Backward Stochastic Partial Differential Equations with Jumps and Application to Optimal Control of Random Jump Fields

Dept. of Math. University of Oslo Pure Mathematics No. 10 ISSN 0806–2439 March 2005

Backward Stochastic Partial Differential Equations with Jumps and Application to Optimal Control of Random Jump Fields

Bernt Øksendal^1,2, Frank Proske¹, Tusheng Zhang^1,3

Revised June 7, 2005

Abstract

We prove an existence and uniqueness result for a general class of backward stochastic partial differential equations with jumps. This is a type of equations which appear as adjoint equations in the maximum principle approach to optimal control of systems described by stochastic partial differential equations driven by L´evy processes.

AMS Subject Classification: Primary 60H15 Secondary 93E20, 35R60.

1 Introduction

Let Bt, t ≥ 0 and η(t) = Rt 0

R

RⁿzNe(ds, dz); t ≥ 0 be an m-dimensional Brownian motion and a pure jump L´evy process, respectively, on a filtered probability space (Ω,F,F_t, P). Fix T >0 and letφ(ω) be an F_T-measurable random variable. Let

b: [0, T]×Rⁿ×R^n×m→Rⁿ

be a given vector field. Consider the problem to find three F_t-adapted processes p(t) ∈ Rⁿ, q(t)∈R^n×m andr(t, z)∈R^n×m such that

dp(t) = b(t, p(t), q(t))dt+q(t)dBt+ Z

Rⁿ

r(t, z)Ne(ds, dz), t∈(0, T)1.1 (1.1)

p(T) = φ a.s.1.2 (1.2)

This is a backward stochastic (ordinary) differential equation (BSDE). It is called backward because it is the terminal valuep(T) =φthat is given, not the initial valuep(0). Stillp(t) is required to be F_t-adapted. In general this is only possible if we also are free to chooseq(t) and r(t, z) (in anF_t-adapted way).

The theory of BSDEs, when η = 0, is now well developed. See e.g. [EPQ], [MY], [PP1], [PP2] and [YZ] and the references therein. In the jump case (η 6= 0) BSDE’s have been studied. See [FØS], [NS], [S] and the references therein.

There are many applications of this theory. Examples include the following:

1CMA, Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-316 Oslo, Norway.

Email: [email protected], [email protected]

2Norwegian School of Economics and Business Administration, Helleveien 30, N-5045 Bergen, Norway

3Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England, U.K. Email: [email protected]

(i) The problem of finding a replicating portfolio of a given contingent claim in a complete financial market can be transformed into a problem of solving a BSDE.

(ii) The maximum principle method for solving a stochastic control problem involves a BSDE for the adjoint processesp(t), q(t), r(t, z).

For more information about these and other applications of BSDEs we refer to [EPQ] and [YZ] and references therein.

The purpose of this paper is to study backward stochastic partial differential equations (BSPDEs) with jumps. They are defined in a similar way as BSDEs, but with the basic equation being a stochastic partial differential equation rather than a stochastic ordinary differential equation. More precisely, we will study a class of BSPDEs which includes the following:

Find adapted processesY(t, x),Z(t, x),Q(t, x, z) such that dY(t, x) =AY(t, x)dt+b(t, x, Y(t, x), Z(t, x))dt

+Z(t, x)dBt+ Z

Rⁿ

Q(t, x, z)Ne(dt, dz),(t, x)∈(0, T)×Rⁿ (1.3)

Y(T, x) =φ(x, ω) (1.4)

Here dY(t, x) denotes the Itˆo differential with respect to t, while A is a partial differen- tial operator with respect to x and Ne(dt, dz) is the compensated Poisson random measure associated with a L´evy processη(·).

