J Optim Theory ApplDOI 10.1007/s10957-012-0166-7

Forward–Backward Stochastic Differential Gamesand Stochastic Control under Model Uncertainty

Bernt Øksendal · Agnès Sulem

Received: 4 January 2012 / Accepted: 20 August 2012© Springer Science+Business Media, LLC 2012

Abstract We study optimal stochastic control problems with jumps under modeluncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximumprinciples for such games, both in the zero-sum case (finding conditions for saddlepoints) and for the nonzero sum games (finding conditions for Nash equilibria). Wethen apply these results to study robust optimal portfolio-consumption problems withpenalty. We establish a connection between market viability under model uncertaintyand equivalent martingale measures. In the case with entropic penalty, we prove ageneral reduction theorem, stating that a optimal portfolio-consumption problem un-der model uncertainty can be reduced to a classical portfolio-consumption problemunder model certainty, with a change in the utility function, and we relate this to risksensitive control. In particular, this result shows that model uncertainty increases theArrow–Pratt risk aversion index.

Keywords Forward–backward SDEs · Stochastic differential games · Maximumprinciple · Model uncertainty · Robust control · Viability · Optimal portfolio ·Optimal consumption · Jump diffusions

1 Introduction

One of the aftereffects of the financial crisis is the increased awareness of the needfor more advanced modeling in mathematical finance, and a focus of attention is on

B. Øksendal (�)Dept. of Mathematics, University of Oslo, Center of Mathematics for Applications (CMA),P.O. Box 1053, Blindern, 0316 Oslo, Norwaye-mail: oksendal@math.uio.no

A. SulemINRIA Paris-Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, Le Chesnay Cedex,78153, Francee-mail: agnes.sulem@inria.fr

J Optim Theory Appl

the problem of model uncertainty. This paper is motivated by a topic of this type. Weconsider a stochastic system described by a general Itô–Lévy process controlled byan agent. The performance functional is expressed as the Q-expectation of an inte-grated profit rate plus a terminal payoff, where Q is a probability measure equivalentto the original probability measure P , which is often called a reference measure. Wemay regard Q as a scenario measure controlled by the market or the environment. IfQ = P , the problem becomes a classical stochastic control problem of the type stud-ied in [1]. If Q is uncertain, however, the agent might seek the strategy which max-imizes the performance in the worst possible choice of Q. This leads to a stochasticdifferential game between the agent and the market. Our approach is the following:We write the performance functional as the value at time t = 0 of the solution ofan associated controlled backward stochastic differential equation (BSDE). Thus, wearrive at a (zero-sum) stochastic differential game of a system of forward–backwardSDEs (FBSDEs) that we study by the maximum principle approach.

There are several papers of related content. Stochastic control of forward–backward SDEs (FBSDEs) has been studied in [2, 3], and in [4] a maximum principlefor stochastic differential g-expectation games of SDEs is developed. The recent pa-pers [5–9] also study optimal portfolio under model uncertainty by means of BSDEs.The approaches in the three latter papers are strongly linked to the exponential utilitycase. A key feature of the current paper is that it applies to general utility functionsand also general dynamics for the state process.

Our paper is organized as follows. In Sect. 2, we state general stochastic maximumprinciples for stochastic differential games with partial information, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (find-ing conditions for Nash equilibria). The proofs are given in Appendix A. In Sect. 3,we consider stochastic control problems under model uncertainty, also called robustcontrol problems. We formulate these problems as (zero-sum) stochastic differentialgames of forward–backward SDEs (FBSDEs), and we study them by the maximumprinciple approach of Sect. 2. In Sect. 4, we apply these techniques to study a robustoptimal portfolio-consumption problem with penalty. We establish a connection be-tween market viability under model uncertainty and equivalent martingale measures.Finally, we study the case with entropic penalty, and we prove a general reductiontheorem, stating that any optimal portfolio-consumption problem under model un-certainty can be reduced to a classical portfolio-consumption problem under modelcertainty, with a change in the utility function. In particular, we obtain a connection torisk-sensitive control, and we show that model uncertainty increases the Arrow–Prattrisk aversion index.

2 Maximum Principles for Stochastic Differential Gamesof Forward–Backward Stochastic Differential Equations

2.1 The Case of General (Nonzero) Stochastic Differential Games

In this section, we formulate and prove a sufficient and necessary maximum prin-ciple for general stochastic differential games (not necessarily zero-sum games) of

J Optim Theory Appl

forward–backward SDEs. Let (Ω, F , {Ft }t≥0,P ) be a filtered probability space,where P is a reference probability measure. Consider a controlled forward SDE ofthe form

dX(t) = dXu(t)

= b(t,X(t), u(t),ω

)dt + σ

(t,X(t), u(t),ω

)dB(t)

+∫

R

γ(t,X

(t−

), u(t), ζ,ω

)N(dt, dζ ), 0 ≤ t ≤ T ,

X(0) = x ∈ R,

(1)

where B is a Brownian motion, and N(dt, dζ ) = N(dt, dζ ) − ν(dζ ) dt is an inde-pendent compensated Poisson random measure, where ν is the Lévy measure of N

such that∫

Rζ 2ν(dζ ) < ∞. We assume that F = {Ft , t ≥ 0} is the P -augmentation

of the natural filtration associated with B and N . Here u = (u1, u2), where ui(t) isthe control of player i, i = 1,2. We assume that we are given two subfiltrations

E (i)t ⊆ Ft ; t ∈ [0, T ], (2)

representing the information available to player i at time t ; i = 1,2. We let Ai de-note a given set of admissible control processes for player i, contained in the set ofE (i)

t -predictable processes; i = 1,2, with values in Ai ⊂ Rd , d ≥ 1. Denote U :=A1 × A2. We assume that b(·, x,u,ω), σ(·, , x,u,ω), γ (·, , x,u, ζ,ω) are given pre-dictable processes for each x in R, u in U, and ζ in R0 := R\{0} such that (1) has aunique solution for each u in U.

We consider the associated controlled backward SDEs (i.e., BSDEs) in the un-knowns (Y u

i (t),Zui (t),Ku

i (t, ζ )) = (Yi(t),Zi(t),Ki(t, ζ )) of the form

dYi(t) = −gi

(t,X(t), Yi(t),Zi(t),Ki(t, ·), u(t),ω

)dt

+ Zi(t) dB(t) +∫

R

Ki(t, ζ )N(dt, dζ ), 0 ≤ t ≤ T ,

Yi(T ) = hi

(X(T ),ω

), i = 1,2.

(3)

Here gi(·, x, y, z, k,u,ω) are given predictable processes for each x in R, y in R,z in R, k in R

R0 , u in U, ζ , and hi(x,ω) is FT -measurable for each given x in R,such that the BSDEs (3) have unique solutions for each u in U.

Let fi(t, x,u) : [0, T ]×R×U → R, ϕi(x) : R → R, and ψi(x) : R → R be givenprofit rates, bequest functions, and “risk evaluations” respectively, of player i, i =1,2. Define

Ji(u) := E

[∫ T

0fi

(t,Xu(t), u(t),ω

)dt + ϕi

(Xu(T ),ω

) + ψi

(Yu

i (0))]

, i = 1,2,

(4)provided that the integrals and expectations exist. We call Ji(u) the performancefunctional of player i, i = 1,2. We assume that b,σ, γ, gi, hi, fi, ϕi , and ψi are C1

with respect to x, y, z,u and that

ψ ′i (x) ≥ 0 for all x, i = 1,2. (5)

J Optim Theory Appl

A Nash equilibrium for the FBSDE game (1)–(4) is a pair (u1, u2) ∈ A1 × A2 suchthat

J1(u1, u2) ≤ J1(u1, u2) for all u1 ∈ A1 (6)

and

J2(u1, u2) ≤ J2(u1, u2) for all u2 ∈ A2. (7)

Heuristically, this means that player i has no incentive to deviate from the con-trol ui , as long as player j (j = i) does not deviate from uj , i = 1,2. Therefore aNash equilibrium is in some cases a likely outcome of a game. Suppose that thereexists a Nash equilibrium (u1, u2). We now present a method to find it, based on themaximum principle for stochastic control. Our result may be regarded as an extensionof the maximum principles for FBSDEs in [2] and for (forward) SDE games in [4].

Define the Hamiltonians

Hi(t, x, y, z, k,u1, u2, λ,p, q, r) : [0, T ] × R3 × R × U × R

3 × R → R

of this game by

Hi(t, x, y, z, k,u1, u2, λ,p, q, r)

:= fi(t, x,u1, u2) + λgi(t, x, y, z, k,u1, u2) + pb(t, x,u1, u2)

+ qσ(t, x,u1, u2) +∫

R

r(ζ )γ (t, x,u1, u2, ζ )ν(dζ ), i = 1,2, (8)

where R is the set of functions from R0 into R such that the integral in (8) converges.We assume that Hi is Fréchet differentiable (C 1) in the variables x, y, z, k,u, i =

1,2, and that ∇kHi(t, ζ ) as a random measure is absolutely continuous with respectto ν, i = 1,2. We also assume that Hi and its derivatives with respect to u1 and u2are integrable with respect to P , i = 1,2.

