx if it satisfies

∫

M

AJ (t)f (x) · g(x) dx =∫

M

f (x) · AJ (t)∗g(x) dx

for any f, g ∈ C∞0 (M). It is computed directly, using formulas of the integration

by parts and the change of variables. Indeed, similarly to the dual in the Euclideanspace (Sect. 4.6), it is written for any g ∈ C∞

0 (M) as

AJ (t)∗g = 1

2

d ′∑

k=1

(Vk(t)+ divVk(t))2g − (V0(t)+ divV0(t))g (7.13)

+∫

|z|>0+

{|J

φ−1t,z|g ◦ φ−1

t,z − g}ν(dz).

Here, divV is the divergence of the vector field V with respect to the volumeelement dx (Appendix). Further, for a diffeomorphic map φ;M → M , Jφ is definedas a C∞-function such that the pullback φ∗ dx coincides with Jφ dx. It coincideswith the Jacobian determinant of φ if M is a Euclidean space and dx is the Lebesguemeasure. We call Jφ the Jacobian of φ.

Let Ψs,t (x), s > τ∞(x, t) be the backward flow of local diffeomorphismsgenerated by the backward SDE on the manifold M .

f (Xs) = f (x)−d ′∑

k=0

∫ t

s

Vk(r)f (Xr )◦dWkr +

∫ t

s

∫

|z|>0+{f (φ−1

r,z (Xr ))−f (Xr )}N(dr dz)

for t > s > τ∞ for any f ∈ C∞0 (M). Let Φ−1

s,t be the inverse map of the flow

Φs,t . We saw in Sect. 7.1 that Φ−1s,t (x) = Ψs,t (x) holds for any t < τ∞(x, s). For

s < τ∞(t, x), we set Ψs,t (x) = ∞ and Ψs,t (∞) = ∞ for any 0 < s < t . ThenXs = Ψs,t (x) is a backward Markov process. Its generator is given by

AJ (t)g = 1

2

d ′∑

k=1

Vk(t)2g − V0(t)g +

∫

|z|>0+{g ◦ φ−1

t,z − g}ν(dz). (7.14)


Then the dual operator AJ (t)∗ is obtained by the transformation of the operator

AJ (t) as

AJ (t)∗g = AJ (t)

−div V,|Jφ−1 |g. (7.15)

For the construction of the dual process, we need an additional condition.Condition (D) (Dual condition)

(i) divVk(t), Vk(t)(divVk(t)), k = 0, . . . , d ′ are bounded C∞,1-functions on thespace M × T.

(ii) The improper integral∫|z|>0+

{|Jφ−1t,z| − 1

}ν(dz) is a bounded function of x, t

and is smooth with respect to x.

The above condition is always satisfied if the manifold M is compact.In the case where the manifold M is an Euclidean space and dx is the

Lebesgue measure, Condition (D) tells us that coefficients Vk(x, t), φt,z(x)−x =g(x, t, z) of the SDE might be unbounded, but divVk(x, t), Vk(t)(divVk(t))(x) and∫|z|>0(| det∇φt,z(x)| − 1)ν(dz) should be bounded.

The function c∗(t) = c∗(x, t) = AJ (t)∗1(x) is written as

c∗(t) = −divV0(t)+ 1

2

d ′∑

k=1

{Vk(t)(divVk(t))+ (divVk(t))2}

+∫

|z|>0+(|J

φ−1t,z| − 1)ν(dz). (7.16)

Then the dual condition makes sure that the function c∗(x, t) is a bounded functionon M × T. Then we can construct the dual jump-diffusion with generator (7.13).

We apply the change-of-variable formula (A.4) in the Appendix for opensubmanifolds Ds,t ≡ {x; τ∞(x, s) > t}. Replace f (x) by f (Φs,t (x))g(x) and φ

by Ψs,t . Note that Ψs,t are diffeomorphic maps from Ds,t ≡ {x; τ∞(x, t) < s} toDs,t by Theorem 7.1.2, and the equality Φs,t (Ψs,t (x)) = x holds. Then we get theformula of the change of variables:

∫

Ds,t

f (Φs,t (x))g(x) dx =∫

Ds,t

f (x)g(Ψs,t (x))|JΨs,t(x)| dx, a.s. P ,

for any functions f, g ∈ C0(M). On the set Dcs,t , we have f (Φs,t (x)) = f (∞) = 0,

and on the set Dcs,t we have g(Ψs,t (x)) = g(∞) = 0. If τ∞(x, t) > s, we set

JΨs,t

(x) = 0. Then we get the equality

∫

M

f (Φs,t (x))g(x) dx =∫

M

f (x)g(Ψs,t (x))|JΨs,t(x)| dx, a.s. P . (7.17)


Lemma 7.2.1 |JΨs,t

|, τ∞(x, t) < s < t is written as a backward exponentialfunctional with coefficients −div V = (−divV0(x, t), . . . ,−divVd ′(x, t)) and|Jφ−1 | = (|J

φ−1t,z

(x)|):

exp{ d ′∑

k=0

∫ t

s

−divVk(t)(Ψr,t ) ◦dWkr +

∫ t

s

∫

|z|>0+log |J

φ−1r,z

(Ψr,t )| dN}. (7.18)

Further, under condition (D), E[|JΨs,t

(x)|] is bounded with respect to s < t and x.

Proof We showed the above lemma for the stochastic flow on a Euclidean space(Proposition 4.6.2). What we need in the proof are the following three facts:

1. Differential rule for the flow {Φs,t } with respect to the backward variable s.2. Change-of-variable formula.3. Formula of integration by parts.

We know that the first one holds for stochastic flows on manifold. The secondand the third are known as we have mentioned above. Therefore, discussions in theproof of Lemma 4.6.1 are valid to the present case and we get the expression (7.18).

Further, (7.18) is written as the product of three backward exponential functionalsG

(0)s,t , G

(1)s,t and G

(2)s,t as in Sect. 4.6. Since c∗(x, t) is a bounded function, we find that

E[|JΨs,t

(x)|1τ∞(x,t)<s] is bounded. ��In view of the above lemma,

P ∗s,t g(x) := E

[g(Ψs,t (x))|JΨs,t

(x)|] (7.19)

is well defined for any g ∈ C0(M). It is a linear transformation from C0(M) toCb(M). Taking expectations for both terms of (7.14), we have

∫

M


M

E[f (Φs,t (x))g(x)

]dx =

∫

M

f (x) · P ∗s,t g(x) dx,

for any f, g ∈ C∞0 (M). Therefore P ∗

s,t is the dual of Ps,t with respect to dx.

Theorem 7.2.1 Assume that jump-maps φt,z in SDE (7.9) on the orientablemanifold M satisfy (J.1)’, (J.2)’ and Condition (D). Let {Ps,t } be the semigroupof the stochastic flow {Φs,t } defined by the SDE (7.9). Then the semigroup {Ps,t } hasthe dual semigroup {P ∗

s,t } with respect to a given volume element dx. It coincides

with the backward semigroup of the inverse flow {Ψs,t } transformed by the backwardexponential functional with coefficients −div V and |Jφ−1 |.

We will apply the above theorem to the case where M is a Euclidean space.Suppose that coefficients Vk(t), k = 0, . . . , d ′ of the operator AJ (t) of (4.52)are C∞-class. These may be unbounded and may have unbounded derivatives.However, adjoint operator AJ (t)

∗;C∞0 (Rd) → C∞

0 (Rd) is well defined. It is given

7.3 Brownian Motion, Lévy Process and Their Duals on Lie Group 317

by (4.68). Further, if Condition (D) is satisfied, then the semigroup of the jump-diffusion has a dual semigroup {P ∗

s,t }.

7.3 Brownian Motion, Lévy Process and Their Duals on LieGroup

Let Xt ; t ∈ [0,∞) be a cadlag process with values in a Lie group G, continuousin probability, and X0 = e. It is said to have (left) independent increments if G-valued random variables X−1

tm−1Xtm,m = 1, . . . , n are independent for any 0 ≤

t0 < · · · < tn. It is (left) time homogeneous if the law of X−1s Xt coincides with

the law of Xt−s for any s < t . If Xt is (left) time homogeneous and has (left)independent increments, it is called a Lévy process on the Lie group G. Further, if itis a continuous process, it is called a Brownian motion on the Lie group G. Relatednotations and terminologies for Lie groups are collected in the Appendix at the endof this chapter.

We shall construct a Brownian motion on a Lie group G by solving an SDE onG. Let V0, V1, . . . , Vd ′ be left-invariant vector fields on the Lie group G. Regardingthat G is a C∞-manifold, we consider a continuous SDE on G with coefficientsVk, k = 0, . . . , d ′. A solution Xt starting from X0 at time t0 should satisfy

f (Xt ) = f (X0)+d ′∑

k=0

∫ t

t0

Vkf (Xr) ◦ dWkr , (7.20)

for any f ∈ C∞0 (G). We denote the solution by X

X0,t0t . The solution can be defined

up to the explosion time τ∞ = τ∞(X0, t0).Let us consider a solution starting from the unit element e at time 0. We denote

the solution by Xt, t < τ∞, simply. We want to show τ∞ = ∞ a.s. Let U be anopen neighborhood of the origin e of G. We can define a sequence of stopping timesσn such that σn < τ∞ a.s. for any n and satisfies

σ1 = inf{t > 0;Xt /∈ U}, . . . , σn = inf{t > σn−1;X−1σn−1

Xt /∈ U}. (7.21)

Lemma 7.3.1 The sequence of random variables σ1, σ2 − σ1, . . . , σn − σn−1 isindependent and identically distributed.

Proof Set G′ = G∪{∞} and g ·∞ = ∞ for any g ∈ G. We set Xt = ∞ if t > τ∞.Then for a function f on G′ such that f (∞) = 0 and is C∞ on G, we have

f (Xt ) = f (Xs∧σ1)+d ′∑

k=0

∫ t

s∧σ1

Vkf (Xr) ◦ dWkr .