The function b: [0, T]×Rⁿ×R×R→ R is given and so is the terminal value function φ(x) =φ(x, ω). We assume thatφ(x) isF_T-measurable for allx and that

E[

Z

Rⁿ

φ(x)²dx]<∞, (1.5)

whereE denotes expectation with respect toP. We are seeking the processesY(t, x),Z(t, x) and Q(t, x, z) such that (1.3) and (1.4) hold. The processes Y(t, x), Z(t, x) and Q(t, x, z) are required to be F_t-adapted, i.e., Y(t, x), Z(t, x) and Q(t, x, z) are F_t-measurable for all x∈Rⁿ, z∈Rand we also require that

E Z

Rⁿ

Z T 0

{Y(t, x)²+Z(t, x)²+ Z

Rⁿ

Q(t, x, z)²ν(dz)}dt dx

<∞, (1.6) where ν(·) is the L´evy measure of the underlying L´evy process. Equations of this type are of interest because they appear as adjoint equations in a maximum principle approach to optimal control of stochastic partial differential equations. See Section 2.

Example 1.1 Consider the following BSPDE:

dY(t, x) =−¹₂∆Y(t, x)dt+Z(t, x)dB_t+ Z

Rⁿ

Q(t, x, z)N(dt, dz),e (t, x)∈(0, T)×Rⁿ (1.7)

Y(T, x) =φ(x) (1.8)

Here∆Y(t, x) =Pn i=1

∂²Y(t,x)

∂x²_i is the Laplacian with respect tox applied to Y(t, x), andφ(x) satisfiesE[R

Rⁿφ(x)²dx]<∞.

In this simple case, we are able to find the solution explicitly:

We first use the Itˆo representation theorem to write, for almost allx, φ(x) =h(x) +

Z T 0

g(s, x, ω)dBs+ Z T

0

Z

Rⁿ

k(s, x, z, ω)Ne(ds, dz) where

h(x) =E[φ(x)], g(s, x,·) andk(s, x, z) are F_s-measurable for all s, x and

E[

Z

Rⁿ

Z T 0

g²(s, x,·) + Z

Rⁿ

k²(s, x, z)ν(dz)

dsdx]<∞.

Let

R_tf(x) = Z

Rⁿ

(2πt)⁻ⁿ²f(y)exp(−|x−y|²

2t )dy, t >0

be the transition operator for Brownian motion defined for all measurable f :Rⁿ →R such that the integral converges. Then it is well known that

∂

∂t(Rtf(x)) = ¹₂∆(Rtf(x)) (1.9) Now define

Y(t, x) =RT−t( Z t

0

g(s,·, ω)dB_s+ Z t

0

Z

Rⁿ

k(s,·, z, ω)N(ds, dz) +e h(·))(x)

= Z t

0

(R_T−tg(s,·, ω))(x)dB_s+ Z t

0

Z

Rⁿ

(R_T−tk(s,·, z, ω))(x)Ne(ds, dz) + (R_T−th)(x) (1.10) Then

dY(t, x)

= Z t

0

−¹₂∆(R_T−tg(s,·, ω))(x)dB_s−¹₂ Z t

0

Z

Rⁿ

∆(R_T−tk(s,·, z, ω))(x)Ne(ds, dz)− ¹₂∆R_T−th(·)(x)

dt

+ (R_T−tg(t,·, ω))(x)dB_t+ Z

Rⁿ

R_T−t(k(t,·, z, ω))(x)Ne(dt, dz)

=−¹₂∆Y(t, x)dt+Z(t, x)dB_t+ Z

Rⁿ

Q(t, x, z)Ne(dt, dz), where

Z(t, x) = (R_T−tg(t,·, ω))(x) (1.11) and

Q(t, x, z) = (R_T−tk(t,·, z))(x). (1.12) Hence the processes Y(t, x), Z(t, x) and Q(t, x, z) given by (1.10)–(1.12) solve the BSPDE (1.7)–(1.8).

In the general case it is not possible to find explicit solutions of a BSPDE. However, in Section 3 we will prove an existence and uniqueness result for a general class of such equations. We will achieve this by regarding the BSPDE of type (1.3)–(1.4) as a special case of a backward stochastic evolution equation for Hilbert space valued processes. This, in turn, is studied by taking finite dimensional projections and then taking the limit. This is the well known Galerkin approximation method which has been used by several authors in other connections. See e.g. [B1], [B2] and [P]. We also refer readers to [PZ] for the general theory of stochastic evolution equations on Hilbert spaces.