In the following, we are using the shorthand notation

∂Hi

∂y(t) = ∂Hi

∂y

(t,X(t), Yi(t),Zi(t),Ki(t, ·), u1(t), u2(t), λi(t),pi(t), qi(t), ri(t, ·)

)

and similarly for the other partial derivatives of Hi .To these Hamiltonians we associate a system of FBSDEs in the adjoint processes

λi(t), pi(t), qi(t), and ri(t, ζ ) as follows:

1. Forward SDE in λi(t):⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dλi(t) = ∂Hi

∂y(t) dt + ∂Hi

∂z(t) dB(t) +

∫

R

d

dν∇kHi(t, ζ )N(dt, dζ )

= λi(t)

[∂gi

∂y(t) dt + ∂gi

∂z(t) dB(t) +

∫

R

d

dν∇kgi(t, ζ )N(dt, dζ )

],

0 ≤ t ≤ T ,

λi(0) = ψ ′i

(Yi(0)

)(

= dψi

dy

(Yi(0)

))

≥ 0,

(9)

J Optim Theory Appl

where ddν

∇kgi(t, ζ ) is the Radon–Nikodym derivative of ∇kgi(t, ζ ) with respectto ν(ζ ).

2. Backward SDE in pi(t), qi(t), ri(t, ζ ):

⎧⎪⎨

⎪⎩

dpi(t) = −∂Hi

∂x(t) dt + qi(t) dB(t) +

∫

R

ri(t, ζ )N(dt, dζ ), 0 ≤ t ≤ T ,

pi(T ) = ϕ′i

(X(T )

) + h′i

(X(T )

)λi(T ).

(10)

See Appendix A for an explanation of the gradient operator ∇kHi(t, ζ ) = ∇kHi(t, ζ )(·).

Theorem 2.1 (Sufficient maximum principle for FBSDE games) Let (u1, u2) ∈ A1 ×A2 with corresponding solutions X(t), Yi (t), Zi(t), Ki(t), λi (t), pi(t), qi (t), ri (t, ζ )

of Eqs. (1), (3), (9), and (10) for i = 1,2. Suppose that the following holds:

• (Concavity I) The functions x → hi(x), x → ϕi(x), x → ψi(x) are concave, i =1,2.

• (The conditional maximum principle)

ess supv∈A1E

[H1

(t, X(t), Y1(t), Z1(t), K1(t, ·),

v, u2(t), λ1(t), p1(t), q1(t), r1(t, ·)) | E (1)

t

]

= E[H1

(t, X(t), Y1(t), Z1(t), K1(t, ·),

u1(t), u2(t), λ1(t), p1(t), q1(t), r1(t, ·)) | E (1)

t

](11)

and similarly

ess supv∈A2E

[H2

(t, X(t), Y2(t), Z2(t), K2(t, ·),

u1(t), v, λ2(t), p2(t), q2(t), r2(t, ·)) | E (2)

t

]

= E[H2

(t, X(t), Y2(t), Z2(t), K2(t, ·),

u1(t), u2(t), λ2(t), p2(t), q2(t), r2(t, ·)) | E (2)

t

]. (12)

• (Concavity II) (The Arrow conditions) The functions

H1(x, y, z, k)

:= ess supv1∈A1E

[H1

(t, x, y, z, k, v1, u2(t), λ1(t), p1(t), q1(t), r1(t, ·)

) | E (1)t

]

and

H2(x, y, z, k)

:= ess supv2∈A2E

[H2

(t, x, y, z, k, u1(t), v2, λ2(t), p2(t), q2(t), r2(t, ·)

) | E (2)t

]

are concave for all t , a.s.

J Optim Theory Appl

• Assume that

d

dν∇kgi(t, ζ ) > −1

for i = 1,2.• Moreover, assume that the following growth conditions hold:

E

[∫ T

0

{p2

i (t)

[(σ(t) − σ (t)

)2 +∫

R

(ri(t, ζ ) − ri (t, ζ )

)2ν(dζ )

]

+ (X(t) − X(t)

)2[q2i (t) +

∫

R

r2i (t, ζ )ν(dζ )

]

+ (Yi(t) − Yi (t)

)2[(

∂Hi

∂z

)2

(t) +∫

R

∥∥∇kHi(t, ζ )∥∥2

ν(dζ )

]

+ λ2i (t)

[(Zi(t) − Zi(t)

)2 +∫

R

(Ki(t, ζ ) − Ki(t, ζ )

)2ν(dζ )

]}dt

]< ∞

for i = 1,2. (13)

Then u(t) = (u1(t), u2(t)) is a Nash equilibrium for (1)–(4).

Above and in the proof in Appendix A, we have used the following shorthandnotation:

If i = 1, then X(t) = X(u1,u2)(t) and Y1(t) = Y(u1,u2)1 (t) are the processes corre-

sponding to the control u(t) = (u1(t), u2(t)), while X(t) = Xu(t) and Y1(t) = Y u1 (t)

are those corresponding to the control u(t) = (u1(t), u2(t)). An analogue notation isused for i = 2. Moreover, we put

∂Hi

∂x(t) = ∂Hi

∂x

(t, X(t), Yi(t), Zi(t), Ki(t, ·), u(t), λi (t), pi(t), qi (t), ri (t, ·)

)

and similarly with ∂Hi

∂z(t) and ∇kHi(t, ζ ), i = 1,2.

Proof See Appendix A. �

It is also of interest to prove a version of the maximum principle which doesnot require the concavity conditions. One such version is the following necessarymaximum principle (Theorem 2.2) which requires the following assumptions:

• For all t0 ∈ [0, T ] and all bounded, E (i)t -measurable random variables αi(ω), the

control

βi(t) := χ(t0,T )(t)αi(ω) belongs to Ai , i = 1,2. (14)

• For all ui,βi ∈ Ai with βi bounded, there exists δi > 0 such that the control

ui (t) := ui(t) + sβi(t), t ∈ [0, T ],belongs to Ai for all s ∈ (−δi, δi), i = 1,2. (15)

J Optim Theory Appl

• The following derivative processes exist and belong to L2([0, T ] × Ω):

x1(t) = d

dsX(u1+sβ1,u2)(t) |s=0,

y1(t) = d

dsY

(u1+sβ1,u2)

1 (t) |s=0,

z1(t) = d

dsZ

(u1+sβ1,u2)

1 (t) |s=0,

k1(t, ζ ) = d

dsK

(u1+sβ1,u2)

1 (t) |s=0,

(16)

and, similarly,

x2(t) = d

dsX(u1,u2+sβ2)(t) |s=0, etc.

Note that since Xu(0) = x for all u, we have xi(0) = 0 for i = 1,2.In the following we write

∂b

∂x(t) for

∂b

∂x

(t,X(t), u(t)

), etc.

By (1) and (3) we have, using the estimates in [4],

dx1(t) ={

∂b

∂x(t)x1(t) + ∂b

∂u1(t)β1(t)

}dt +

{∂σ

∂x(t)x1(t) + ∂σ

∂u1(t)β1(t)

}dB(t)

+∫

R

{∂γ

∂x(t, ζ )x1(t) + ∂γ

∂u1(t, ζ )β1(t)

}N(dt, dζ ), (17)

dy1(t) = −{

∂g1

∂x(t)x1(t) + ∂g1

∂y(t)y1(t) + ∂g1

∂z(t)z1(t)

+∫

R

∇kg1(t, ζ )k1(t, ζ )ν(dζ ) + ∂g1

∂u1(t)β1(t)

}dt

+ zi(t) dB(t) +∫

R

k1(t, ζ )N(dt, dζ ), 0 ≤ t ≤ T ,

y1(T ) = h′1

(X(u1,u2)(T )

)x1(T ),

(18)

and similarly for dx2(t), dy2(t).We are now ready to state a necessary maximum principle, which is an extension

of Theorem 3.1 in [4] and Theorem 3.1 in [2]. In the sequel, ∂H∂v

means ∇vH .

Theorem 2.2 (Necessary maximum principle) Suppose that u ∈ A with correspond-ing solutions X(t), Yi(t),Zi(t),Ki(t, ζ ), λi(t),pi(t), qi(t), ri(t, ζ ) of Eqs. (1), (3),(9) and (10). Suppose that (14), (15), and (16) hold.

J Optim Theory Appl

Moreover, assume that

E

[∫ T

0

{p2

i (t)

[(∂σ

∂x(t)xi(t) + ∂σ

∂ui

(t)βi(t)

)2

+∫

R

(∂γ

∂x(t, ζ )xi(t) + ∂γ

∂ui

(t, ζ )βi(t)

)2

ν(dζ )

]

+ x2i (t)

(q2i (t) +

∫

R

r2i (t, ζ )ν(dζ )

)+ λ2

i (t)

(z2i (t) +

∫

R

k2i (t, ζ )ν(dζ )

)

+ y2i (t)

((∂Hi

∂z

)2

(t) +∫

R

∥∥∇kHi(t, ζ )∥∥2

ν(dζ )

)}]dt < ∞ for i = 1,2.

(19)

Then the following are equivalent:

(1)d

dsJ1(u1 + sβ1, u2) |s=0= d

dsJ2(u1, u2 + sβ2) |s=0= 0

for all bounded β1 ∈ A1, β2 ∈ A2.

(2) E

[∂

∂v1H1

(t,X(t), Y1(t),Z1(t),K1(t, ·),

v1, u2(t), λ1(t),p1(t)q1(t), r1(t, ·)) | E (1)

t

]

v1=u1(t)

= E

[∂

∂v2H2

(t,X(t), Y2(t),Z2(t),K2(t, ·),

u1(t), v2, λ2(t),p2(t), q2(t), r2(t, ·)) | E (2)

t

]

v2=u2(t)

= 0.

Proof See Appendix A. �

2.2 The Zero-Sum Game Case

In the zero-sum case we have

J1(u1, u2) + J2(u1, u2) = 0. (20)

Then the Nash equilibrium (u1, u2) ∈ A1 × A2 satisfying (6)–(7) becomes a saddlepoint for J (u1, u2) := J1(u1, u2). To see this, note that (6)–(7) imply that

J1(u1, u2) ≤ J1(u1, u2) = −J2(u1, u2) ≤ −J2(u1, u2)

and hence

J (u1, u2) ≤ J (u1, u2) ≤ J (u1, u2) for all u1, u2.