Since Vk are left-invariant vector fields, we have

f (xXt ) = f (xXs∧σ1)+d ′∑

k=0

∫ t

s∧σ1

Vkf (xXr) ◦ dWkr

for any x ∈ G. Setting x = X−1s∧σ1

, we get

f (X−1s∧σ1

Xt) = f (e)+d ′∑

k=0

∫ t

s∧σ1

Vkf (X−1s∧σ1

Xr) ◦ dWkr .

Consequently, X−1s∧σ1

Xt coincides with the solution Xe,s∧σ1t of equation (7.20), in

view of the pathwise uniqueness of the solution. This means that, for any s and σ1,stochastic processes {Xt, t ≤ s ∧ σ1} and {X−1

s∧σ1Xt−s∧σ1 , s ∧ σ1 < t < 2s ∧ σ2}

are independent and have the same law. Then random variables σ1 and σ2 − σ1are independent and identically distributed. Repeating this argument, we find thatσ1, . . . , σn − σn−1 are independent and identically distributed. ��

Now, the expectation of σ1 is positive. Then by the law of large numbers,limn→∞ σn = ∞ holds a.s. Therefore we have τ∞ = ∞ a.s. Further, since thesolution Xt is defined for all t , we have the formula

f (X−1s Xt ) = f (e)+

d ′∑

k=0

∫ t

s

Vkf (X−1s Xr) ◦ dWk

r . (7.22)

Then X−1s Xt is independent of Fs for any s < t . It shows that Xt has left

independent increments. Therefore we get the following theorem.

Theorem 7.3.1 Any continuous SDE on a Lie group G with left-invariant vectorfields Vk, k = 0, . . . , d ′ is strongly complete. The solution is a time homogeneousdiffusion on G with the generator Af = 1

2

∑d ′k=1 V

2k f +V0f. Let Xt be the solution

starting from the unit element e ∈ G at time 0. Then it is a Brownian motion on theLie group G. Further, Φs,t (x) := xX−1

s Xt is the stochastic flow of diffeomorphisms.

We will consider the dual of Xt with respect to a left Haar measure m. Letexp sVk, s ∈ (−∞,∞) be the one-parameter subgroup of G generated by the left-invariant vector field Vk . Then the modular function Δ(exp sVk) is a one-parametersubgroup of the multiplicative group (0,∞). We set ck = d

dsΔ(exp sVk)|s=0 for

k = 0, . . . , d ′. These constants satisfy

∫Vkf (x)g(x) dm = −

∫f (x)Vkg(x) dm− ck

∫f (x)g(x) dm

for functions f, g ∈ C∞0 (M). Therefore, the constant ck coincides with divVk . Then

the dual of A is represented by

7.3 Brownian Motion, Lévy Process and Their Duals on Lie Group 319

A∗g(x) = 1

2

d ′∑

k=1

(Vk + ck)2g(x)− (V0 + c0)g(x).

In particular, if the Lie group G is unimodular, then ck = 0 holds for all k. Thereforethe dual is represented by A∗f = 1

2

∑k≥1 V

2k f−V0f. Hence the dual is a backward

Brownian motion.We shall next consider an SDE with jumps. We assume that the Lévy measure

of the Poisson random measure has a weak drift. Let Vk, k = 0, . . . , d ′ be left-invariant vector fields on the Lie group G. Let φ(z) be a smooth map R

d ′ → G

such that φ(0) = e. We consider an SDE on G with characteristics (Vk, k =0, . . . , d ′, Rφ(z), ν), where Rφ(z) is the right translation. Let Xt, t < τ∞ be thesolution starting from e at time 0, where τ∞ is the terminal time. It satisfies, for anysmooth function f and 0 ≤ s < t < τ∞,

f (Xt ) = f (Xs)+d ′∑

k=0

∫ t

s

Vkf (Xr) ◦ dWkr (7.23)

+∫ t

s

∫

|z|>0+{f (Xr−φ(z))− f (Xr−)}N(dr dz).

For the process Xt , we define a sequence of stopping times σ1, σ2, . . . by (7.21).Then the assertion of Lemma 7.3.1 is valid. Therefore τ∞ = ∞ holds a.s.

We will show that Xt has independent increments. Since Vk are left-invariantvector fields, we have

f (xXt ) = f (xXs)+d ′∑

k=0

∫ t

s

Vkf (xXr) ◦ dWkr

+∫ t

s

∫

|z|>0+{f (xXr−φ(z))−f (xXr−)}N(dr dz).

The above equality holds for x = X−1s . Then we find that X−1

s Xt is a solution ofthe SDE with characteristics (Vk, k = 0, . . . , d ′, Rφ(z), ν) starting from e at time s.Then Xs and X−1

s Xt are independent. This proves that Xt is a Lévy process on theLie group G.

Theorem 7.3.2 SDE (7.23) on a Lie group G with left-invariant vector fieldsVk, k = 0, . . . , d ′ and right translation Rφ(z) is strongly complete.

Let Xt be the solution starting from the unit element e ∈ G at time 0. Then it isa Lévy process on the Lie group G. Further, Φs,t (x) := xX−1

s Xt is the stochasticflow of diffeomorphisms generated by the SDE (7.23). Its generator is given by


AJf (x) = 1

2

d ′∑

k=1

V 2k f (x)+V0f (x)+

∫

|z|>0+

{f (xφ(z))− f (x)

}ν(dz). (7.24)

We will consider the dual of the operator AJ with respect to the left Haar measurem. Consider the transformation Rφ(z). Regarding m as a positive d-form, we definedthe function JRφ(z)

by the relation R∗φ(z) dm = JRφ(z)

dm in Sect. 4.8. Since m is aHaar measure, we have R∗

φ(z) dm = Δ(φ(z)) dm. Therefore JRφ(z)(x) is a constant

function of x and we have JRφ(z)(x) = Δ(φ(z)). If

∫

|z|>δ

Δ(φ(z))ν(dz) < ∞ (7.25)

holds for some δ > 0, the dual of AJ is well defined. It is written as

A∗J g(x) =

1

2

d ′∑

k=1

(Vk + ck)2g(x)− (V0 + c0)g(x)

+∫

|z|>0+

{Δ(φ(z))g(xφ(z)−1)− g(x)

}ν(dz).

For a unimodular Lie group, it holds that ck = 0 and Δ(φ(z)) = 1. Then theoperator A∗

J is written simply as

A∗J g(x) =

1

2

d ′∑

k=1

V 2k g(x)− V0g(x)+

∫

|z|>0+

{g(xφ(z)−1)− g(x)

}ν(dz).

Denote the inverse X−1s of the Lévy process Xs by Xs . Then Xs is a backward

jump-diffusion with the generator AJ given by (7.14). It coincides with the aboveA∗

J . Thus if the Lie group G is unimodular, the inverse process Xs coincides withthe dual process with respect to the left Haar measure m.

Note It was shown by Itô [49] that a stationary Brownian motion on a Lie groupis a stationary diffusion with the generator A represented by Af = ∑

k akXkf +

12

∑i,k a

ikXiXkf, where {Xi} is a basis of L(G) and (aik) is a nonnegative definitesymmetric matrix. By the method of the diagonalization, the above operator isrewritten as Af = V0f + 1

2

∑k≥1 V

2k f , where V0, . . . , Vd ′ are elements of L(G).

Therefore all Brownian motions on a Lie group can be obtained by solving SDEs ofthe form (7.20).

A Lévy process on a Lie group is studied in Hunt [39]. He obtained its generatorexplicitly, using left-invariant vector fields and a Lévy measure on the Lie group.

7.4 Smooth Density for Diffusion on Manifold 321

7.4 Smooth Density for Diffusion on Manifold

We will consider a diffusion on the orientable manifold M determined by SDE

f (Xt ) = f (X0)+d ′∑

k=0

∫ t

t0

Vk(r)f (Xr) ◦ dWkr , (7.26)

where f are C∞-functions on M . Let {Φs,t (x), t < τ∞(x, s)} be the stochasticflow generated by SDE (7.26), where τ∞(x, s) is the explosion time of the processX

x,st := Φs,t (x). We set X

x,st = Φs,t (x) = ∞ if t > τ∞(x, s). Then X

x,st

is a (not necessarily conservative) diffusion process on M . If the tangent vectorsVk(x, t), k = 1, . . . , d ′ span the tangent space Tx(M) for any x ∈ M and t , theSDE is called elliptic and the above X

x,st is called an elliptic diffusion.

Let ck(x, t), k = 0, . . . , d ′ be C∞,1-functions on M×T. We assume that ck(x, t)and Vk(t)ck(x, t), k = 0, . . . , d ′ are bounded functions. Define the differentialoperator Ac(t) by

Ac(t)f (x) = 1

2

d ′∑

k=1

(Vk(t)+ ck(t))2f (x)+ (V0(t)+ c0(t))f (x). (7.27)

Its potential part c(t) = Ac(t)1 is a bounded function, since it is equal to

c(x, t) = c0(x, t)+ 1

2

{ d ′∑

k=1

Vk(t)ck(x, t)+d ′∑

k=1

ck(x, t)2}.

We set ck(∞, t) = 0 for k = 0, . . . , d ′ and define the exponential functionalGs,t (x) by


{ d ′∑

k=0

∫ t

s


}. (7.28)

It satisfies sups<t,x E[|Gs,t (x)|p] < ∞ for any p > 1 (see the proof ofLemma 4.2.1).

Let f be a bounded continuous function on M . We set f (∞) = 0. Thenthe semigroup P c

s,t f (x) = E[f (Φs,t (x))Gcs,t (x)] satisfies Kolmogorov’s forward

equation associated with the differential operator Ac(t) defined by (7.27).We will fix a volume element of M and will denote it by dx, dy etc. We will

study the smooth density of the weighted transition function of an elliptic diffusiondefined by P c

s,t (x, E) = E[1E(Φs,t (x))Gcs,t (x)], with respect to dy.

For a positive integer i, diφ is the i-th differential of φ and for i = 0, d0φ ≡ φ.When differentials are operated to the variable x of the C∞(M × M)-function


φ(x, y), these are denoted by dixφ and their localization at the point (x, y) are

denoted by dixφ(x, y). An objective of this section is to prove the following.