The rest of the paper is organized as follows: In Section 2 we prove a (sufficient) maximum principle for optimal control of random jump fields, i.e. solutions of SPDE’s driven by L´evy processes (Theorem 2.1). This principle involves a BSPDE of the form (1.3)-(1.4) in the associated adjoint processes. In Section 3 we give the precise framework of our general existence and uniqueness result. The existence and uniqueness result and its proof are given in Section 4.

2 The stochastic maximum principle

In this section we prove a verification theorem for optimal control of a process described by a stochastic partial differential equation (SPDE) driven by a Brownian motion B(t) and a Poisson random measureNe(dt, dz).We call such a process a (controlled)random jump field.

The verification theorem has the form of a sufficient stochastic maximum principle and the adjoint equation for this principle turns out to be a backward SPDE driven by B(·) and Ne(·,·). This part of the paper is an extension of [Ø] to the case including jumps and an extension of [FØS] to SPDE control.

Let Γ(t, x) = Γ^(u)(t, x); t ∈ [0, T], x ∈ R^k be the solution of a (controlled) stochastic reaction-diffusion equation of the form

dΓ(t, x) = [(LΓ)(t, x) +b(t, x,Γ(t, x), u(t, x))]dt+σ(t, x,Γ(t, x), u(t, x))dB(t) +

Z

R

θ(t, x,Γ(t, x), u(t, x), z)Ne(dt, dz); (t, x)∈[0, T]×G (2.1)

Γ(0, x) =ξ(x);x∈G (2.2)

Γ(t, x) =η(t, x); (t, x)∈(0, T)×∂G (2.3)

Here dΓ(t, x) = d_tΓ(t, x) is the differential with respect to t and L =L_x is a given partial differential operator of ordermacting on the variable x∈R^k.We assume thatG⊂R^k,f⊂ R^l are given open and closed sets, respectively, and that b : [0, T]× G×R×f → R, σ: [0, T]×G×R×f→R, θ: [0, T]×G×R×f×R→R,ξ:G→Randη: (0, T)×∂G→R are given measurable functions. The process

u: [0, T]×G→f

is called an admissible control if the equation (2.1)-(2.3) has a unique continuous solution Γ(t, x) = Γ^(u)(t, x) which satisfies

E Z T

0

Z

G

|f(t, x,Γ(t, x), u(t, x))|dx

dt+ Z

G

|g(x,Γ(T, x))|dx

<∞, (2.4)

wheref : [0, T]×G×R×f→R and g:G×R→Rare givenC¹ functions. The set of all admissible controls is denoted byA.Ifu∈ Awe define theperformance of u,J(u),by

J(u) =E Z T

0

Z

G

f(t, x,Γ(t, x), u(t, x))dx

dt+ Z

G

g(x,Γ(T, x))dx

. (2.5)

We consider the problem to findJ^∗ ∈Rand u^∗ ∈ Asuch that J^∗ := sup

u∈A

J(u) =J(u^∗). (2.6)

J^∗ is called thevalue of the problem andu^∗∈ A(if it exists) is called an optimal control.

In the following we let L^∗ denote the adjoint of the operator L, defined by

(L^∗ϕ₁, ϕ₂) = (ϕ₁, Lϕ₂) ; ϕ₁, ϕ₂ ∈C₀^∞(G), (2.7) where (ϕ, ψ) = (ϕ, ψ)_L2(G) =R

Gϕ(x)ψ(x)dx and C₀^∞(G) is the set of infinitely many times differentiable functions with compact support inG.