J Optim Theory Appl

From this we deduce that

infu2∈A2

supu1∈A1

J (u1, u2) ≤ supu1∈A1

J (u1, u2) ≤ J (u1, u2)

≤ infu2∈A2

J (u1, u2) ≤ supu1∈A1

infu2∈A2

J (u1, u2). (21)

Since we always have inf sup ≥ sup inf, we conclude that

infu2∈A2

supu1∈A1

J (u1, u2) = supu1∈A1

J (u1, u2) = J (u1, u2)

= infu2∈A2

J (u1, u2) = supu1∈A1

infu2∈A2

J (u1, u2), (22)

i.e., (u1, u2) ∈ A1 × A2 is a saddle point for J (u1, u2).In this case, only one Hamiltonian is needed, and only one set of adjoint equations.

Indeed, let g1 = g2 =: g, h1 = h2 =: h , f1 = −f2 := f,ϕ1 = −ϕ2 := ϕ, and ψ1 =−ψ2 := ψ . Then

H1(t, x, y, z, k,u1, u2, λ,p, q, r)

= f (t, x,u1, u2) + λg(t, x, y, z, k,u1, u2) + pb(t, x,u1, u2)

+ qσ(t, x,u1, u2) +∫

R

r(ζ )γ (t, x,u1, u2, ζ )ν(dζ ) (23)

and

H2(t, x, y, z, k,u1, u2, λ,p, q, r)

= −f (t, x,u1, u2) + λg(t, x, y, z, k,u1, u2) + pb(t, x,u1, u2)

+ qσ(t, x,u1, u2) +∫

R

r(ζ )γ (t, x,u1, u2, ζ )ν(dζ ). (24)

For u = (u1, u2) ∈ A1 × A2, let Xu(t) be defined by (1), and (Y u(t),Zu(t),Ku(t, ζ ))

be defined by (3) with gi = g and hi = h. The adjoint processes λi, i = 1,2, satisfy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dλi(t) = λi(t)

[∂g

∂y

(t,X(t), Y (t),Z(t),K(t, ·), u1(t), u2(t)

)dt

+ ∂g

∂z

(t,X(t), Y (t),Z(t),K(t, ·), u1(t), u2(t)

)dB(t)

+∫

R

d∇kg(t, ζ )

dν(ζ )N(dt, dζ )

], 0 ≤ t ≤ T ,

λi(0) = ψ ′i (Y (0)).

(25)

It follows that

λ2(t) = −λ1(t), t ∈ [0, T ]. (26)

J Optim Theory Appl

The adjoint processes for pi, qi , and ri, i = 1,2, become by (10)⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dp1(t) = −[∂f

∂x(t) + λ1(t)

∂g

∂x(t) + p1(t)

∂b

∂x(t) + q1(t)

∂σ

∂x(t)

+∫

R

r1(t, ζ )∂γ

∂x(t, ζ )ν(dζ )

]dt

+ q1(t) dB(t) +∫

R

r1(t, ζ )N(dt, dζ ), 0 ≤ t ≤ T ,

p1(T ) = ϕ′1

(X(T )

) + h′1

(X(T )

)λ1(T ),

(27)

and, by (10) and (26),⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dp2(t) = −[−∂f

∂x(t) − λ1(t)

∂g

∂x(t) + p2(t)

∂b

∂x(t) + q2(t)

∂σ

∂x(t)

+∫

R

r2(t, ζ )∂γ

∂x(t, ζ )ν(dζ )

]dt

+ q2(t) dB(t) +∫

R

r2(t, ζ )N(dt, dζ ), 0 ≤ t ≤ T ,

p2(T ) = −ϕ′1

(X(T )

) − h′1

(X(T )

)λ1(T ).

(28)

Thus, we see that(p2(t), q2(t), r2(t)

) = −(p1(t), q1(t), r1(t)

), t ∈ [0, T ]. (29)

Consequently,

− H2(t,X(t), Y (t),Z(t),K(t, ·), u1(t), u2(t), λ2(t),p2(t), q2(t), r2(t, ·)

)

= H1(t,X(t), Y (t),Z(t),K(t, ·), u1(t), u2(t), λ1(t),p1(t), q1(t), r1(t, ·)

).

We thus conclude that in the zero-sum game case, we only need one Hamiltonianand one quadruple of controlled adjoint processes. In the following, we set:

J (u1, u2) := E

[∫ T

0f

(t,Xu(t), u(t)

)dt + ϕ

(X(u)(T )

) + ψ(Yu(0)

)],

H := H1 as defined in (23), and(λ(t),p(t), q(t), r(t, ·)) := (

λ1(t),p1(t), q1(t), r1(t, ·))

as defined in (25), (27).

(30)

We can now state the necessary and sufficient maximum principles for the zero-sum game:

Theorem 2.3 (Necessary maximum principle for zero-sum forward–backwardgames) Assume that the conditions of Theorem 2.2 hold. Then the following areequivalent:

(1)d

dsJ (u1 + sβ1, u2) |s=0= d

dsJ (u1, u2 + sβ2) |s=0= 0 (31)

for all bounded β1 ∈ A1, β2 ∈ A2.

J Optim Theory Appl

(2) E

[∂

∂v1H

(t,X(t), Y (t),Z(t),K(t, ·),

v1, u2(t), λ(t),p(t), q(t), r(t, ·)) | E (1)t

]

v1=u1(t)

= E

[∂

∂v2H

(t,X(t), Y (t),Z(t),K(t, ·),

u1(t), v2, λ(t),p(t), q(t), r(t, ·)) | E (2)t

]

v2=u2(t)

= 0. (32)

Proof The proof is similar to that of Theorem 2.2 and is omitted. �

Corollary 2.1 Let u = (u1, u2) ∈ A1 × A2 be a Nash equilibrium (saddle point) forthe zero-sum game in Theorem 2.3. Then (32) holds.

Proof This follows from Theorem 2.3 by noting that if u = (u1, u2) is a Nash equi-librium, then (31) holds by (22). �

Similarly, we get

Theorem 2.4 (Sufficient maximum principle for zero-sum forward–backward games)Let (u1, u2) ∈ A1 × A2, with corresponding solutions X(t), Y (t), Z(t), K(t), λ(t),

p(t), q(t), r(t, ζ ). Suppose that the following hold:

• The functions x → ϕ(x) and x → ψ(x) are affine, and the function x → h(x) isconcave.

• (The conditional maximum principle)

ess supv1∈A1E

[H

(t, X(t), Y (t), Z(t), K(t, ·),

v1, u2(t), λ(t), p(t), q(t), r(t, ·)) | E (1)t

]

= E[H

(t, X(t), Y (t), Z(t), K(t, ·),

u1(t), u2(t), λ(t), p(t), q(t), r(t, ·)) | E (1)t

](33)

and

ess infv2∈A2 E[H

(t, X(t), Y (t), Z(t), K(t, ·),

u1(t), v2, λ(t), p(t), q(t), r(t, ·)) | E (2)t

]

= E[H

(t, X(t), Y (t), Z(t), K(t, ·),

u1(t), u2(t), λ(t), p(t), q(t), r(t, ·)) | E (2)t

]. (34)

J Optim Theory Appl

• (The Arrow conditions) The function

H(x, y, z, k)

:= ess supv1∈A1E

[H

(t, x, y, z, k, v1, u2(t), λ(t), p(t), q(t), r(t, ·)) | E (1)

t

]

is concave, and the function

∨H (x, y, z, k)

:= ess infv2∈A2 E[H

(t, x, y, z, k, u1(t), v2, λ(t), p(t), q(t), r(t, ·)) | E (2)

t

]

is convex for all t ∈ [0, T ], a.s.• Assume that

d∇kg(t, ζ )

dν(ζ )≥ −1.

• The growth condition (13) holds with pi = p, etc.

Then u(t) = (u1(t), u2(t)) is a saddle point for J (u1, u2).

Proof The proof is similar to that of Theorem 2.1 and is omitted. �

3 Stochastic Control under Model Uncertainty

Let X(t) = Xvx(t) be a controlled Itô–Lévy process of the form

dX(t) = b(t,X(t), v(t),ω

)dt + σ

(t,X(t), v(t),ω

)dB(t)

+∫

R

γ(t,X

(t−

), v(t), ζ,ω

)N(dt, dζ ), 0 ≤ t ≤ T ,

X(0) = x ∈ R, (35)

where v(·) is the control process, and b,σ , and γ are as in Sect. 2.1.We consider a model uncertainty setup, represented by a probability measure Q =

Qθ which is equivalent to P , with the Radon–Nikodym derivative on Ft given by

d(Q | Ft )

d(P | Ft )= Gθ(t), (36)

where, for 0 ≤ t ≤ T , Gθ(t) is a martingale of the form

dGθ(t) = Gθ(t−

)[θ0(t) dB(t) +

∫

R

θ1(t, ζ )N(dt, dζ )

]; Gθ(0) = 1. (37)

Here θ = (θ0, θ1) may be regarded as a scenario control. Let A1 denote a given familyof admissible controls v, and A2 denote a given set of admissible scenario controlsθ such that E[∫ T

0 {|θ20 (t)| + ∫

Rθ2

1 (t, ζ )ν(dζ )}dt] < ∞ and θ1(t, ζ ) ≥ −1 + ε for

J Optim Theory Appl

some ε > 0. Let E (1)0≤t≤T and E (2)

0≤t≤T be given subfiltrations of F0≤t≤T , representingthe information available to the controllers at time t . It is required that v ∈ A1 beE 1

t -predictable, and θ ∈ A2 be E 2t -predictable. We set u = (v, θ).