Theorem 7.4.1 Consider an elliptic diffusion on the orientable manifold M deter-mined by SDE (7.26). Assume that ck(x, t), Vk(t)ck(x, t) are bounded C∞,1-functions. Then, for any 0 ≤ s < t and x ∈ M , the transition function P c

s,t (x, E)

weighted by Gcs,t (x) of (7.28) has a C∞-density pc

s,t (x, y), y ∈ M with respect tothe volume element. Further, for any compact subset K of M , nonnegative integer jand T > 0, there exists a positive constant c such that

supx,y∈K

∣∣∣djy p

cs,t (x, y)

∣∣∣ ≤ c

(t − s)j+d

2

, 0 ≤ s < t ≤ T . (7.29)

Assume further Condition (D) (i). Then the density function is a C∞,1-functionof x, s (< t) for any y, t . Further, for any compact subset K of M , nonnegativeinteger i and T > 0, there exists a positive constant c′ such that

supx,y∈K

∣∣∣dixp

cs,t (x, y)

∣∣∣ ≤ c′

(t − s)i+d

2

, 0 ≤ s < t ≤ T . (7.30)

We will first show the existence of the smooth density on each local chart of M .Let V,U,D be a triple of local charts such that V ⊂ U ⊂ U ⊂ D and D is compact.Then D is diffeomorphic to a bounded open set D of R

d by a diffeomorphismψ : D → D. SDE (7.26) on D is transformed to an SDE on D by the diffeomorphicmap ψ . Indeed, vector fields Vk(t) on the local chart D are transformed to vectorfields dψ(Vk(t)) on D by the map ψ . Since D is relatively compact in R

d , we canextend them to C

∞,1b -vector fields Vk(x, t) on the whole space R

d such that theseare uniformly elliptic. We consider an equation on R

d :

dXt =d ′∑

k=0

Vk(Xt , t) ◦ dWkt . (7.31)

We denote the solution starting from x at time s by Φs,t (x). Let ck(x, t) be C∞,1b -

functions on Rd such that ψ(ck(x, t)) = ck(ψ(x), t) holds on the local chart D. We

define a positive functional Gs,t (x) similarly to Gs,t (x), making use of functionsck, k = 0, . . . , d ′ on R

d and Rd -valued process Φs,t (x).

Let τ (x, s) be the first leaving time of the process Xx,st = Φs,t (x) from D. We

shall consider the law of the killed process given by

Qcs,t (x, E) = E

[1E(Φs,t (x))Gs,t (x)1τ (x,s)>t

]. (7.32)

We can apply Lemma 6.10.1 for the killed process. Let U = ψ(U). Then the closureof U in R

d is included in the open set D. Then the measure Qcs,t (x, ·) restricted to


U has a C∞-density qcs,t (x, y), y ∈ U with respect to the Lebesgue measure by

Lemma 6.10.1. It satisfies (6.111) and (6.112).For a given T > 0, take 0 ≤ s < t ≤ T . Let Φs,t (x) be the solution on the

manifold M . We shall consider its killed process at Dc. Let τ(x, s) be the firstleaving time of the process X

x,st = Φs,t (x) from D. Then the weighted law

Qcs,t (x, E) = E

[1E(Φs,t (x))G

cs,t (x)1τ(x,s)>t

]

has also a C∞-density qcs,t (x, y), y ∈ U with respect to the volume element dy,

since Φs,t (x) = ψ−1(Φs,t (ψ(x))) holds for t < τ(x, s) a.s. P. Further, since qcs,t

satisfies (6.111) and (6.112), for any j, i there are positive constants c, c′ such thatqcs,t satisfies

supx∈D,y∈U

∣∣∣djy q

cs,t (x, y)

∣∣∣ ≤ c

(t − s)j+d

2

, (7.33)

supx∈Uc∩D,y∈V

∣∣∣diyq

cs,t (x, y)

∣∣∣ ≤ c′ (7.34)

for all 0 ≤ s < t ≤ T .We want to show that for any x ∈ M the weighted transition function P c

s,t (x, E)

has a C∞-density with respect to the volume element dy. This will be done bypiecing together the above density functions qc

s,t (x, y) of the killed process. Forthis purpose, we will fix a pair (U,D) of the above local charts. For the processXt = X

x,st := Φs,t (x) we define stopping times τm = τm(x, s),m = 0, 1, 2, . . .

and σm = σm(x, s),m = 1, 2, . . . by induction as τ0 = s and

σm = inf{t ≥ τm−1; Xt ∈ Dc} (= ∞ if {· · · } is empty),

τm = inf{t ≥ σm; Xt ∈ U} (= ∞ if {· · · } is empty). (7.35)

Then we have for E ⊂ U ,

E[1E(Φs,t (x))Gcs,t (x)] =

∞∑

m=0

E[1E(Φs,t (x))G

cs,t (x)1τm<t≤σm+1

].

Using the strong Markov property, we can rewrite the above equality as

P cs,t (x, E) =

∞∑

m=0

E[Qc

τm,t (Φs,τm(x), E)Gcs,τm

(x)1τm<t

]. (7.36)

Now, Qcs,t (x, E ∩ U) has a C∞-density function qc

s,t (x, y), y ∈ U for any s, t, x ∈D. We set qc

s,t (x, y) ≡ 0 if x ∈ Dc. Consider the finite sum for x ∈ M and y ∈ V :


p(n)s,t (x, y) := qc

s,t (x, y)+n∑

m=1

E[qcτm,t (Φs,τm(x), y)G

cs,τm

(x)1τm<t

]. (7.37)

We want to show that the above converges to a function pcs,t (x, y) uniformly with

respect to x ∈ M and y ∈ V . We need three lemmas.

Lemma 7.4.1 Let Φs,t (x) be the solution of equation (7.31) on the Euclidean spaceR

d . Let Ba(x) = {y; |y − x| ≤ a} and σ(x, s) be the hitting time of the processX

x,st = Φs,t (x) to Ba(x)

c. Then for any s < T , there exists 0 < c < 1 such that

supx∈Rd

P (σ (x, s) < T ) ≤ c. (7.38)

Proof Let f be a C2-function on Rd such that f (x) = |x|2 if |x| ≤ 1 and f (x) =

|x| if |x| ≥ 2. By Itô’s formula, we have f (Φs,t (x) − x) = Mt + At , where Mt

is a continuous martingale with Ms = 0 and At is continuous process of boundedvariation with As = 0; these are represented by

Mt =d ′∑

k=1

∫ t

s

Vk(r)f (Φs,r (x)) dWkr ,

At =∫ t

s

A(r)f (Φs,r (x)) dr,

where A(r) is the generator. Therefore 〈M〉t =∫ t

s

∑k Vk(r)f (Φs,r (x))

2 dr . Since∑k Vk(r)f (Φs,r (x))

2 is bounded and strictly positive a.s., 〈M〉t is a strictly increas-ing continuous function of t . Further, the inverse function 〈M〉−1

t is also strictlyincreasing and continuous. Consider the time-changed process Bt = M〈M〉−1

t. It

is a continuous martingale with respect to the filtration F〈M〉t , in view of Doob’soptional stopping theorem. Its quadratic variation 〈B〉t is equal to t . Therefore Bt isa Wiener process (Proposition 2.2.1).

Let c1, c2 ≥ 1 be constants such that

d ′∑

k=1

Vk(r)f (Φs,r (x))2 ≤ c1, |A(r)f (Φs,r (x))| ≤ c2

hold for all x, s < r < T a.s. Then we have

{σ(x, s) < T } ⊂ {Mt > a − At for some s < t < T }⊂ {Bt > a − c2t for some s < t < c1T }.

Therefore,


supx,s

P (σ (x, s) < T ) ≤ P(Bt > a − c2t for some t < c1T ).

Since Bt is a Wiener process, the last probability denoted by c is less than 1. Thisproves the assertion of the lemma. ��Lemma 7.4.2 Let U,D be a pair of local charts satisfying U ⊂ D. Let τm =τm(x, s) be stopping times defined by (7.35). Then there exists a positive constant0 < c < 1 such that

sups<t,x∈U

P (τm(x, s) < t) ≤ cm (7.39)

holds for any positive integer m.

Proof Let σ(x, s) be the hitting time of the process Xx,st = Φs,t (x) to the set Dc,

where d(ψ(U),ψ(D)c) ≥ a > 0. Then, τ1(x, s) ≥ σ(x, s) a.s. Therefore, we haveby Lemma 7.4.1

supx∈U

P (τ1(x, s) < t) ≤ supx∈U

P (σ (x, s) < t) ≤ c.

Then we have P(τl − τl−1 < t) ≤ c < 1. Further, since τm = ∑ml=1(τl − τl−1),

we have {τm < t} ⊂ ⋂ml=1{τl − τl−1 < t}. Therefore, using the strong Markov

property, we have

P(τm < t) ≤ P(

m⋂

l=1

{τl − τl−1 < t})

≤ E[P(τm − τm−1 < t |Fτm−1);

m−1⋂

l=1

{τl − τl−1 < t}]

≤ E[P(τ1(y) < t)

∣∣y=Φτm−1 (x)

;m−1⋂

l=1

{τl − τl−1 < t}]

≤ cP (

m−1⋂

l=1

{τl − τl−1 < t})

≤ c2P(

m−2⋂

l=1

{τl − τl−1 < t})

≤ cm,

proving (7.39). ��


Lemma 7.4.3 The sequence {p(n)s,t (x, y)} of functions defined by (7.37) converges

uniformly with respect to x ∈ M,y ∈ V . Let pcs,t (x, y) be its limit. It is a C∞-

function of y and it is the density of transition function P cs,t (x, E ∩ V ), so that it

does not depend on the choice of local charts U,D. Furthermore, for any j ∈ N

and T > 0, there exists a constant cj > 0 such that

supx∈D,y∈V

∣∣∣djy p

cs,t (x, y)

∣∣∣ ≤ cj

(t − s)j+d

2

, ∀0 ≤ s < t ≤ T . (7.40)

Proof Since Φs,τm(x) ∈ ∂U , qcτm,t (Φs,τm(x), y) is uniformly bounded with respect

to y ∈ V by (7.34). Further, E[|Gcs,τm

(x)|p] is bounded uniformly with respect tox, s and m for any p > 1. Consequently, there is a positive constant K such that

E[qcτm,t (Φs,τm(x), y)G

cs,τm

(x)1τm<t

]

≤ E[(

qcτm,t (Φs,τm(x), y)G

cs,τm

(x))2] 1

2P(τm < t)

12 ≤ Kc

m2

holds for all x ∈ M,y ∈ V and m ∈ N, where 0 < c < 1 is the positive constantof Lemma 7.4.2. Therefore p

(n)s,t (x, y) of (7.37) converges uniformly with respect to

x ∈ M,y ∈ V . The limit function pcs,t (x, y) is the density function of P c

s,t (x, E∩V )

in view of (7.36). Hence pcs,t (x, y) does not depend on the choice of U,D.