We now formulate a (sufficient) stochastic maximum principle for the problem (2.6):

LetR denote the set of functionsr : [0, T]×G×R×Ω→R. Define the Hamiltonian

H: [0, T]×G×R×f×R×R×R →R by

H(t, x, γ, u, p, q, r(t, x,·))

= f(t, x, γ, u) +b(t, x, γ, u)p+σ(t, x, γ, u)q+ Z

R

θ(t, x, γ, u, z)r(t, x, z)ν(dz)2.8 (2.3)

For each u ∈ A consider the following adjoint backward SPDE in the 3 unknown adapted processesp(t, x), q(t, x) andr(t, x, z):

dp(t, x) =−{∂H

∂γ (t, x,Γ^(u)(t, x), u(t, x), p(t, x), q(t, x), r(t, x,·)) +L^∗p(t, x)}dt+q(t, x)dB(t) +

Z

R

r(t, x, z)Ne(dt, dz);t < T. (2.9) p(T, x) = ∂g

∂γ(x,Γ^(u)(t, x)); x∈G (2.10)

p(t, x) = 0; (t, x)∈(0, T)×∂G. (2.11)

The following result may be regarded as a synthesis of Theorem 2.1 in [Ø] and Theorem 2.1 in [FØS]:

Theorem 2.1(Sufficient SPDE maximum principle for optimal control of reaction- diffusion jump fields)

Let bu∈ Awith corresponding solution bΓ(t, x)of (2.1)-(2.3) and let p(t, x),b q(t, x),b br(t, x,·) be a solution of the associated adjoint backward SPDE (2.9)-(2.11). Suppose the following, (2.12)-(2.15), holds:

The functions2.12 (2.4)

(γ, u) 7→ H(γ, u) :=H(t, x, γ, u,p(t, x),b q(t, x),b r(t, x,b ·)); (γ, u)∈R×f and

γ 7→ g(x, γ); γ ∈R are concave for all(t, x) ∈ [0, T]×G.

H(t, x,Γ(t, x),b u(t, x),b p(t, x),b q(t, x),b r(t, x,b ·))2.13 (2.5)

= sup

v∈f

H(t, x,bΓ(t, x), v,p(t, x),b q(t, x),b r(t, x,b ·)) for all(t, x) ∈ [0, T]×G.

For all u∈ A E

Z

G

Z T 0

Γ(t, x)−Γ(t, x)b 2

bq(t, x)²+ Z

R

br(t, x, z)²ν(dz)

dtdx

<∞ (2.14) and

E Z

G

Z T 0

p(t, x)b ²

σ(t, x,Γ(t, x), u(t, x))²+ Z

R

θ(t, x,Γ(t, x), u(t, x), z)²ν(dz)

dtdx

<∞ (2.15) Thenu(t, x)b is an optimal control for the stochastic control problem (2.6).

Proof. Let u be an arbitrary admissible control with corresponding solution Γ(t, x) = Γ^(u)(t, x) of (2.1)-(2.3). Then by (2.5)

J(u)b −J(u) =E Z T

0

Z

G

n fb−f

o

dxdt+ Z

G

{bg−g}dx

, (2.16)

where

fb = f(t, x,bΓ(t, x),u(t, x)), fb =f(t, x,Γ(t, x), u(t, x)) bg = g(x,Γ(T, x)) andb g=g(x,Γ(T, x)).

Similarly we put

bb = b(t, x,bΓ(t, x),u(t, x)), bb =b(t, x,Γ(t, x), u(t, x)) bσ = σ(t, x,bΓ(t, x),u(t, x)), σb =σ(t, x,Γ(t, x), u(t, x)) and

θb=θ(t, x,Γ(t, x),b u(t, x), z), θb =θ(t, x,Γ(t, x), u(t, x), z).

Moreover, we set

Hb = H(t, x,bΓ(t, x),u(t, x),b p(t, x),b q(t, x),b br(t, x,·)) H = H(t, x,Γ(t, x), u(t, x),p(t, x),b q(t, x),b br(t, x,·)).

Combining this notation with (2.8) and (2.16) we get

J(u)b −J(u) =I1+I2, (2.17) where

I1=E Z T

0

Z

G

Hb−H−(bb−b)pb−(bσ−σ)qb− Z

R

(θb−θ)brν(dz)

dxdt

(2.18) and

I2 =E Z

G

{bg−g}dx

. (2.19)

Sinceγ 7→g(x, γ) is concave we have g−bg≤ ∂g

∂γ(x,bΓ(T, x))·(Γ(T, x)−Γ(T, x)).b Therefore, if we put

Γ(t, x) = Γ(t, x)e −bΓ(t, x)

we get, by the Itˆo formula (or integration by parts formula) for jump diffusions ([ØS, Ex.