We consider the stochastic differential game to find (v, θ ) ∈ A1 × A2 such that

supv∈A1

infθ∈A2

EQθ

[W(v, θ)

] = EQθ

[W(v, θ)

] = infθ∈A2

supv∈A1

EQθ

[W(v, θ)

], (38)

where

W(v, θ) = U(Xv(T )

) +∫ T

0F

(s,Xv(s), v(s), θ(s)

)ds, (39)

where U and F are given functions.For example, U is a given utility function, and F(t, x, v, θ) = U1(t, x, v) + ρ(θ),

with U1 a utility function and ρ a convex function. The term EQθ [∫ T

0 ρ(θ(t)) dt] canthen be seen as a penalty term, penalizing the difference between Qθ and the originalprobability measure P .

We have

EQθ

[W(v, θ)

] = E

[Gθ(T )U

(Xv(T )

) +∫ T

0Gθ(s)F

(s,Xv(s), u(s)

)ds

]. (40)

We now define Y(t) = Yv,θ (t) by

Y(t) = E

[Gθ(T )

Gθ(t)U

(Xv(T )

) +∫ T

t

Gθ (s)

Gθ (t)F

(s,Xv(s), u(s)

)ds | Ft

], t ∈ [0, T ].

(41)Then we recognize Y(t) as the solution of the linear BSDE (see Lemma B.1)

dY (t) = −[F

(t,Xv(t), u(t)

) + θ0(t)Z(t) +∫

R

θ1(t, ζ )K(t, ζ )ν(dζ )

]dt

+ Z(t) dB(t) +∫

R

K(t, ζ )N(dt, dζ ), 0 ≤ t ≤ T , (42)

Y(T ) = U(Xv(T )

).

Note that

Y(0) = Yv,θ (0) = EQθ

[W(v, θ)

]. (43)

Therefore, problem (38) can be written as

supv∈A1

infθ∈A2

Yv,θ (0) = Y v,θ (0) = infθ∈A2

supv∈A1

Yv,θ (0), (44)

where Yv,θ (t) is given by the forward–backward system (35) and (42). This is a zero-sum stochastic differential game (SDG) of forward–backward SDEs of the form (30)with f = ϕ = 0, ψ = Id, and h(x) = U(x).

J Optim Theory Appl

Proceeding as in Sect. 2, define the Hamiltonian

H : [0, T ] × R × R × R0 × R × A1 × A2 × R × R × R × R → R

by

H(t, x, y, z, k, v, θ, λ,p, q, r)

:=[F(t, x,u) + θ0z +

∫

R

θ1(t, ζ )k(t, ζ )ν(dζ )

]λ

+ b(t, x, v)p + σ(t, x, v)q +∫

R

γ (t, x, v, ζ )r(ζ )ν(dζ ). (45)

Define a pair of FBSDEs in the adjoint processes λ(t),p(t), q(t), r(t, ζ ) as follows.Forward SDE for λ(t):

dλ(t) = ∂H

∂y(t) dt + ∂H

∂z(t) dB(t) +

∫

R

d∇kHi(t, ζ )

dν(ζ )N(dt, dζ )

= λ(t)θ0(t) dB(t) + λ(t)

∫

R

θ1(t, ζ )(·)N(dt, dζ ), t ∈ [0, T ],

λ(0) = 1.

(46)

Backward SDE for (p(t), q(t), r(t, ζ )):

dp(t) = −∂H

∂x(t) dt + q(t) dB(t) +

∫

R

r(t, ζ )N(dt, dζ )

= −{

∂F

∂x(t)λ(t) + p(t)

∂b

∂x(t) + q(t)

∂σ

∂x(t) +

∫

R

r(t, ζ )∂γ

∂x(t, ζ )ν(dζ )

}dt

+ q(t) dB(t) +∫

R

r(t, ζ )N(dt, dζ ), t ∈ [0, T ],

p(T ) = λ(T )U ′(X(T )).

(47)

Here we have used the abbreviated notation

∂H

∂y(t) = ∂H

∂y

(t,X(t), Y (t),Z(t),K(t, ·), v(t), θ(t), λ(t),p(t), q(t), r(t, ·))

and similarly for the other partial derivatives. We now present a necessary maximumprinciple for the forward–backward stochastic differential game (35), (42), (44) byadapting Theorem 2.3 to this case.

Theorem 3.1 Suppose that the conditions of Theorem 2.2 hold. Let u = (v, θ ) ∈A1 ×A2, with corresponding solutions X(t), Y (t), Z(t), K(t, ·), λ(t), p(t), q(t), r(t, ·)

J Optim Theory Appl

of Eqs. (35), (42), (47), and (46). Suppose that (44) holds, together with (13). Thenthe following holds:

E

[λ(t)

∂F

∂v

(t, X(t), u(t)

) + p(t)∂b

∂v

(t, X(t), v(t)

)

+ q(t)∂σ

∂v

(t, X(t), v(t)

) +∫

R

r(t, ζ )∂γ

∂v

(t, X(t), v(t), ζ

)ν(dζ ) | E (1)

t

]= 0,

E

[λ(t)

(∂F

∂θ0

(t, X(t), u(t)

) + Z(t)

)| E (2)

t

]= 0,

E

[λ(t)

(∇θ1F

(t, X(t), u(t)

) +∫

R

(·)K(t, ζ )ν(dζ )

)| E (2)

t

]= 0.

Note that both ∇θ1F and∫

R(·)K(t, ζ )ν(dζ ) are linear functionals, the latter being

defined by the action

ϕ →∫

R

ϕ(ζ )K(t, ζ )ν(dζ )

for all bounded continuous functions ϕ : R0 → R.

4 Robust Optimal Portfolio and Consumption with Penalty

We now apply this to the following portfolio problem under model uncertainty. Werestrict here ourselves to the case E (1)

t = E (2)t = Ft , t ∈ [0, T ].

Consider a financial market consisting of a bond with unit price S0(t)=1,0≤t≤T ,and a stock, with unit price S(t) given by

dS(t) = S(t−

)[b0(t) dt + σ0(t) dB(t) +

∫

R

γ0(t, ζ )N(dt, dζ )

], (48)

where b0(t) = b0(t,ω), σ0(t) = σ0(t,ω), and γ0(t, ζ ) = γ0(t, ζ,ω) are given {Ft }-predictable processes such that γ0 ≥ −1 + ε for some ε > 0, and

E

[∫ T

0

{∣∣b0(t)∣∣ + σ 2

0 (t) +∫

R

γ 20 (t, ζ )ν(dζ )

}dt

]< ∞.

Note that this system is non-Markovian since the coefficients are random processes.Let X(t) = Xπ,c(t) be the wealth process corresponding to a portfolio π(t) and a

consumption rate c(t), i.e.,⎧⎪⎪⎪⎨

⎪⎪⎪⎩

dX(t) = π(t)

[b0(t) dt + σ0(t) dB(t) +

∫

R

γ0(t, ζ )N(dt, dζ )

]− c(t) dt,

t ∈ [0, T ],X(0) = x ∈ R.

(49)For π and c to be admissible, we require that Xπ,c ≥ 0 for all t ∈ [0, T ].

J Optim Theory Appl

We consider the stochastic differential game to find (π , c, θ ) ∈ A1 × A2 such that

supπ,c∈A1

infθ∈A2

EQθ

[W(π, c, θ)

] = EQθ

[W(π, c, θ )

] = infθ∈A2

supπ,c∈A1

EQθ

[W(π, c, θ)

]

(50)with

W(π, c, θ) = U(Xπ,c(T )

) +∫ T

0U1

(c(s)

)ds +

∫ T

0ρ(θ(s)

)ds, (51)

where U and U1 are utility functions, and ρ is a convex function. We have seen inSect. 3 that this problem can be written as

supπ,c∈A1

infθ∈A2

Yπ,c,θ (0) = Y π,c,θ (0) = infθ∈A2

supπ,c∈A1

Yπ,c,θ (0), (52)

where Y(t) = Yπ,c,θ (t) is given by

dY (t) = −[U1

(c(t)

) + ρ(θ(t)

) + θ0(t)Z(t) +∫

R

θ1(t, ζ )K(t, ζ )ν(dζ )

]dt

+ Z(t) dB(t) +∫

R

K(t, ζ )N(dt, dζ ), t ∈ [0, T ], (53)

Y(T ) = U(X(T )

). (54)

In particular,

Yπ,c,θ (0) = E

[Gθ(T )U

(X(T )

) +∫ T

0Gθ(s)

[U1

(c(s)

) + ρ(θ(s)

)]ds

]. (55)

The Hamiltonian for the problem (52) is, by (45),

H(t, x, y, z, k,π, c, θ, λ,p, q, r)

=[U1(c) + ρ(θ) + θ0z +

∫

R

θ1(ζ )k(ζ )ν(dζ )

]λ

+ (πb0(t)

) − c)p + πσ0(t)q + π

∫

R

γ0(t, ζ )r(ζ )ν(dζ ). (56)

The forward SDE for λ(t) = λθ (t) is

dλ(t) = λ(t)

[θ0(t) dB(t) +

∫

R

θ1(t, ζ )N(dt, dζ )

], t ∈ [0, T ],

λ(0) = 1.