Now, differentiating each term of (7.37) with respect to y, we get

djy p

(n)s,t (x, y) = d

jy q

cs,t (x, y)+

n∑

m=1

E[djy q

cτm,t (Φs,τm(x), y)G

cs,τm

(x)1τm<t

]

(7.41)for any j ∈ N. Then d

jy p

(n)s,t (x, y) converges uniformly to a function p

js,t (x, y) with

respect to x ∈ M and y ∈ V , similarly to the convergence of p(n)s,t (x, y). Then

pcs,t (x, y) is a C∞-function of y ∈ V and d

jy p

cs,t (x, y) = p

js,t (x, y) holds.

djy p

cs,t (x, y) has the short-time estimate (7.40), since d

jy q

cs,t (x, y) satisfies the

inequality (7.33) and the term∑n

m=1(· · · ) in (7.41) is bounded with respect to x, y

and n. ��Proof of Theorem 7.4.1 We saw the existence of the density pc

s,t (x, y) and itssmoothness with respect to y ∈ V and its short-time estimate (7.40) in Lemma 7.4.3.The function pc

s,t (x, y) can be extended to x, y ∈ M , since M is covered by acountable number of the above local charts V . It is a C∞-function of y. Further,since the compact set K is covered by finite numbers of local charts V , the short-time estimate (7.29) holds.

We apply the same argument to the dual transition function, assuming the dualcondition. We saw in Sect. 7.2 that the dual transition function of P c

s,t (x, E) is given

by Pc,∗s,t (y, F ) = E[1F (Ψs,t (y))Gc

s,t (y)|JΨs,t(y)|], where Ψs,t (y) is the backward

flow generated by a continuous backward SDE with coefficients −Vk(x, t), k =


0, . . . , d ′. It is an elliptic backward diffusion. Further, Gcs,t (y) = Gc

s,t (Ψs,t (y)) and

JΨs,t

(y) is the Jacobian of the map Ψs,t on the manifold (see Sect. 7.2). Then the dual

transition function Pc,∗s,t (y, F ) has a smooth density p

c,∗s,t (y, x) for any y. It should

coincide with pcs,t (x, y) for any x, y (see the proof of Lemma 6.10.2). Consequently,

the density pcs,t (x, y) is smooth with respect to x. It satisfies (7.30). ��

Corollary 7.4.1 Assume Condition (D)(i).

1. The semigroup {P cs,t } maps C0(M) to C∞(M) ∩ Cb(M). Further, P c

s,t f satisfiesKolmogorov’s backward equation if f ∈ C0(M).

2. If f1 ∈ C0(M), then v(x, s) := ∫M

pcs,t1

(x, y)f1(y) dy is a C∞,1-function of x, sand is a solution of the final value problem of the backward heat equation on themanifold associated with the differential operator Ac(t).

Proof The dual semigroup {P c,∗s,t } satisfies P

c,∗s,t g = g + ∫ t

sP

c,∗r,t Ac(r)∗g dr . Then

we can show that the density function pcs,t (x, y) satisfies

∂

∂spcs,t (x, y) = −Ac(s)xp

cs,t (x, y),

similarly to the proof of (6.116). Let f ∈ C0(M). Then the function v(x, s) :=∫pcs,t (x, y)f (y) dy is a bounded C∞,1-function and satisfies

Ac(s)v(x, s)=∫Ac(s)pc

s,t (x, y)f (y) dy=−∫

∂

∂spcs,t (x, y)f (y) dy=− ∂

∂sv(x, s),

since we can change the order of differentials and the integrals for v(x, s).Therefore the semigroup {P c

s,t } satisfies Kolmogorov’s backward equation. Thesecond assertion will be obvious. ��

Let us discuss briefly the backward SDE with coefficients Vk(x, t), k = 0, . . . , d ′on the manifold M . Let {Φs,t } be the backward flow defined by the backward SDE.Then P c

s,t (x, E) := E[1E(Φs,t (x))Gcs,t (x)] has a density pc

s,t (x, y), which is a C∞-function of x and y. Further, for any f0 ∈ C0(M), u(x, t) := ∫

Mpct0,t

(x, y)f0(y) dy

is a C∞,1-function and it is a solution of the initial value problem of the heatequation on the manifold M associated with the differential operator Ac(t).

In Sect. 6.3, we showed the existence of the smooth density of transitionfunction of an elliptic diffusion on R

d , if its coefficients Vk(t), k = 0, . . . , d ′ areC∞,1b -functions. The fact can be extended to elliptic diffusions with unbounded

coefficients, if we apply Theorem 7.4.1 to the case where the manifold M is anEuclidean space Rd . Indeed, if coefficients Vk(t) are (not necessarily bounded) C∞-functions, the transition function has a C∞-density.

It seems that assertions of Theorem 7.4.1 except short-time estimates (7.29)and (7.30) should be valid for hypo-elliptic diffusions on a manifold. However, forthe proof, short-time estimates of the density function like Theorem 6.7.1 will beneeded for hypo-elliptic diffusions on Euclidean space.


Finally, we will apply Theorem 7.4.1 to a Brownian motion on a Lie group.

Theorem 7.4.2 Let Xt, t ∈ [0,∞) be a Brownian motion on a Lie group G withthe generator Af (x) = 1

2

∑d ′k=1 V

2k f (x) + V0f (x), where Vk; k = 0, . . . , d ′ are

left-invariant vector fields. Suppose that Vk; k = 1, . . . , d ′ spans L(G). Then thelaw μt of Xt has a C∞-density μt(y) with respect to the Haar measure m. Further,the transition probability has a C∞-density and it is given by pt (x, y) = μt(x

−1y).For any compact subset K , a positive integer j and T > 0, there exists a positive

constant cj such that the following holds for all 0 < t < T :

supy∈K

∣∣∣djμt (y)

∣∣∣ ≤ cj

tj+d

2

. (7.42)

Note For the existence of the smooth density of the law of a continuous SDE on amanifold, Taniguchi [112] applied the Whitney embedding theorem of a manifoldinto a higher-dimensional Euclidean space. Assumptions required in the paper aredifferent from ours.

The existence of the smooth density for a diffusion with the explosion seems tobe new.

7.5 Density for Jump-Diffusion on Compact Manifold

We will consider a jump-diffusion on the compact and orientable manifold M

defined by SDE (7.9). The equation generates a stochastic flow of diffeomor-phisms {Φs,t (x)}. See Sect. 7.2. We define vector fields Vk(t), k = 1, . . . , d ′ byVk(t)f (x) = ∂

∂zkf (φt,z(x))|z=0 as before. Let Vk(x, t) and Vk(x, t) be coefficients

of vector fields Vk(t) and Vk(t) when these are represented by a local coordinate.Set V (x, t) = (V1(x, t), . . . , Vd ′(x, t)) and V (x, t) = (V1(x, t), . . . , Vd ′(x, t)). LetΓ0 be a lower bound of matrices {Γε} given by (1.21). If, at the tangent space TxM ,V (x, t)V (x, t)T + V (x, t)Γ0V (x, t) is nondegenerate for any x, t , the SDE andjump-diffusion generated by the SDE are called pseudo-elliptic.

Let Gs,t (x) = Gc,ds,t (x) be an exponential Wiener–Poisson functional defined by

Gs,t := exp{ d ′∑

k=0

∫ t

s

ck(Φs,r , r) ◦ dWkr +

∫ t

s

∫

|z|>0+log dr,z(Φs,r−)N(dr dz)

},

where ck(x, t), k = 0, . . . , d ′ are C∞,1-functions of x, t on M × Rd ′ and dt,z(x) is

a C∞,1,2-function of (x, t, z) on M × T× Rd ′ satisfying dt,0(x) = 1 for any x, t .

Suppose we are given a volume element dx on the manifold M . In the followingwe will fix it. It is also written as dy. We will study the existence of the smoothdensity of P c,d

s,t (x, E) = E[1E(Φs,t (x))Gc,ds,t (x)] with respect to the volume element

7.5 Density for Jump-Diffusion on Compact Manifold 329

dy. In order to construct the smooth density, we need more careful discussionsthan those for diffusions, since sample paths may jump to other local charts. Ourdiscussion will be divided into two. First, we show that if the size of jumps issufficiently small, say smaller than δ, we can construct the density function by amethod similar to that of diffusion discussed in Sect. 7.4. Next, we adjoin jumpswhich are bigger than δ and show the existence of the smooth density by the methodof perturbation discussed in Sect. 6.9.

Let δ be a positive number. We will truncate jumps which are bigger than δ > 0from the above equation: The equation is written as

f (Xt ) = f (X0)+d ′∑

k=0

∫ t

t0

Vk(r)f (Xr) ◦ dWkr (7.43)

+∫ t

t0

∫

δ≥|z|>0+{f (φr,z(Xr−))− f (Xr−)}N(dr dz).