1.7])

I₂ ≥ −E Z

G

∂g

∂γ(x,bΓ(T, x))·Γ(T, x)dxe

= −E

Z

Gp(T, x)b ·Γ(T, x)dxe

= −E

Z

G

p(0, x)b ·eΓ(0, x) +

Z T 0

n

Γ(t, x)de p(t, x) +b p(t, x)db Γ(t, x) + (σe −σ)b bq(t, x) o

dt

+ Z T

0

Z

R

(θ−bθ)br(t, x, z)N(dt, dz)

dx

= −E

Z

G

Z T 0

Γ(t, x)e ·

− ∂H

∂γ ∧

−L^∗bp(t, x)

+p(t, x)b h

LΓ(t, x) + (be −bb) i

+ (σ−bσ)q(t, x)b +

Z

R

(θ−θ)bbr(t, x, z)ν(dz)

dt

dx

,2.20 (2.6)

where

∂H

∂γ ∧

= ∂H

∂γ (t, x,Γ(t, x),b u(t, x),b p(t, x),b bq(t, x),br(t, x,·)).

Combining (2.17)-(2.20) we get J(u)b −J(u) ≥ E

Z T 0

Z

G

n

eΓL^∗pb−pLb eΓ o

dx

dt

+E Z

G

Z T 0

Hb−H+ ∂H

∂γ ∧

·Γ(t, x)e

dt

dx

.2.21 (2.7) SinceΓ(t, x) =e p(t, x) = 0 for all (t, x)b ∈(0, T)×∂G we get by an extension of (2.7) that

Z

G

n

ΓLe ^∗pb−pLb Γe o

dx= 0 for allt∈(0, T).

Combining this with (2.21) we get J(u)b −J(u)≥E

Z

G

Z T 0

Hb−H+ ∂H

∂γ ∧

·eΓ(t, x)

dt

dx

. (2.22)

From the concavity assumption in (2.12) we deduce that H−Hb ≤ ∂H

∂γ(Γ,b u)b ·(Γ−Γ) +b ∂H

∂u(Γ,b bu)·(u−u).b (2.23) From the maximality assumption in (2.13) we get that

∂H

∂u(Γ,b u)b ·(u−bu)≤0. (2.24) Combining (2.23) and (2.24) we get

H−Hb −∂H

∂γ (Γ,b bu)·(Γ−bΓ)≤0, which substituted in (2.22) gives

J(u)b −J(u)≥0.

Sinceu∈ Awas arbitrary this proves Theorem 2.1.

3 Framework

We now present the general setting in which we will prove our main existence and uniqueness result for backward SPDE’s with jumps:

LetV,H be two separable Hilbert spaces such thatV is continuously, densely imbedded inH. IdentifyingH with its dual we have

V ⊂H∼=H^∗⊂V^∗, (3.1)

whereV^∗ stands for the topological dual of V. LetA be a bounded linear operator from V toV^∗ satisfying the following coercivity hypothesis: There exist constants α >0 and λ≥0 such that

2hAu, ui+λ|u|²_H ≥α||u||²_V for all u∈V , (3.2)

wherehAu, ui=Au(u) denotes the action of Au∈V^∗ on u∈V.

Remark thatAis generally not bounded as an operator fromH intoH. LetKbe another separable Hilbert space. Let (Ω,F, P) be a probability space. Let{B_t, t≥0}be a cylindrical Brownian motion with covariance space K on the probability space (Ω,F, P), i.e., for any k ∈ K,hB_t, ki is a real valued-Brownian motion with E[hB_t, ki²] = t|k|²_K. Let (X,B(X)) be a measurable space, where X is a topological vector space. Further let η(t) be a L´evy process onX.Denote byν(dx) the L´evy measure ofη. Denote byL²(ν) theL²-space of square integrableH-valued measurable functions associated withν. Setp(t) = ∆η(t) =η(t)−η(t−).