(57)

By comparing (37) and (57), we see that

λ(t) = Gθ(t), t ∈ [0, T ]. (58)

J Optim Theory Appl

The BSDE for (pπ,c,θ (t), qπ,c,θ (t), rπ,c,θ (t, ζ )) = (p(t), q(t), r(t, ζ )) is (see (46)–(47))

dp(t) = q(t) dB(t) +∫

R

r(t, ζ )N(dt, dζ ), t ∈ [0, T ],p(T ) = λ(T )U ′(X(T )

) = Gθ(T )U ′(X(T )).

(59)

We get

p(t) = E[Gθ(T )U ′(X(T )

) | Ft

]> 0, t ∈ [0, T ]. (60)

4.1 Viability and Martingale Measures

We now apply the necessary maximum principle given by Theorem 3.1. MaximizingH with respect to π and c and minimizing H with respect to θ = (θ0, θ1) gives thefollowing first-order conditions for the optimal portfolio π , the optimal consumptionrate c, and the optimal scenario parameter θ = (θ0, θ1):

b0(t)p(t) + σ0(t)q(t) +∫

R

γ0(t, ζ )r(t, ζ )ν(dζ ) = 0, (61)

U ′1

(c(t)

)Gθ(t) = p(t) = E

[Gθ(T )U ′(X(T )

) | Ft

], (62)

∂ρ

∂θ0

(θ(t)

) = −Z(t), (63)

∇θ1ρ(θ(t)

)(·) = −

∫

R

(·)K(t, ζ )ν(dζ ). (64)

Equation (61) can be written as

b0(t) + σ0(t)q(t)

p(t)+

∫

R

γ0(t, ζ )r(t, ζ )

p(t)ν(dζ ) = 0. (65)

By the Girsanov theorem (see, e.g., [1], Chap. 1), this means that the measure Q

on FT , defined by

dQ(ω) := R(T )dP (ω), (66)

with R(t) = Rπ,c,θ (t) , t ∈ [0, T ], given by

dR(t) = R(t−

)[ q(t)

p(t−)dB(t) +

∫

R

r(t, ζ )

p(t−)N(dt, dζ )

], t ∈ [0, T ],

R(0) = 1,

(67)

is an equivalent local martingale measure (ELMM) for the market price process S(t)

given by (48). By (59) and (67) we get

R(t−

)dp(t) = R

(t−

)[q(t) dB(t) +

∫

R

r(t, ζ )N(dt, dζ )

]

= p(t−

)dR(t),

J Optim Theory Appl

i.e.,

dp(t)

p(t−)= dR(t)

R(t−).

We conclude that

p(t) = p(0)R(t) = E[Gθ(T )U ′(X(T )

)]R(t), 0 ≤ t ≤ T .

Therefore,

dQ(ω) = Gθ(T )U ′(Xπ,c(T ))

E[Gθ(T )U ′(Xπ,c(T ))] dP (ω) on FT . (68)

This proves the first part of the following result:

Theorem 4.1 (a) Suppose that there exists an optimal portfolio π , an optimal con-sumption rate c, and an optimal scenario parameter θ for the model uncertaintyportfolio-consumption optimization problem (52). Then (62) holds, and the measureQ = Qπ,c,θ defined by (68) is an ELMM for the market (48).

(b) Conversely, suppose that there exists a portfolio π , a consumption rate c, anda scenario parameter θ such that (62) holds and Q = Qπ,c,θ defined by (68) is anELMM for the market (48). Suppose that there exists a unique solution Y(t) of theBSDE (54), with θ satisfying (63)–(64).

Define θ = θ (z, k) = (θ0(z, k), θ1(z, k, ·)) as the solution of the equation

∇ρ(θ) :=(

∂ρ

∂θ0(θ),∇θ1ρ(θ)

)=

(−z,−

∫

R

(·)k(ζ )ν(dζ )

). (69)

Suppose that the function

H(z, k) := ρ(θ (z, k)

) + θ0(z, k)z +∫

R

θ1(z, k, ζ )k(ζ )ν(dζ ) (70)

is concave. Then π is an optimal portfolio, c is an optimal consumption rate, and θ

is an optimal scenario parameter for the problem (52).

Proof of (b) If Q is defined by (68), then, by (59),

dQ | Ft

dP | Ft

= E[Gθ(T )U ′(Xπ,c(T )) | Ft ]E[Gθ(T )U ′(Xπ,c(T ))] = p(t)

p(0), (71)

where

dp(t) = p(t−

)[ q(t)

p(t)dB(t) +

∫

R

r(t, ζ )

p(t−)N(dt, dζ )

], t ∈ [0, T ],

p(T ) = Gθ(T )U ′(Xπ,c(T )).

(72)

It follows by the Girsanov theorem that if Q is an ELMM for S(t),we must have

b0(t) + σ0(t)q(t)

p(t)+

∫

R

γ0(t, ζ )r(t, ζ )

p(t)ν(dζ ) = 0.

J Optim Theory Appl

This implies (61). We conclude that all Eqs. (61)–(64) are satisfied. Moreover, sinceθ (z, k) defined by (69) is the minimizer w.r.t. θ of H(t, x, y, z, k,π, c, θ, λ,p, q, r),it follows by (70) and the Arrow condition in Theorem 2.4 applied to our situation thatall conditions of the sufficient maximum principle are satisfied, and we can concludethat π, c, and θ are optimal. �

Remark 4.1 In agreement with the terminology used elsewhere in similar situations,we call the market (48) model uncertainty viable if the problem (52) has a solutionπ, c, θ . Then, under the assumptions of Theorem 4.1, we have proved that the marketis model uncertainty viable with optimal π, c, and θ if and only if (62) holds and themeasure

dQπ,c,θ := Gθ(T )U ′(Xπ,c(T ))

E[Gθ(T )U ′(Xπ,c(T ))] dP

is an ELMM. This is an extension to model uncertainty markets of the followingresult which is well known in classical types of financial markets, mainly the equiva-lence between (i) the existence of an optimal portfolio (viability) and (ii) the measuredQ := U ′(X(T ))/E[U ′(X(T ))]dP being an ELMM. See, e.g., [10–13].

Remark 4.2 Using the same method, we can also consider performances of the form(39) with U(Xπ,c(T )) replaced by

∫ T

0 U(Xπ,c(t)) dt .

4.2 The Entropic Penalty Case

We consider now the case where the penalty function has the form

ρa(θ) := 1

aρ1(θ), (73)

where a > 0 is a given parameter, and

ρ1(θ0, θ1)(t) := 1

2θ2

0 (t)+∫

R

{(1 + θ1(t, ζ )

)ln

(1 + θ1(s, ζ )

)− θ1(s, ζ )}ν(dζ ). (74)

Note that, by (37),

Gθ(t) = exp

(∫ t

0θ0(s) dB(s) − 1

2

∫ t

0θ2

0 (s) ds +∫ t

0

∫

R

ln(1 + θ1(s, ζ )

)N(ds, dζ )

+∫ t

0

∫

R

{ln

(1 + θ1(s, ζ )

) − θ1(s, ζ )}ν(dζ )

). (75)

Therefore, the relative entropy E (Qθ | P) of Qθ with respect to P defined as

E(Qθ | P ) := E

[dQθ

dPln

(dQθ

dP

)]

is seen to be

E(Qθ | P ) = E

[Gθ(T ) lnGθ(T )

] = E

[∫ T

0Gθ(t)ρ1

(θ(t)

)dt

].

J Optim Theory Appl

We call ρa the entropic penalty function.Then, the optimality conditions (63) and (64) for θ0, θ1 become

1

aθ0(t) = −Z(t),

1

a

∫

R

ln(1 + θ1(t, ζ )

)(·)ν(dζ ) = −

∫

R

(·)K(t, ζ )ν(dζ ),

(76)

i.e.,

1

aln

(1 + θ1(t, ζ )

) = −K(t, ζ ). (77)

Substituted into (54), this gives

dY (t) =[−U1

(c(t)

) + 1

2aθ2

0 (t) − 1

a

∫

R

{ln

(1 + θ1(t, ζ )

) − θ1(t, ζ )}ν(dζ )

]dt

− 1

aθ0(t) dB(t) − 1

a

∫

R

ln(1 + θ1(t, ζ )

)N(dt, dζ );

Y(T ) = U(X(T )

).

(78)

It follows that (78) can be written as

Y(t) = Y(0) − 1

alnGθ(t) −

∫ t

0U1

(c(s)

)ds, t ∈ [0, T ].

Taking exponentials gives

Gθ(t) = exp

(−aY (t) − a

∫ t

0U1

(c(s)

)ds

)exp

(aY (0)

), t ∈ [0, T ].

In particular, if we put t = T , we get

Gθ(T ) = exp(−aU(X(T )) − a∫ T

0 U1(c(s)) ds)

E[exp(−aU(X(T )) − a∫ T

0 U1(c(s)) ds)]. (79)

This gives the optimal scenario Gθ(T ) expressed in terms of the optimal terminalwealth X(T ) = Xπ,c(T ) and the optimal consumption rate c.