We denote the stochastic flow of diffeomorphisms generated by the above equationby {Φδ

s,t }. We set

Gδs,t := exp

{ d ′∑

k=0

∫ t

s

ck(Φδs,r , r) ◦ dWk

r +∫ t

s

∫

δ≥|z|>0+log dr,z(Φ

δs,r−)N(dr dz)

},

and define P δs,t (x, E)(= P

c,d,δs,t (x, E)) := E[1E(Φ

δs,t (x))G

δs,t (x)]. For a given n0 ∈

N, we want to show that the measure P δs,t (x, E) has a Cn0 -density with respect to

the volume element dy, if δ (depending on n0) is taken sufficiently small. We willdiscuss how such δ can be taken.

Let D be a local chart of M and let τ δ(x, s) be the first leaving time of the processX

x,st = Φδ

s,t (x) from D. We consider the transition function of the killed processdefined by

Qδs,t (x, E) = E[1E(Φ

δs,t (x))G

δs,t (x)1τ δ(x,s)>t ]. (7.44)

In the first step, we want to show the existence of the smooth density of the aboveQδ

s,t (x, E). Let α0 be any fixed positive number satisfying α < α0 < 2.

Lemma 7.5.1 Let n0 be a positive integer. Let V,U,D be a triple of local chartssuch that V ⊂ U ⊂ U ⊂ D. Then there exists a positive constant δn0,V ,U,D

(depending on n0 and V,U,D) such that for any 0 < δ < δn0,V ,U,D , the killedtransition function Qδ

s,t (x, E ∩U) has a Cn0-density qδs,t (x, y), y ∈ U with respect

to the volume element dy for any s < t and x ∈ D. Further, for any integer0 ≤ j ≤ n0 and T > 0, there are positive constants cT ,δ,j and c′T ,δ,j such that


supx∈D,y∈U

∣∣∣djy q

δs,t (x, y)

∣∣∣ ≤ cT ,δ,j

(t − s)j+d

2−α0

, ∀0 ≤ s < t ≤ T , (7.45)

supx∈Uc∩D,y∈V

sup0≤s<t≤T

∣∣∣diyq

δs,t (x, y)

∣∣∣ ≤ c′T ,δ,j . (7.46)

Proof Note that D is diffeomorphic to a bounded open set D of Rd by a

diffeomorphism ψ : D → D. SDE (7.43) on D is transformed to an SDE on D bythe diffeomorphic map ψ . Indeed, for the vector fields Vk(t) on the local chart D,there are vector fields Vk(x, t) on R

d of C∞,1b -class, which coincide with dψ(Vk(t))

on D. Further, there exists δ0 > 0 such that for any |z| ≤ δ0 and t ∈ T, we canconstruct a family of diffeomorphic maps φt,z on R

d such that ψ(φt,z(x)) = φt,z(x)

holds if x ∈ D (Lemma 7.1.3).Let N be a positive integer such that N >

n0+d2−α0

. Let U = ψ(U) and V = ψ(V ).There exists a positive constant δN,V,U,D , less than or equal to δ0, such that for any0 < δ < δN,V,U,D inequalities

d(φNδ (U), φ−N

δ (Dc)) > 0, d(φNδ (V ), φ−N

δ (U c)) > 0 (7.47)

hold, where φδ(U ) = ⋃t∈T,|z|<δ φt,z(U ) and φN

δ (U) are defined by iteration.

Indeed, note that φNδ (U) ↓ U and φ−N

δ (Dc) ↓ Dc as δ ↓ 0. Then

d(φNδ (U), φ−N

δ (Dc)) ↑ d(U , Dc) > 0.

Therefore the first inequality of (7.47) is valid. The second inequality is verified inthe same way. Now, take 0 < δ < δN,V,U,D and consider an SDE on R

d ;

dXt =d ′∑

k=0

Vk(Xt , t) ◦ dWkt +

∫

δ≥|z|>0+{φt,z(Xt−)− Xt−}N(dt dz). (7.48)

It generates a stochastic flow {Φδs,t } of diffeomorphisms on R

d . Let τ δ(x, s) be the

first leaving time of Xt = Φδs,r (x) from D. We shall consider the law of the killed

process given by

Qδs,t (x, E) = E

[1E(Φ

δs,t (x))G

δs,t (x)1τ δ (x,s)>t

]. (7.49)

Since the process Φδs,t (x) is pseudo-elliptic, the measure Qδ

s,t (x, ·) restricted to U =ψ(U) has a Cn0 -density qδ

s,t (x, y), y ∈ U with respect to the Lebesgue measure by

Lemma 6.10.3. For a given T > 0, it satisfies (6.123) and (6.124) in place of qc,ds,t .


Now let Φδs,t (x) be the solution of equation (7.43) on the manifold M . Let

τ δ(x, s) be the first leaving time of Xδ,x,st = Φδ

s,t (x) from D. Then the weightedlaw Qδ

s,t (x, E) of the killed process Φδs,t (x), t < τδ(x, s) has also a Cn0 -density

qδs,t (x, y), y ∈ U with respect to the volume element dy, since Φδ

s,t (x) =ψ−1(Φδ

s,t (ψ(x))) holds for t < τδ(x, s) almost surely. Further, since qδs,t satis-

fies (6.123) and (6.124) (in place of qc,ds,t ), qδ

s,t satisfies (7.45) and (7.46). ��Next, we will show that for a sufficiently small δ > 0, the law P δ

s,t (x, E) has aCn0 -density with respect to the volume element dy for any x ∈ M, s < t , by piecingtogether the above density function qδ

s,t (x, y) of the killed process. Our discussionproceeds in parallel with the discussion for diffusion processes in Sect. 7.4, but weneed some rectifications due to jumps of the process.

Given a pair of local charts V,D such that V ⊂ D, we can choose open sets U

and W such that V ⊂ U ⊂ U ⊂ W ⊂ W ⊂ D and a positive constant δN,V,U,W,D

with 0 < δN,V,U,W,D ≤ δN,V,U,D such that

d(φNδ (V ), φ−N

δ (Uc)) > 0, d(φδ(U),Wc) > 0, d(φNδ (W), φ−N

δ (Dc)) > 0

hold for any 0 < δ < δN,V,U,W,D , where N ≥ n0+d2−α0

. We want to prove that

P δs,t (x;E ∩ V ) has a Cn0 -density for any x ∈ M and 0 ≤ s < t ≤ T .

For this purpose, we define stopping times τ δm = τ δ

m(x, s),m = 0, 1, 2, . . ., andσ δm = σ δ

m(x, s),m = 1, 2, . . . for the process Xδt = X

δ,x,st = Φδ

s,t (x) by inductionas τ δ

0 = s and

σ δm = inf{t ≥ τ δ

m−1; Xδt ∈ Dc} (= ∞ if {· · · } is empty),

τ δm = inf{t ≥ σ δ

m; Xδt ∈ W } (= ∞ if {· · · } is empty).

Then we have for E ⊂ U ,

E[1E(Φδs,t (x))G

δs,t (x)] =

∞∑

m=0

E[1E(Φ

δs,t (x))G

δs,t (x)1τ δm<t≤σδ

m+1

].

Using the strong Markov property, we can rewrite the above equality as

P δs,t (x, E) =

∞∑

m=0

E[Qδ

τδm,t(Φδ

s,τ δm(x), E)Gδ

s,τ δm(x)1τ δm<t

]. (7.50)

Now take 0 < δ < δN,V,U,W,D . Then in view of Lemma 7.5.1, Qδs,t (x;E ∩ V )

has a Cn0 -density function qδs,t (x, y), y ∈ V for any 0 ≤ s < t ≤ T , x ∈ D. For

x ∈ Dc and y ∈ V , we set qδs,t (x, y) = 0, conventionally.


Lemma 7.5.2 There is a positive constant TN,U,V,W,D such that if 0 ≤ s < t ≤ T

and 0 < t − s ≤ TN,U,V,W,D , the infinite sum

pδs,t (x, y) := qδ

s,t (x, y)+∞∑

m=1

E[qδτδm,t

(Φδs,τ δm

(x), y)Gδs,τ δm

(x)1τ δm<t

](7.51)

is uniformly convergent with respect to x ∈ M and y ∈ V . Further, it is the densityof P δ

s,t (x, E ∩ V ) with respect to the volume element dy and it does not depend onthe choice of sets U,V,W,D.

Proof We need an inequality similar to (7.39). Let us consider Lemma 6.5.2 again.The lemma states a fact for a process on Euclidean space. But the fact is validfor a process on a manifold. Indeed, it tells us that for 0 < c < 1 there existsTU,V,W,D > 0 such that the inequality supx∈U P (τ δ

1 (x, s) < t) ≤ c holds for any0 < t − s < TU,V,W,D . Then we have the inequality

supx∈U

P (τ δm(x, s) < t) ≤ cm, ∀0 < t − s < TU,V,W,D,

similarly to the proof of Lemma 7.4.2.Suppose m ≥ 1. Note that qδ

s,t satisfies (7.45). Since Φδs,τ δm

(x) ∈ Uc,

qδτδm,t

(Φδs,τ δm

(x), y) is uniformly bounded with respect to y ∈ V in view of

Lemma 6.10.3. Furthermore, E[supt |Gδs,t (x)|p] is bounded with respect to x, since

ck(x, t), Vk(t)ck(t)(x) are bounded on compact set M . Therefore the infinite sumof (7.51) converges uniformly with respect to x ∈ M,y ∈ V , if 0 ≤ s < t ≤ T

and 0 < t − s < TU,V,W,D . Denote it by pδs,t (x, y). It is the density function of

P δs,t (x, E ∩ V ) for any x ∈ M in view of (7.50). Hence pδ

s,t (x, y) does not dependon the choice of U,V,W,D. ��Proposition 7.5.1 Given a positive integer n0, there exists δ1 > 0 such that for any0 < δ < δ1, the weighted transition function P δ

s,t (x, E) of the truncated processΦδ

s,t (x) has a Cn0 -density pδs,t (x, y) for any s < t and x ∈ M with respect to the

volume element dy. Further, for any nonnegative integer j ≤ n0 and T > 0, thereexists a positive constant c = cδ,n0 such that the inequality

∣∣∣djy p

δs,t (x, y)

∣∣∣ ≤ c

(t − s)j+d

2−α0

, ∀0 ≤ s < t ≤ T (7.52)

holds for all x, y ∈ M .