Then p = (p(t), t ∈ D_p) is a stationary Poisson point process on X with characteristic measureν. See [IW] for details on Poisson point processes. Denote by N(dt, dx) the Poisson counting measure associated with the L´evy process, i.e., N(t, A) =P

s∈Dp,s≤tI_A(p(s)). Let Ne(dt, dx) :=N(dt, dx)−dtν(dx) be the compensated Poisson mesasure. Define

F_t=σ(Bs, N(s, A), A∈ B(X), s≤t).

Recall that a linear operatorS from K intoH is called Hilbert-Schmidt if P∞

i=1|Sk_i|²_H <∞ for some orthonormal basis {k_i, i ≥ 1} of K. L2(K, H) will denote the Hilbert space of Hilbert-Schmidt operators fromK intoH equipped with the inner producthS₁, S2i_L2(K,H)= P∞

i=1hS₁k_i, S₂k_ii_H. Letb(t, y, z, q, ω) be a measurable mapping from [0, T]×H×L₂(K, H)× L²(ν) ×Ω into H such that b(t, y, z, q, ω) is F_t-adapted,i.e., b(t, y, z, q,·) is F_t-measurable for all t, y, z, q. Suppose we are given an F_T -measurable, H-valued random variable φ(ω).

We are looking for F_t-adapted processes Y_t, Z_t, Q_t with values in H, L₂(K, H) and L²(ν) respectively, such that the following backward stochastic evolution equation holds:

dYt = AYtdt+b(t, Yt, Zt, Qt)dt+ZtdBt

+ Z

X

Q_t(x)Ne(dt, dx), t∈(0, T)3.3 (3.8)

Y_T = φ(ω) a.s.3.4 (3.9)

From now on we assume that the following, (3.5) and (3.6), hold:

There exists a constantc <∞ such that

|b(t, y₁, z₁, q₁)(ω)−b(t, y₂, z₂, q₂)(ω)|_H

≤c(|y₁−y2|_H +|z₁−z2|_L2(K,H)+|q₁−q2|_L2(ν)) (3.5) for allt, y1, z1, q1, y2, z2, q2.

E[

Z T 0

|b(t,0,0,0)|²_Hdt]<∞ (3.6)

4 Existence and Uniqueness

We now state and prove the main existence and uniqueness result of this paper.

Body Math Theorem 4.1 Assume that E[|φ|²_H]<∞. Then there exists a unique H× L2(K, H)×L²(ν)-valued progressively measurable process (Yt, Zt, Qt) such that

(i) E[RT

0 |Y_t|²_Hdt]<∞, E[RT 0 |Z_t|²_L

2(K,H)dt]<∞, E[RT

0 |Q_t|²_L2(ν)dt]<∞.

(ii) φ=Yt+RT

t AYsds+RT

t b(s, Ys, Zs, Qs)ds+RT

t ZsdBs+RT t

R

XQs(x)N(ds, dx);e 0≤t≤T.

As the proof is long, we split it into lemmas.

Body Math Lemma 4.2 Assume that E[|φ|²_H]< ∞, and that b(t, y, z, q, ω) =b(t, ω) is independent of y, z and q, and E[RT

0 |b(t)|²_Hdt] < ∞. Then there exists a unique H × L₂(K, H)×L²(ν)-valued progressively measurable process (Y_t, Z_t, Q_t) such that

(i) E[RT

0 |Y_t|²_Hdt]<∞, E[RT 0 |Z_t|²_L

2(K,H)dt]<∞, E[RT

0 |Q_t|²_L2(ν)dt]<∞.

(ii) φ=Y_t+RT

t AY_sds+RT

t b(s)ds+RT

t Z_sdB_s+RT t

R

XQ_s(x)Ne(ds, dx); 0≤t≤T.

Proof.

Existence of solution.