Combining this with Theorem 4.1a), we get that

dQ := exp(−aU(X(T )) − a∫ T

0 U1(c(s)) ds)U ′(X(T ))

E[exp(−aU(X(T )) − a∫ T

0 U1(c(s)) ds)U ′(X(T ))]dP (80)

is an ELMM.With our choice of ρ, we see that the concavity condition for the function H de-

fined in (70) is satisfied. Therefore, if we combine Theorem 4.1 with the calculationsabove, we get the following:

J Optim Theory Appl

Theorem 4.2 The following, (a) and (b), are equivalent:

(a) There exists an optimal (π, c, θ) for the problem (52)–(54).(b) The measure Q defined by (80) is an ELMM, and

U ′1

(c(t)

)Gθ(t) = E

[Gθ(T )U ′(Xπ,c(T )

) | Ft

], 0 ≤ t ≤ T , (81)

with

Gθ(t) = E[exp{−aU(Xπ,c(T )) − a∫ T

0 U1(c(s)) ds} | Ft ]E[exp{−aU(Xπ,c(T )) − a

∫ T

0 U1(c(s)) ds}], 0 ≤ t ≤ T . (82)

An equivalent formulation involving only π and c is as follows:

Theorem 4.3 The following, (a) and (b), are equivalent:

(a) There exists an optimal (π, c, θ) for the problem (52)–(54).(b) The measure Q defined by (80) is an ELMM, and

U ′1

(c(t)

)E

[exp

{−aU

(Xπ,c(T )

) − a

∫ T

0U1

(c(s)

)ds

}| Ft

]

= E

[exp

{−aU

(Xπ,c(T )

) −∫ T

0aU1

(c(s)

)ds

}U ′(Xπ,c(T )

) | Ft

],

0 ≤ t ≤ T . (83)

Proof (a) ⇒ (b): Suppose that a) holds. Then, using Theorem 4.2 and substituting(82) into (81), we get (83).

(b) ⇒ (a): Conversely, let (π, c) be such that the measure Q defined by (80) is anELMM and (83) holds. Define G(t) by

G(t) := E[exp(−aU(Xπ,c(T )) − a∫ T

0 U1(c(s)) ds) | Ft ]E[exp(−aU(Xπ,c(T )) − a

∫ T

0 U1(c(s)) ds)]> 0, 0 ≤ t ≤ T , (84)

and let θ0, θ1 be such that

dG(t) = G(t−

)[θ0(t) dB(t) +

∫

R

θ1(t, ζ )N(dt, dζ )

], 0 ≤ t ≤ T , (85)

is the Itô representation of the martingale G(t).Then, with Gθ(t) = G(t) we see that (π, c, θ) satisfies all the requirements in part

(b) of Theorem 4.2, and hence (π, c, θ) is optimal. �

We now compare this result to a model certainty problem, as follows.Let X(t) be as in (49) and choose a utility function V . With U1,U as above, define

the performance functional

J0(π, c) := E

[V

(U

(X(T )

) +∫ T

0U1

(c(t)

)dt

)]. (86)

J Optim Theory Appl

We want to find (π , c) ∈ A1 such that

sup(π,c)∈A1

J0(π, c) = J0(π, c). (87)

To put this problem into the context of our maximum principle, we define X2(t) =X(t) and

dX1(t) = U1(c(t)

)dt, X1(0) = 0. (88)

Then

J0(π, c) = E[V

(U

(X(T )

) + X1(T ))]

,

and the corresponding Hamiltonian becomes

H = U1(c)p1 + (πb0(t) − c

)p2 + πσ0(t)q2 +

∫

R

π r2(ζ )γ0(t, ζ )ν(dζ ). (89)

The adjoint equations are

⎧⎨

⎩

dp1(t) = q1(t) dB(t) +∫

R

r1(t, ζ )N(dt, dζ ), 0 ≤ t ≤ T ,

p1(T ) = V ′(U(X(T )

) + X1(T )),

(90)

⎧⎨

⎩

dp2(t) = q2(t) dB(t) +∫

R

r2(t, ζ )N(dt, dζ ), 0 ≤ t ≤ T ,

p2(T ) = V ′(U(X(T )

) + X1(T ))U ′(X(T )

).

(91)

Arguing as before, we now deduce that (π, c) is optimal for (87) if and only if

U ′1

(c(t)

) = E[V ′(U(X(T )) + X1(T ))U ′(X(T )) | Ft ]E[V ′(U(X(T )) + X1(T )) | Ft ] (92)

and the measure

dQ0 := V ′(U(X(T )) + X1(T ))U ′(X(T ))

E[V ′(U(X(T )) + X1(T ))U ′(X(T ))] dP (93)

is an ELMM for the market (48).Therefore, if we choose

V (x) := −1

aexp(−ax) (94)

and compare (92) and (93) with (83) and (80), respectively, we obtain the following:

J Optim Theory Appl

Theorem 4.4 (Model uncertainty reduction theorem) Suppose that (π, c) ∈ A is op-timal for the (model certainty) portfolio-consumption problem

sup(π,c)∈A1

−1

aE

[exp

(−aU

(Xπ,c(T )

) − a

∫ T

0U1

(c(t)

)dt

)]. (95)

Then (π, c) is optimal for the model uncertainty portfolio-consumption problem

supπ,c∈A1

infθ∈A2

J (π, c, θ) = infθ∈A2

supπ,c∈A1

J (π, c, θ) (96)

with

J (π, c, θ) = E

[Gθ(T )U

(Xπ,c(T )

)+∫ T

0Gθ(s)U1

(c(s)

)ds

]+ 1

aE(Qθ | P )

, (97)

where E (Qθ | P) is the relative entropy of Qθ with respect to P . Moreover, theRadon–Nikodym derivative of the optimal probability measure Qθ is given by

Gθ(t) := E[exp(−aU(Xπ,c(T )) − a∫ T

0 U1(c(s)) ds) | Ft ]E[exp(−aU(Xπ,c(T )) − a

∫ T

0 U1(c(s)) ds)], 0 ≤ t ≤ T . (98)

Remark 4.3 When the optimal Gθ(T ) is known, we can find the corresponding θ0(t)

and θ1(t, ζ ) in feedback form as follows: By the Clark–Ocone theorem combinedwith (37), we get

θ0(t) = 1

Gθ(t)E

[DtG

θ(T ) | Ft

], 0 ≤ t ≤ T ,

θ1(t, ζ )(t) = 1

Gθ(t)E

[Dt,ζ G

θ (T ) | Ft

], 0 ≤ t ≤ T ,

where Dt and Dt,ζ denote the Malliavin derivatives with respect to B(·) and N(·, ·),respectively (see [14]). We refer to [15] for more information on Malliavin calculusfor Lévy processes.

Theorem 4.4 shows that the problem of optimal portfolio and consumption undermodel uncertainty with entropic penalty can be reduced to a corresponding modelcertainty problem, but with a different performance functional (different utilities).

Model Uncertainty and Risk Aversion The Arrow–Pratt coefficient of absolute riskaversion at x of a utility function U is defined by

αU(x) := −U ′′(x)

U ′(x).

Define

W(x) = −1

aexp

(−aU(x)), where a > 0.

J Optim Theory Appl

Then

αW(x) = −W ′′(x)

W ′(x)= −U ′′(x) − aU ′(x)2

U ′(x).

Hence,

αW(x) = αU(x) + aU ′(x).

We conclude that the risk aversion of W is bigger that the risk aversion of U . Hence,in view of Theorem 4.4, we can say that, in this sense, model uncertainty increases therisk aversion. For more discussion on this topic, see [16] and the references therein.

4.3 Relation of Robust Portfolio-Consumption Problem with Entropic Penalty withRisk-Sensitive Control

Fix π , c and let X(T ) and X1(T ) = ∫ T

0 U1(c(t)) dt be the corresponding terminalwealth and total utility from consumption, respectively.

Consider the problem

I := infθ

{EQθ

[U

(X(T )

) +∫ T

0U1

(c(t)

)dt +

∫ T

0ρa

(θ(t)

)dt

]}, (99)

where ρa is the entropic penalty defined in (73), so that

E

[∫ T

0Gθ(t)ρa

(θ(t)

)dt

]= 1

aE

[Gθ(T ) lnGθ(T )

]. (100)

We have seen that (99) can be written as

I = infθ

Y θ (0), (101)

where Y θ (t) solves the BSDE

dY (t) = −[U1

(c(t)

) + ρa

(θ(t)

) + θ0(t)Z(t) +∫

R

θ1(t, ζ )K(t, ζ )ν(dζ )

]dt

+ Z(t) dB(t) +∫

R

K(t, ζ )N(dt, dζ )], t ∈ [0, T ], (102)

Y(T ) = U(X(T )

). (103)

By the comparison theorem for BSDEs (see [17] and [18]) , we see that to solve(101), all we need is to minimize

θ → ρa

(θ(t)

) + θ0(t)Z(t) +∫

R

θ1(t, ζ )K(t, ζ )ν(dζ ) dt.

The first-order conditions for optimal θ are

∂ρa

∂θ0

(θ(t)

) = −Z(t), ∇θ1ρa

(θ(t)

)(·) = −

∫

R

(·)K(t, ζ )ν(dζ ), (104)

J Optim Theory Appl

i.e.,

Z(t) = −1

aθ0(t),

K(t, ζ ) = −1

aln

(1 + θ1(t, ζ )

).

(105)

Substituting this into (103) and arguing as before, we obtain the formula (79), i.e.,

Gθ(T ) = exp(−aU(X(T )) − a∫ T

0 U1(c(s)) ds)

E[exp(−aU(X(T )) − a∫ T

0 U1(c(s)) ds)]. (106)

Therefore, since θ is optimal, we get

I = E

[Gθ(T )

(U

(X(T )

) + X1(T ) + 1

alnGθ(T )

)]

= E

[Gθ(T )

(U

(X(T )

) + X1(T ) − U(X(T )

) − X1(T )

− 1

alnE

[exp

(−aU(X(T )

) − aX1(T ))])

]

= −1

aE

[Gθ(T )

]lnE

[exp

(−aU(X(T )

) − aX1(T ))]

= −1

alnE

[exp

(−aU(X(T )

) − aX1(T ))]

.