Proof Let {Vα ⊂ Uα ⊂ Wα ⊂ Dα} be a family of quaternions of local chartsdiscussed above, where {Vα} covers the manifold M . Since M is compact, it iscovered by a finite number of local charts {Vαi

, i = 1, 2, . . . , n}. We set δ1 =


mini δVαi,Uαi

,Wαi,Dαi

and T1 = mini TUαi,Vαi

,Wαi,Dαi

. Suppose 0 ≤ s < t ≤ T .

Then, if 0 < δ < δ1, pδs,t (x, y) of (7.51) is uniformly convergent for all x, y ∈ M

and 0 < t − s < T1. It is the density function of P δs,t (x, E) with respect to the

volume element dy for any x ∈ M, 0 < t − s < T1.We will show that pδ

s,t (x, y) is a Cn0 -function of y. Similarly to the proof ofLemma 7.5.2, we can show that for any j ≤ n0, the infinite sum

djy q

δs,t (x, y)+

∞∑

m=1

E[djy q

δτδm,t

(Φδs,τ δm

(x), y)Gδs,τ δm

(x)1τ δm<t

](7.53)

is uniformly convergent with respect to x, y ∈ M if 0 < t − s < T1.Next, if T1 < t − s < 2T1, using the Chapman–Kolmogorov equality, we can

define pδs,t (x, y) by

pδs,t (x, y) =

∫

M

pδs,s+T ′

1(x, y′)pδ

s+T ′1,t

(y′, y) dy′.

It is a Cn0 -function of y and is a density function of P δs,t (x, dy). Repeating this

argument, we can verify that P δs,t (x, dy) has a Cn0 -density for any 0 ≤ s < t ≤ T ,

��We have seen so far that the transition function of a pseudo-elliptic jump-

diffusion on a compact orientable manifold has a Cn0 -density, if jumps aresufficiently small. We will show next that the same fact holds for any pseudo-ellipticjump-diffusion on compact orientable manifold, which may allow big jumps.

Theorem 7.5.1 Suppose that the jump-diffusion on a compact orientable manifoldM defined by SDE (7.9) is pseudo-elliptic. Let P c,d

s,t (x, E) be its weighted transition

function. Then it has a C∞-density pc,ds,t (x, y), y ∈ M with respect to a given

volume element dy. Further, it is a C∞,1-function of x, s for any y and t .Let α0 be any fixed number satisfying α < α0 < 2, where α is the exponent of

the Lévy measure. For any i, j ∈ N and T > 0 there exist positive constants c, c′such that inequalities

supx,y∈M

∣∣∣djy p

c,ds,t (x, y)

∣∣∣ ≤ c

(t − s)j+d

2−α0

, ∀0 ≤ s < t ≤ T , (7.54)

supx,y∈M

∣∣∣dixp

c,ds,t (x, y)

∣∣∣ ≤ c′

(t − s)i+d

2−α0

, ∀0 ≤ s < t ≤ T . (7.55)

hold.


Further, p(x, s; y, t) := pc,ds,t (x, y) is the fundamental solution of the backward

heat equation on the manifold M , associated with the operator Ac,dJ (t) defined by

Ac,dJ (t)f (x) = 1

2

d ′∑

k=1

(Vk(t)+ ck(t))2f (x)+ (V0(t)+ c0(t))f (x)

+∫

|z|>0+{dt,z(x)f (φt,z(x))− f (x)}ν(dz). (7.56)

Proof We want to apply Proposition 7.5.1 for the proof of the theorem. Given apositive integer n0, let δ1 > 0 be a number such that for any 0 < δ < δ1, theweighted transition function of the truncated process Φδ

s,t (x) has a Cn0 -density. Wewill show the existence of the smooth density of the weighted law of Φs,t (x) bythe method of perturbation studied at Sect. 6.9. For u = ∅ ∈ U

0 (empty set), wedefine Φ

δ,φs,t (x) = Φδ

s,t (x). Further, for u = ((t1, z1), . . . , (tn, zn)) ∈ Un such that

s < t1 < · · · < tn ≤ t and zi ∈ Rd ′0 , we define

Φδ,us,t (x) := Φδ

tn,t◦ φtn,zn ◦ · · · ◦Φδ

t1,t2◦ φt1,z1 ◦Φδ

s,t1(x). (7.57)

In the following arguments, we will let n run on n = 0, 1, . . .. Let q be a Poissonpoint process with intensity dtνδ(dz), where νδ(dz) = 1|z|≥δν(dz). Then Dq ∩(s, t]is an empty set or it is written as {τ1 < · · · < τn} (stopping times) as in Sect. 6.9. Weset u(q) = ∅ or u(q) = ((τ1, q(τ1)), . . . , (τn, q(τn))). Then it holds that Φs,t (x) =Φ

δ,u(q)s,t (x) a.s. See Sect. 6.9.Let pδ

s,t (x, y), x, y ∈ M be the density function of the weighted transition

function P δs,t (x, E) of Φδ

s,t (x). For u = ∅, we define pδ,φs,t (x, y) = pδ

s,t (x, y) and

for u = ((t1, z1), . . . , (tn, zn)) ∈ Un, we define p

δ,us,t (x, y) by

pδ,us,t (x, y) =

∫

M

· · ·∫

M

pδs,t1

(x, x1) dt1,z1(x1)pδt1,t2

(φt1,z1(x1), x2)× · · ·

× dtn,zn(xn)pδtn,t

(φtn,zn(xn), y) dx1 · · · dxn. (7.58)

Set

pc,ds,t (x, y) := E[pδ,u(q)

s,t (x, y)]. (7.59)

Then it is a density function of the weighted transition function Pc,ds,t (x, E).

We want to show that the function pc,ds,t (x, y) is of Cn0 -class with respect to

y. Our discussion is close to those in Theorem 6.9.1. Condition (6.96) should bemodified as


supt,x

∫

|z|>δ

|Jφ−1t,z|ν(dz) < ∞.

However, the above condition is satisfied in the present case, since M is a compactmanifold and the support of the Lévy measure ν is compact. Therefore, pc,d

s,t (x, y)

is a continuous function of y. See the proof of Theorem 6.9.1, (1) The n0-timesdifferentiability with respect to y can be verified similarly to Theorem 6.9.1, (2)Finally, remark that the above fact is valid for any positive integer n0. Then thedensity p

c,ds,t (x, y) is in fact a C∞-function of y. It satisfies (7.54), since qδ

s,t (x, y)

satisfies (7.45) and the term∑

i≥1 · · · in (7.51) is bounded.

Let Ψs,t be the inverse map of Φs,t and let {P c,d,∗s,t } be the dual semigroup

of the semigroup {P c,ds,t } with respect to a volume element dx. Apply the above

discussion to the backward jump-diffusion Ψs,t . Then we find that its transitionfunction P

c,d,∗s,t (y, E) has a C∞-density p

c,d,∗s,t (y, x) for any s < t and y. Further,

we can show as before that pc,d,∗s,t (y, x) = p

c,ds,t (x, y) holds for any s < t and

x, y ∈ M . Therefore pc,ds,t (x, y) is a C∞-function of x and satisfies (7.55). We can

verify that it satisfies ∂∂sp

c,ds,t (x, y) = −A

c,dJ (s)xp

c,ds,t (x, y), making use of the dual

semigroup as in Lemma 6.10.2. Then v(x, s) := ∫p

c,ds,t1

(x, y)f1(y) dy is a C∞,1-function of x, s and in fact a solution of the final value problem of the backwardheat equation associated with the operator Ac,d

J (t), ��The next corollary is an immediate consequence of Theorem 7.5.1.

Corollary 7.5.1 Let {P c,ds,t } be the semigroup of Theorem 7.5.1. Then it maps C(M)

to C∞(M) for any s < t . Further it satisfies Kolomogorov’s backward equation.

We will study the smooth density of the law of a Lévy process on a compact Liegroup. We will consider a Lévy process on a Lie group G with the generator (7.23),where Vk are left-invariant vector fields and xg(z) is the right translation by g(z).

We set Vk = ∂Rg(z)

∂zk

∣∣z=0. Then Vk, k = 1, . . . , d ′ are left-invariant vector fields.

Theorem 7.5.2 Let Xt be a Lévy process on a compact Lie group. We assume thatleft-invariant vector fields {Vk, k = 1, . . . , d ′} ∪ {Vk; k = 1, . . . , d ′} span the Liealgebra of left-invariant vector fields. Then the law of Xt has a C∞-density withrespect to the Haar measure for any t > 0.

Note It is expected that similar assertions of Theorem 7.5.1 and Theorem 7.5.2should hold for non-compact manifold and non-compact Lie groups, respectively. Itremains open. Picard–Savona [95] studied the smooth density for Markov processof pure jumps on non-compact manifolds or non-compact Lie group with anotheradditional conditions. An analytic approach is taken in Applebaum [2] for theproblem on a compact Lie group.


Appendix: Manifolds and Lie Groups

Let M be a Hausdorff topological space with the second countability. It is calleda manifold of dimension d if each point of M has an open neighborhood that ishomeomorphic to an open set in R

d . If U is an open neighborhood of x ∈ M andψ is a homeomorphism from M to an open subset of Rd , we call (U,ψ) a chartat x. For y ∈ U , we write ψ(y) = (x1(y), . . . , xd(y)) and call x1, . . . , xd localcoordinates at U . An atlas is a collection of charts {(Uα,ψα);α ∈ I } so that (Uα)

covers M . If M has an atlas for which mappings ψα ◦ ψ−1β ;Rd → R

d are C∞ forall α, β ∈ I , then M is said to be a C∞-manifold.