We conclude that our robust portfolio-consumption problem (101) with entropicpenalty is related to risk-sensitive control as follows:

infθ

{EQθ

[U

(X(T )

) + X1(T ) +∫ T

0ρa

(θ(t)

)dt

]}

= −1

alnE

[exp

(−aU(X(T )

) − aX1(T ))]

. (107)

This is an extension to consumption-portfolio settings of the risk-sensitive controlresult in [19]. See also the references therein.

Remark 4.4 Equation (107) shows in particular that the sup–inf part of the modeluncertainty problem (50) can be reduced to an optimal consumption-portfolio prob-lem with model certainty. Note that the proof of this is relatively easy and does notrequire the whole machinery that we have set up in the previous sections. However, itseems that the inf–sup part of the same problem cannot be proved so easily; this partrequires techniques for optimal control of forward–backward SDE control/games, asdeveloped in this paper.

J Optim Theory Appl

5 Concluding Remarks

This paper has two main parts:

• The first is a general maximum principle for forward–backward stochastic differ-ential games for Itô–Lévy processes with partial information. This is a result ofindependent interest, and it has a potential for being useful in many situations.

• The second is an application of the general theory in the first part to optimal port-folio and consumption problems under model uncertainty, in markets modeled byItô–Lévy processes. The model uncertainty is represented by a family of equiva-lent probability measures, with a penalty for being “far away” from the originalmeasure P . We obtain a characterization of market viability under model uncer-tainty in terms of equivalent local martingale measures. If the penalty function isentropic, we prove a reduction theorem saying that the model uncertainty problemcan be transformed into a problem without model uncertainty, but with differentutility functions.

It is natural to ask if similar results could be obtained with other representations ofmodel uncertainty. For example, one could consider uncertainties in the noise termsor put constraints on the families of probability measures. In particular, can our reduc-tion theorem for the entropic penalty be extended to more general model uncertaintycontexts?

Acknowledgements We thank Olivier Menoukeu Pamen and Marie-Claire Quenez for helpful com-ments.

The research leading to these results has received funding from the European Research Council underthe European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no[228087]

Appendix A: Proofs of the Maximum Principles for FBSDE Games

We first recall some basic concepts and results from Banach space theory. Let V bean open subset of a Banach space X with norm ‖ · ‖, and let F : V → R.

(i) We say that F has a directional derivative (or Gâteaux derivative) at x ∈ X inthe direction y ∈ X if

DyF(x) := limε→0

1

ε

(F(x + εy) − F(x)

)

exists.(ii) We say that F is Fréchet differentiable at x ∈ V if there exists a linear map

L : X → R

such that

limh→0h∈X

1

‖h‖∣∣F(x + h) − F(x) − L(h)

∣∣ = 0.

J Optim Theory Appl

In this case, we call L the gradient (or Fréchet derivative) of F at x, and wewrite

L = ∇xF.

(iii) If F is Fréchet differentiable, then F has a directional derivative in all directionsy ∈ X , and

DyF(x) = ∇xF (y).

Proof of Theorem 2.1 (Sufficient maximum principle) We first prove that

J1(u1, u2) ≤ J1(u1, u2) for all u1 ∈ A1.

To this end, fix u1 ∈ A1 and consider

� := J1(u1, u2) − J1(u1, u2) = I1 + I2 + I3, (A.1)

where

I1 = E

[∫ T

0

{f1

(t,X(t), u(t)

) − f1(t, X(t), u(t)

)}dt

], (A.2)

I2 = E[ϕ1

(X(T )

) − ϕ1(X(T )

)], (A.3)

I3 = E[ψ1

(Y1(0)

) − ψ1(Y1(0)

)]. (A.4)

By (8) we have

I1 = E

[∫ T

0

{H1(t) − H1(t) − λ1(t)

(g1(t) − g1(t)

) − p1(t)(b(t) − b(t)

)

− q1(t)(σ(t) − σ (t)

) −∫

R

r1(t, ζ )(γ (t, ζ ) − γ (t, ζ )

)ν(dζ )

}dt

]. (A.5)

By the concavity of ϕ1, (10), and the Itô formula,

I2 ≤ E[ϕ′

1

(X(T )

)(X(T ) − X(T )

)]

= E[p1(T )

(X(T ) − X(T )

)] − E[λ1(T )h′

1

(X(T )

)(X(T ) − X(T )

)]

= E

[∫ T

0p1

(t−

)(dX(t) − dX(t)

) +∫ T

0

(X

(t−

) − X(t−

))dp1(t)

+∫ T

0q1(t)

(σ(t) − σ (t)

)dt

+∫ T

0

∫

R

r1(t, ζ )(γ (t, ζ ) − γ (t, ζ )

)ν(dζ ) dt

]

− E[λ1(T )h′

1

(X(T )

)(X(T ) − X(T )

)]

J Optim Theory Appl

= E

[∫ T

0p1(t)

(b(t) − b(t)

)dt +

∫ T

0

(X(t) − X(t)

)(−∂H1

∂x(t)

)dt

+∫ T

0q1(t)

(σ(t) − σ (t)

)dt +

∫ T

0

∫

R

r1(t, ζ )(γ (t, ζ ) − γ (t, ζ )

)ν(dζ ) dt

]

− E[λ1(T )h′

1

(X(T )

)(X(T ) − X(T )

)]. (A.6)

By the concavity of ψ1, (5), (9), and the concavity of ϕ,

I3 = E[ψ1

(Y1(0)

) − ψ1(Y1(0)

)]

≤ E[ψ ′

1

(Y1(0)

)(Y1(0) − Y1(0)

)]

= E[λ1(0)

(Y1(0) − Y1(0)

)]

= E[(

Y1(T ) − Y1(T ))λ1(T )

]

−{E

[∫ T

0

(Y1

(t−

) − Y1(t−

))dλ1(t) +

∫ T

0λ1

(t−

)(dY1(t) − dY1(t)

)

+∫ T

0

∂H1

∂z(t)

(Z1(t) − Z1(t)

)dt

+∫ T

0

∫

R

∇kH1(t, ζ )(K1(t, ζ ) − K1(t, ζ )

)ν(dζ ) dt

]}

= E[(

h1(X(T )

) − h1(X(T )

))λ1(T )

]

−{E

[∫ T

0

∂H1

∂y(t)

(Y1(t) − Y1(t)

)dt

+∫ T

0λ1(t)

(−g1(t) + g1(t))dt

+∫ T

0

∂H1

∂z(t)

(Z1(t) − Z1(t)

)dt

+∫ T

0

∫

R

∇kH1(t, ζ )(K1(t, ζ ) − K1(t, ζ )

)ν(dζ ) dt

]}

≤ E[λ1(T )h′

1

(X(T )

)(X(T ) − X(T )

)]

−{E

[∫ T

0

∂H1

∂y(t)

(Y1(t) − Y1(t)

)dt

+∫ T

0λ1(t)

(−g1(t) + g1(t))dt +

∫ T

0

∂H1

∂z(t)

(Z1(t) − Z1(t)

)dt

+∫ T

0

∫

R

∇kH1(t, ζ )(K1(t, ζ ) − K1(t, ζ )

)ν(dζ ) dt

]}. (A.7)

J Optim Theory Appl

Adding (A.5), (A.6), and (A.7), we get

� = I1 + I2 + I3

≤ E

[∫ T

0

{H1(t) − H1(t) − ∂H1

∂x(t)

(X(t) − X(t)

)

− ∂H1

∂y(t)

(Y1(t) − Y1(t)

) − ∂H1

∂z(t)

(Z1(t) − Z1(t)

)

−∫

R

∇kH1(t, ζ )(K1(t, ζ ) − K1(t, ζ )

)ν(dζ )

}dt

]. (A.8)

Since H1(x, y, z, k) is concave, it follows by a standard separating hyperplaneargument (see, e.g., [20], Chap. 5, Sect. 23) that there exists a supergradient a =(a0, a1, a2, a3(·)) ∈ R

3 × R for H1(x, y, z, k) at x = X(t), y = Y1(t), z = Z1(t−),

and k = K1(t−, ·) such that if we define

ϕ1(x, y, z, k) := H1(x, y, z, k) − H1(X

(t−

), Y1

(t−

), Z1

(t−

), K1(t, ·)

)

−[a0

(x − X(t)

) + a1(y − Y1(t)

) + a2(z − Z1(t)

)

+∫

R

a3(ζ )(k(ζ ) − K(t, ζ )

)ν(dζ )

],

then

ϕ1(x, y, z, k) ≤ 0 for all x, y, z, k.

On the other hand, we clearly have

ϕ1(X(t), Y (t), Z(t), K1(t, ·)

) = 0.

It follows that

∂H1

∂x(t) = ∂H1

∂x

(X(t), Y1(t), Z1(t), K1(t, ·)

) = a0,

∂H1

∂y(t) = ∂H1

∂y

(X(t), Y1(t), Z1(t), K1(t, ·)

) = a1,

∂H1

∂z(t) = ∂H1

∂z

(X(t), Y1(t), Z1(t), K1(t, ·)

) = a2,

∇kH1(t, ζ ) = ∇k H1(X(t), Y1(t), Z1(t), K1(t, ·)

) = a3.

Combining this with (A.8), we get

� ≤ H1(X(t), Y1(t),Z1(t),K1(t, ·)

) − H1(X(t), Y1(t), Z1(t), K1(t, ·)

)

− ∂H1

∂x

(X(t), Y1(t), Z1(t), K1(t, ·)

)(X(t) − X(t)

)

J Optim Theory Appl

− ∂H1

∂y

(X(t), Y1(t), Z1(t), K1(t, ·)

)(Y1(t) − Y1(t)

)

− ∂H1

∂z

(X(t), Y1(t), Z1(t), K1(t, ·)

)(Z1(t) − Z1(t)

)

−∫

R

∇k H1(X(t), Y1(t), Z1(t), K1(t, ·)

)(K1(t, ζ ) − K1(t, ζ )

)ν(dζ )

≤ 0 since H1 is concave.