A function f : M → R is called smooth or C∞, if f ◦ ψ−1α are C∞ from

ψα(Uα) to R for any α. Let C∞(M) be the collection of smooth functions on M .A tangent vector V (x) at x ∈ M is a linear functional on C∞(M) which satisfiesV (x)(fg) = f (x)V (x)(g)+ (V (x)f )g(x) for any f, g ∈ C∞(M). In local charts,we can write V (x) = ∑d

i=1 Vi(x) ∂

∂xi. The set of all tangent vectors at x ∈ M forms

a d-dimensional vector space called the tangent space and it is denoted by Tx(M).The set T (M) = ⋃

x∈M Tx(M) can be regarded as a 2d-dimensional manifold,called the tangent bundle to M .

A C∞-map V from M to T (M) is called a vector field. In each local chart (U,ψ),we can write V (x) = ∑d

i=1 Vi(x) ∂

∂xi, where V i(x) are C∞-functions from ψ(U)

to Rd . The collection of all vector fields is denoted by L(M). For V,W ∈ L(M),

its Lie bracket [V,W ] is defined by [V,W ]f = V (Wf )−W(Vf ). It is an elementof L(M). Thus L(M) is an Lie algebra.

Let M1 and M2 be C∞-manifolds of dimension d1 and d2, respectively and let{(Uα,ψα);α ∈ I } be an atlas of M1 and let {(Uβ,ψβ);β ∈ J } be an atlas of M2. Amapping f ;M1 → M2 is said to be C∞ if ψβ ◦ f ◦ ψ−1

α is C∞ from Rd1 to R

d2 .A C∞-mapping f ;M1 → M2 is said to be a diffeomorphism if it is a bijection andthe inverse f−1 is also a C∞-mapping.

Let ϕt , t ∈ R be a family of diffeomorphisms of M such that ϕt ◦ ϕs = ϕt+s

and (t, x) → ϕt (x) is a C∞-map. {ϕt } is called a one–parameter group oftransformations. For a given {ϕt }, we set for f ∈ C∞(M)

Vf (x) = d

dtf ◦ ϕt (x)

∣∣∣t=0

. (A.1)

Then V is a vector field. Conversely, for a given vector field V , if there exists aone–parameter group of transformations ϕt satisfying (A.1), the vector field V iscalled complete. The associated ϕt (x) is denoted by exp(tV )(x). If the manifold M

is compact, it is known that any vector field is complete, but if M is not compact,non-complete vector fields can exist.

Let M be a connected manifold of dimension d and let L(M) be the linear spaceof vector fields on M . For f ∈ C∞(M), we define a linear map df : L(M) →C∞(M) by df (V ) = Vf . We call df a one-form or the differential of f . Localizingdf at x ∈ M , dfx is a linear mapping from the tangent space Tx(M) to R, i.e., it is

Appendix: Manifolds and Lie Groups 337

the dual of Tx(M). It is denoted by Tx(M)∗ and is called the cotangent space. LetT ∗x (M)⊗r

be the r-times tensor product to T ∗x (M). Its element is called an r-tensor.

The alternating r-tensor of v1, . . . , vr ∈ T ∗x (M) is defined by

Alt(v1, . . . , vr ) = 1

r!∑

σ∈Σ(r)

sign(σ )vσ(1) ⊗ · · · ⊗ vσ(r), (A.2)

where v1, . . . , vr ∈ T ∗x (M), vσ(1) ⊗ · · · ⊗ vσ(r) is an r-tensor, and Σ(r) is the

permutation group of r letters.If an r-tensor v satisfies v = Alt(v1, . . . , vr ), v is called an alternating tensor.

Let Λr(T ∗x (M)) be the collection of alternating r-tensors in T ∗

x (M)⊗r. It is a vector

space of dimension

(d

r

)if r ≤ d and it is equal to {0} if r > d. We set Λr(M) =

⋃x∈M Λr(T ∗

x (M)). A smooth r-form ω is a C∞-mapping from x ∈ M to Λr(M).If v ∈ Λr(M) and w ∈ Λs(M), the exterior product (wedge product) v ∧ w is

defined by

v ∧ w = (r + s)!r!s! Alt(v1, . . . , vr , w1, . . . , ws).

At a local chart U , let x1, . . . , xd be a local coordinate and let dxi, i = 1, . . . , d be1-forms. Then a smooth r-form ω can be written at x ∈ U as

ω(x) =∑

{i1,...,ir }⊂{1,...,d}hωi1,...,ir

(x) dxi1 ∧ · · · ∧ dxir ,

where hωi1,...,ir

are smooth maps from U to R. In the case r = d, ω(x) is written as

ω(x) = hω(x) dx1 ∧ · · · ∧ dxd . (A.3)

If there exists a continuous d-form ω such that ω(x) is not 0 everywhere, then themanifold is called orientable. If hω(x) > 0, the d-form ω is called a positive d-formor a volume element.

Let Ωd(M) be the collection of smooth d-form on M . Let φ;M → M be aC∞-map. The pullback φ∗ is the linear map from Ωd(M) into itself defined by

φ∗(ω)(V1, . . . , Vd)(x) = ωφ(x)(dφx(V1(x)), . . . , dφx(Vd(x))),

for all V1, . . . , Vd ∈ L(M) and x ∈ M .We will define the integral by a positive d-form (volume element) ω. Let C∞

0 (M)

be the collection of C∞-functions on M with compact supports. For f ∈ C∞0 , we

will define the integral∫M

fω by the formula

∫

M

fω =∫

Rd

f ◦ φ−1(x1, . . . , xd)hω(x1, . . . , xd) dx1 · · · dxd,


if the support of f is included in a chart U . If the support of f is not included ina chart, we need a partition of unity. It is a collection of functions (ψi, i ∈ I ) inC∞(M) for which:

(1) At each x ∈ M , only finite number of ψi are nonzero.(2) For each i, supp(ψi) is compact.(3) For each x ∈ M, i ∈ I, ψi(x) ≥ 0 and

∑i∈I ψi(x) = 1.

We can define the integral of a continuous function f with compact support by

∫

M

fω =∑

i

∫

M

fφiω.

We will fix a positive d-form (volume element) ω and denote it by dx. Theintegral by dx of a continuous function f on M is denoted by

∫M

f (x) dx if itexists. It is usually denoted by

∫M

f (y) dy etc.Let φ : M → M be a locally diffeomorphic C∞-map. Let φ∗dx be the pullback

of dx by φ. Then there exists a positive or negative C∞-function Jφ on M such thatφ∗ dx = Jφdx. We have a formula of the change of variables:

∫

M

f (x) dx =∫

M

f (φ(x))|Jφ(x)| dx (A.4)

for any continuous functions f of compact supports.Let V be a complete vector field on M and let {ϕt } be the one–parameter group of

diffeomorphisms generated by V . Then the Lie derivative of dx denoted by LV dx isdefined by limt→0

1t(ϕ∗

t dx−dx). Since LV dx is a d-form, it is written as LV dx =h(x) dx with a smooth scalar function h, which we denote by divV (x). We have forany f, g ∈ C∞

0 (M),

∫f (x)(g(ϕt (x))− g(x)) dx =

∫(f ◦ ϕ−1

t (x)− f (x))g(x) dx

+∫

f ◦ ϕ−1t (x)g(x)((ϕ−1

t )∗ dx − dx).

Since the inverse map ϕ−1t is generated by the vector field −V , we have

limt→0

1

t

∫f (x)(g(ϕt (x))− g(x)) dx =

∫f (x)Vg(x) dx,

limt→0

1

t

∫(f ◦ ϕ−1

t (x)− f (x))g(x) dx = −∫

Vf (x)g(x) dx,

limt→0

1

t

∫f ◦ ϕ−1

t (x)g(x)((ϕ−1t )∗ dx − dx) = −

∫f (x)g(x)divV (x) dx.

Appendix: Manifolds and Lie Groups 339

Therefore we have a formula of the integration by parts:

∫

M

f (x)·Vg(x) dx = −∫

M

Vf (x)·g(x) dx−∫

M

divV (x)f (x)g(x) dx (A.5)

for any f, g ∈ C∞0 (M). The above formula holds for any (not necessarily complete)

C∞-vector field on M

We should remark that values of scalar functions Jφ and divV depend on thechoice of the volume element dx.

Remark If M is a Euclidean space Rd , we can take dx as the Lebesgue measure.

Then the C∞-function Jφ coincides with the Jacobian determinant det∇φ of themap φ. Further, equality (A.4) for M = R

d is a well known formula for change ofvariables. Further, div V coincides with

∑i ∂Vi/∂xi . Formula (A.5) for M = R

d iswell known as a formula for the integration by parts.

As a volume element, however, we can take any measure m(dx) = m(x) dx onthe Euclidean space, where m is a positive C∞-function. Then Jφ is no longer a

Jacobian determinant and divV is non longer equal to∑

i∂V i

∂xi .

A set G is called a topological group if it satisfies the following three proper-ties.

(1) G is a group and, at the same time, it is a topological space.(2) The map (x, y) → xy from the product space G×G into G is continuous.(3) The map x → x−1 from G into itself is continuous, where x−1 is the inverse

element of x.

Let G be a topological group and let g ∈ G. Two maps Lg,Rg;G → G aredefined by

Lg(x) = gx, Rg(x) = xg, (x ∈ G).

These two maps are homeomorphisms of G. Lg is called the left translation of G

by g, and Rg is called the right translation of G by g.A topological group G is called a Lie group if G is a C∞-manifold with a

countable basis and maps (x, y) → xy and x → x−1 are differentiable. Let V be aC∞-vector field on G. It is called left-invariant if for any f ∈ C0(G), it satisfies

V (f ◦ Lg)(x) = Vf (Lgx)

for any g ∈ G. It is called right invariant if V (f ◦ Rg)(x) = Vf (Rgx) holds forany g ∈ G. We denote by L(G) the set of all left-invariant vector fields on the Liegroup G.


Let G be a connected Lie group of dimension d. A nontrivial regular Borelmeasure m on G is called a left Haar measure if m(A) = m(gA) holds for any Borelsubset A of G and g ∈ G. A right Haar measure is defined similarly. It is known thatthe left Haar measure exists and it is unique up to a positive multiplicative constant.

Let m be a left Haar measure on G. For g ∈ G, we define

mg(A) = m(Ag),

for each Borel set A. Then mg is another left Haar measure on G. Then there existsa positive constant Δ(g) such that mg(A) = Δ(g)m(A) holds for all Borel subsetsA. We call the mapping g → Δ(g) from G to (0,∞) the modular function of G. Ifthe modular function is identically equal to 1, the Lie group is called unimodular.Examples of unimodular Lie groups are abelian Lie groups, compact Lie groups,semisimple Lie groups and connected nilpotent Lie groups.

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Symbol Index

| · |�0,n,p , norms, 200‖ · ‖0,n,p;A, Sobolev norms, 197‖ · ‖m,n,p;A, Sobolev norms, 226‖ · ‖m,p , Sobolev norms, 174

‖ · ‖�0,n,p , Sobolev norms, 199

‖ · ‖�m,n,p , Sobolev norms, 227∂ j, differential operator, 3C∞,mb -class, function, 77

C∞,m,nb -class, function, 78

D∞,D∞T

, Sobolev spaces, 174

D∞, D∞T, D∞

U, Sobolev spaces, 227

D∞, D∞U

, Sobolev spaces, 199

det∇Ψs,t , Jacobians of Ψs,t , 141Φs,t , stochastic flow, 79Φs,t , backward stochastic flow, 101KF , Malliavin covariance at the center, 210

KF , Malliavin covarance at the center, 237LT, space, 51LU, space of predictable random fields, 64,

67LT,LT(Λ), spaces of predictable

processes, 48, 49LLip,pT

(Λ),Ln+Lip,pT

(Λ), spaces ofpredictable processes, 56

Ln+Lip,pU

(Λ), space, 72Lp

T, space of predictable processes, 52

Lp

U, space of predictable random fields, 70

Ψs,t , inverse stochastic flow, 80RF

ρ , Malliavin covariance, 238

RFρ , Malliavin covariance, 209

RF , Malliavin covariance, 184〈X〉t , quadratic variation, 31〈X, Y 〉t , quadratic covariation, 35


347

https://doi.org/10.1007/978-981-13-3801-4

Index

Symbolsβ-stable, 10μ-continuous, 4σ -field, 1H -derivative, 170, 223H -differentiable, 170, 223, 249C∞-manifold, 336Cn

b -function, 4F -space, 168i-th H -derivative, 170Lp-martingale, 29n-th moment, 3r-form, 337r-tensor, 337

AAdapted, 28Adjoint operator, 172Admissible, 28Almost surely, 14Alternating r-tensor, 337Alternating tensor, 337Anticipating stochastic integral, 176Atlas, 336

BBackward diffusion, 138Backward exponential functional, 138Backward filtration, 30Backward heat kernel, 256Backward Itô integral, 74Backward local martingale, 31Backward Markov process, 40

Backward martingale, 30, 73Backward predictable processes, 73Backward process, 15Backward semi-group, 40Backward semi-martingale, 31Backward stochastic integral, 73Backward stopping time, 30Backward symmetric integral, 74Backward symmetric SDE, 80Backward transition function, 40Bounded kernel, 37Brownian motion, 16Brownian motion on Lie group, 317Burkholder-Davis-Gundy, 52

CCadlag process, 15Cameron Martin space, 168Center of the intensity n, 201Center of the Lévy measure, 10Chapman–Kolmogorov equation, 37Characteristic function, 3Characteristics of distribution, 10Characteristics of SDE, 122Chart, 336Commutation relation, 172, 194Compensated random measure, 23, 64Complete, 310, 336Compound Poisson distribution, 9Compound Poisson process, 22Conditional expectation, 25Conditional probability, 2, 26Condition (D), 315Condition (J.1), 82, 304


349

https://doi.org/10.1007/978-981-13-3801-4

350 Index

Condition (J.2), 82, 304Condition (J.1)K , 118Condition (J.2)K , 118Condition (R), 210Conditions 1-3 for master equation, 88Conservative, 38Continuous master equation, 88Continuous process, 15Continuous semi-martingale, 30Continuous stochastic flow, 80Converge almost surely, 2Convergence in Lp , 2Convergence in probability, 2Convolution, 3Cotangent space, 337Covariance matrix, 2

DDeterministic flow, 84Diffeomorphism, 78, 336Difference operator, 192Differential, 336Diffusion on manifold, 313Diffusion process, 40Distribution, 2Doob’s inequality, 29

EElliptic, 265Elliptic operator, 246Elliptic SDE, 246Equivalent, 14Equivalent in law, 15Event, 1Expectation, 2Explosion time, 304Exponential distribution, 8Exponential functional, 130Exterior product, 337

FFeynman–Kac–Girsanov formula, 136Feynman–Kac transformation, 135Filtration, 26Filtration with continuous parameter, 28Final value problem, 298Final value problem for backward heat

equation, 133, 256First exit time, 34Fisk–Stratonovitch’s integral, 59

Flow of local diffeomorphisms, 313Formula of the integration by parts, 189Forward process, 15Fourier inversion formula, 240Fundamental solution, 256, 257

GGamma distribution, 8Gaussian distribution, 9Generalized Itô’s formula, 57Generator, 133, 148Girsanov transformation, 55, 136Gronwall’s inequality, 88

HHaar measure, 340Heat kernel, 257Hitting time, 34Hölder’s inequality, 175Hörmander condition, 258Hypo-elliptic, 258

IImproper integral, 119Increasing process, 30Independent, 2Independent copy, 18Independent increments, 15Index, 10Infinitely divisible distribution, 9Initial value problem, 136, 300Integrable random variable, 2Intensity, 18Inverse stochastic flow, 80Itô integral, 47Itô process, 67, 75Itô SDE, 79Itô SDE with jumps, 82Itô’s formula, 49Itô–Wentzell formula, 57

JJacobian determinant, 105Jacobian matrix, 105Jump-diffusion on manifold, 313Jump-diffusion process, 147Jumping times, 20Jump process, 147

Index 351

KKernel, 37Killed process, 164Kolmogorov’s backward equation, 128,

132Kolmogorov’s criterion, 41Kolmogorov’s forward equation, 128, 132Kolmogorov–Totoki’s theorem, 41

LLaw, 2Law or random field, 15Law of random variable, 2Left-invariant, 339Left translation, 339Lévy–Itô decomposition, 25Lévy-Khintchine formula, 10Lévy measure, 9Lévy process, 15Lévy process on Lie group, 317Lévy’s inversion formula, 4Lie derivative, 338Lie group, 339, 340Local coordinate, 336Local semi-martingale, 30

MMalliavin covariance, 183, 209Malliavin covariance at the center, 210Manifold, 336Markovian, 38Markov process, 38Martingale part, 30Martingale transform, 26Martingale with continuous time, 28Martingale with discrete time, 26Master equation, 86Mean vector, 2Measurable, 2, 14Measurable space, 1Measure, 1Meyer’s equivalence, 182Modification, 15Modular function, 340Morrey’s Sobolev inequality, 114, 264

NNegative parameter martingale, 28Nondegenerate, 12, 184, 209, 238, 250Nondegenerate at the center, 237

OOff diagonal short-time, 284One-form, 336Order condition of exponent α, 12Order condition of exponent α at the center,

11Orientable, 337

PParameter of Poisson distribution, 9Parseval’s inequality, 240Point function, 25Point process, 25Poisson distribution, 9Poisson functional, 190Poisson point process, 25Poisson process, 18Poisson random measure, 20Poisson space with the intensity n, 190Poisson variable, 222Polynomial Wiener functional, 171Predictable, 64Predictable process, 45Predictable sequence, 26Predictable σ -field, 45Probability measure, 1Probability space, 1Process of bounded variation, 30Pseudo-elliptic jump-diffusion, 328

Qquadratic covariation, 35quadratic variation, 31

RRademacher system, 16Random field, 14Random variable, 2Rapidly decreasing, 7, 8Regular, 236Regular functional, 210Right continuous stochastic flow, 86Right translation, 339

SSample, 1Schrodinger operator, 137Schwartz space, 240Semigroup of linear transformation, 37Semi-martingale, 30Short-time asymptotics, 277

352 Index

Simple functional, 224Skorohod equation, 177Skorohod integral, 172, 176, 193, 224Slowly increasing, 127Smooth Wiener functional, 174Sobolev norms, 226Sobolev norms for Wiener functionals, 174Stable, 10Star-shaped neighborhood, 12Stochastic Dirichlet condition, 299Stochastic flow of C∞-maps, 79Stochastic flow of diffeomorphisms, 79Stochastic integral, 46, 47Stochastic process, 15Stopping time, 27, 28Strong drift, 119Strongly complete, 310Strongly nondegenerate, 239Strong Markov property, 39Sub-martingale, 28Super-martingale, 28Symmetric integral, 59Symmetric SDE, 78Symmetric stochastic differential equation

on manifold, 304Symmetric stochastic differential equation

with jumps, 82

TTangent bundle, 336Tangent space, 336Tangent vector, 336Tangent vector fields, 304Tangent vector of maps, 82Tensor, 337Terminal time, 304Time homogeneous, 15, 38Topological group, 339

Transition function, 37Transition probability, 38Two-sided filtration, 31, 78, 81

UUniformly elliptic, 265Uniformly elliptic operator, 246Uniformly elliptic SDE, 246Uniformly integrable, 2Uniformly Lipschitz continuous, 87Uniformly Lp-bounded, 87Uniformly nondegenerate, 250Uniformly pseudo-elliptic, 265Unimodular, 340

VVector field, 336Volume element, 337Volume-gaining, 161Volume-gaining in the mean, 163Volume-losing, 161Volume-preserving, 161Volume-preserving in the mean, 163

WWeak drift, 119Weighted law, 3, 183Weighted transition function, 132Wiener functional, 168Wiener measure, 168Wiener–Poisson functional, 222Wiener-Poisson space, 222Wiener process, 16Wiener space, 168Wiener variable, 222

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