Hence,

J1(u1, u2) ≤ J1(u1, u2) for all u1 ∈ A1.

The inequality

J2(u1, u2) ≤ J2(u1, u2) for all u2 ∈ A2

is proved similarly. This completes the proof of Theorem 2.1. �

Proof of Theorem 2.2 (Necessary maximum principle) Consider

D1 := d

dsJ1(u1 + sβ1, u2) |s=0

= E

[∫ T

0

{∂f1

∂x(t)x1(t) + ∂f1

∂u1(t)β1(t)

}dt + ϕ′

1

(X(u1,u2)(T )

)x1(T )

+ ψ ′1

(Y1(0)

)y1(0)

]. (A.9)

By (10), (13), and the Itô formula,

E[ϕ′

1

(X(u1,u2)(T )

)x1(T )

]

= E[p1(T )x1(T )

] − E[h′

1

(X(u1,u2)(T )

)λ1(T )

]

= E

[∫ T

0

{p1

(t−

)dx1(t) + x1

(t−

)dp1(t) + q1(t)

[∂σ

∂x(t)x1(t) + ∂σ

∂u1(t)β1(t)

]dt

+∫

R

r1(t, ζ )

[∂γ

∂x(t, ζ )x1(t) + ∂γ

∂u1(t, ζ )β1(t, ζ )

]ν(dζ ) dt

}]

− E[h′

1

(X(u1,u2)(T )

)λ1(T )

]

= E

[∫ T

0

{p1(t)

[∂b

∂x(t)x1(t) + ∂b

∂u1(t)β1(t)

]

+ x1(t)

(−∂H1

∂x(t)

)+ q1(t)

[∂σ

∂x(t)x1(t) + ∂σ

∂u1(t)β1(t)

]

+∫

R

r1(t, ζ )

[∂γ

∂x(t, ζ )x1(t) + ∂γ

∂u1(t, ζ )β1(t, ζ )

]ν(dζ )

}dt

]

− E[h′

1

(X(u1,u2)(T )

)λ1(T )

]. (A.10)

J Optim Theory Appl

By (9), (13), and the Itô formula,

E[ψ ′

1

(Y1(0)

)y1(0)

]

= E[λ1(0)y1(0)

]

= E[λ1(T )y1(T )

] − E

[∫ T

0{λ1

(t−

)dy1(t) + y1

(t−

)dλ1(t)

+ ∂H1

∂z(t)z1(t) dt +

∫

R

∇kH1(t, ζ )k1(t, ζ )ν(dζ ) dt

]

= E[λ1(T )h′

1

(X(u1,u2)(T )

)]

− E

[∫ T

0

{λ1(t)

[−∂g1

∂x(t)x1(t) − ∂g1

∂y(t)y1(t) − ∂g1

∂z(t)z1(t).

−∫

R

∇kg1(t, ζ )k1(t, ζ )ν(dζ ) − ∂g1

∂u1(t)β1(t)

]

+ ∂H1

∂y(t)y1(t) + ∂H1

∂z(t)z1(t) +

∫

R

∇kH1(t, ζ )k1(t, ζ )ν(dζ )

}dt

]. (A.11)

Adding (A.10) and (A.11), we get, by (A.9),

D1 = E

[∫ T

0

{[∂f1

∂x(t) + p1(t)

∂b

∂x(t) + q1(t)

∂σ

∂x(t)

+∫

R

r1(t, ζ )∂γ

∂x(t, ζ )ν(dζ ) − ∂H1

∂x(t) + λ1(t)

∂g1

∂x(t)

]x1(t)

+[−∂H1

∂y(t) + λ1(t)

∂g1

∂y(t)

]y1(t) +

[−∂H1

∂z(t) + λ1(t)

∂g1

∂z(t)

]z1(t)

+∫

R

[−∇kH1(t, ζ ) + λ1(t)∇kg1(t, ζ )]k1(t, ζ )ν(dζ )

+[

∂f1

∂u1(t) + p1(t)

∂b

∂u1(t) + q1(t)

∂σ

∂u1(t)

+∫

R

r1(t, ζ )∂γ

∂u1(t, ζ )ν(dζ ) + ∂g1

∂u1(t)

]β1(t)

}dt

]

= E

[∫ T

0

∂H1

∂u1(t)β1(t) dt

]

= E

[∫ T

0E

[∂H1

∂u1(t)β1(t) | E (1)

t

]dt

]. (A.12)

If D1 = 0 for all bounded β1 ∈ A1, then this holds in particular for β1 of the formin (a1), i.e.,

β1(t) = χ(t0,T ](t)α1(ω),

J Optim Theory Appl

where α1(ω) is bounded and E (1)t0

-measurable. Hence,

E

[∫ T

t0

E

[∂H1

∂u1(t) | E (1)

t

]α1 dt

]= 0.

Differentiating with respect to t0, we get

E

[∂H1

∂u1(t0)α1

]= 0 for a.a. t0.

Since this holds for all bounded E (1)t0

-measurable random variables α1, we concludethat

E

[∂H1

∂u1(t) | E (1)

t

]= 0 for a.a. t ∈ [0, T ].

A similar argument gives that

E

[∂H2

∂u2(t) | E (2)

t

]= 0,

provided that

D2 := d

dsJ2(u1, u2 + sβ2) |s=0= 0 for all bounded β2 ∈ A2.

This shows that (i) ⇒ (ii). The argument above can be reversed, to give that (ii) ⇒ (i).We omit the details. �

Appendix B: Linear BSDEs with Jumps

Lemma B.1 (Linear BSDEs with jumps) Let Λ be an FT -measurable and square-integrable random variable. Let β and ξ0 be bounded predictable processes, and ξ1 apredictable process such that ξ1(t, ζ ) ≥ C1 with C1 > −1 and |ξ1(t, ζ )| ≤ C2(1∧|ζ |)for a constant C2 ≥ 0. Let ϕ be a predictable process such that E[∫ T

0 ϕ2(t) dt] < ∞.Then the linear BSDE

dY (t) = −[ϕ(t) + β(t)Y (t) + ξ0(t)Z(t) +

∫

R

ξ1(t, ζ )K(t, ζ )ν(dζ )

]dt

+ Z(t) dB(t) +∫

R

K(t, ζ )N(dt, dζ ), 0 ≤ t ≤ T ,

Y (T ) = Λ,

(B.1)

has the unique solution

Y(t) = E

[ΛΥ (t, T ) +

∫ T

t

Υ (t, s)ϕ(s) ds | Ft

], 0 ≤ t ≤ T , (B.2)

J Optim Theory Appl

where Υ (t, s), 0 ≤ t ≤ s ≤ T , is defined by

dΥ (t, s) = Υ(t, s−)[

β(s) ds + ξ0(s) dB(s) +∫

R

ξ1(s, ζ )N(ds, dζ )

], t ≤ s ≤ T ,

Υ (t, t) = 1,

(B.3)

i.e.,

Υ (t, s) = exp

(∫ s

t

{β(u) − 1

2ξ2

0 (u)

}du +

∫ s

t

ξ0(u) dB(u)

+∫ s

t

∫

R

{ln

(1 + ξ1(u)

) − ξ1(u)}ν(dζ ) du

+∫ s

t

∫

R

ln(1 + ξ1(u)

)N(du, dζ )

). (B.4)

Hence,

Υ (t, s) = Υ (0, s)

Υ (0, t).

Proof For completeness, we give the proof, also given in [18]. The existence anduniqueness follow by general theorems for BSDEs with Lipschitz coefficients. See,e.g., [17]. Hence, it only remains to prove that if we define Y(t) to be the solution of(B.1), then (B.2) holds. To this end, define

Υ (s) = Υ (0, s).

Then by the Itô formula (see, e.g., [1], Chap. 1),

d(Υ (t)Y (t)

)

= Υ(t−

)dY (t) + Y

(t−

)dΥ (t) + d[Υ Y ](t)

= Υ(t−

)[−{ϕ(t) + β(t)Y (t) + ξ0(t)Z(t) +

∫

R

ξ1(t, ζ )K(t, ζ )ν(dζ )

}dt

+ Z(t) dB(t) +∫

R

K(t, ζ )N(dt, dζ )

]

+ Y(t−

)Υ

(t−

){β(t) dt + ξ0(t) dB(t) +

∫

R

ξ1(t, ζ )N(dt, dζ )

}

+ Υ (t)ξ0(t)Z(t) dt +∫

R

Υ(t−

)ξ1(t, ζ )K(t, ζ )N(dt, dζ )

= −Υ (t)ϕ(t) dt + (Z(t) + ξ0(t)Y (t)

)Υ (t) dB(t)

+∫

R

ξ1(t, ζ )Υ(t−

)(Y

(t−

) + K(t, ζ ))N(dt, dζ ).

J Optim Theory Appl

Hence, Υ (t)Y (t) + ∫ t

0 Υ (s)ϕ(s) ds is a martingale, and therefore

Υ (t)Y (t) +∫ t

0Υ (s)ϕ(s) ds = E

[ΛΥ (T ) +

∫ T

0Υ (s)ϕ(s) ds | Ft

]

or

Y(t) = E

[Λ

Υ (T )

Υ (t)+

∫ T

t

Υ (s)

Υ (t)ϕ(s) ds | Ft

],

as claimed. �

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