Set D(A) = {v;v ∈ V, Av ∈ H}. Then D(A) is a dense subspace of H. Thus we can choose and fix an orthonormal basis {e₁, ...en, ...} of H such that ei ∈ D(A). Set Vn = span(e1, e2, ..., en). Denote by Pn the projection operator from H into Vn. Put An =PnA.

Then A_n is a bounded linear operator from V_n toV_n. For the cylindrical Brownian motion Bt, it is well known that the following decomposition holds:

Bt=

∞

X

i=1

β_tⁱki (4.1)

where {k₁, k2, ..., ki, ...} is an orthonormal basis of K, and β_tⁱ, i = 1,2,3, ... are indepen- dent standard Brownian motions. Set Bⁿ_t = (β_t¹, ..., βⁿ_t). Define F_tⁿ = σ(B_sⁿ, N(s, A), A ∈ B(X), s ≤ t) completed by the probability measure P, and put φn = E[Pnφ|F_Tⁿ] and bn(t) = E[Pnb(t)|F_tⁿ]. Consider the following backward stochastic differential equation on the finite dimensional space V_n:

dY_tⁿ = A_nY_tⁿdt+b_n(t)dt+Z_tⁿdB_tⁿ +

Z

X

Qⁿ_t(x)Ne(dt, dx) ; t < T4.2 (4.10)

Y_Tⁿ = φ_n(ω) a.s.4.3 (4.11)

As A_n is a bounded linear operator from V_n to V_n, it follows from the results of Situ [S]

that (4.2)–(4.3) admits a unique F_tⁿ- adapted solution (Y_tⁿ, Z_tⁿ, Qⁿ_t), where Y_tⁿ ∈ Vn,Z_tⁿ ∈ L2(Kn, Vn), Kn = span(k1, k2, ..., kn) and Qⁿ_t ∈ L²(ν). We are going to show that the sequence (Y_tⁿ, Z_tⁿ, Qⁿ_t) admits a convergent subsequence. Using Itˆo’s formula, we find that

E[|Y_tⁿ|²_H] =E[|φ_n|²_H]−2E[

Z T t

hY_sⁿ, PnAY_sⁿids]

−2E[

Z T t

< Y_sⁿ, b_n(s)> ds]−E[

Z T t

|Z_sⁿ|²_L

2(Kn,Vn)ds]−E[

Z T t

ds Z

X

|Qⁿ_s(x)|²_Hν(dx)], (4.4) where|Z_sⁿ|²_L

2(Kn,Vn)=Pn

i,j=1(Z_sⁿ(i, j))²stands for the Hilbert-Schmidt norm. It follows from (3.2) that

E[|Y_tⁿ|²_H]≤E[|φ|²_H]−αE[

Z T t

||Y_sⁿ||²_Vds] +λE[

Z T t

|Y_sⁿ|²_Hds]

+E[

Z _T

t

|Y_sⁿ|²_Hds] +E[

Z _T

t

|b(s)|²_Hds]−E[

Z _T

t

|Z_sⁿ|²_L

2(Kn,Vn)ds]−E[

Z _T

t

ds Z

X

|Qⁿ_s(x)|²_Hν(dx)].

Hence,

E[|Y_tⁿ|²_H] +αE[

Z T t

||Y_sⁿ||²_Vds] +E[

Z T t

|Z¯_sⁿ|²_L

2(K,H)ds] +E[

Z T t

ds Z

X

|Qⁿ_s(x)|²_Hν(dx)]

≤E[|φ|²_H] + (λ+ 1)E[

Z T t

|Y_sⁿ|²_Hds] +E[

Z T t

|b(s)|²_Hds]

where ¯Z_sⁿ=Z_sⁿP¯_n, and ¯P_nis the projection fromK intoK_n=span(k₁, ..., k_n). In particular, E[|Y_tⁿ|²_H]≤E[|φ|²_H] + (λ+ 1)E[

Z T t

|Y_sⁿ|²_Hds] +E[

Z T t

|b(s)|²_Hds] (4.5) Set ¯Y_tⁿ=RT

t |Y_sⁿ|²_Hds. Then (4.5) implies that

−d(e^(λ+1)tY¯_tⁿ)

dt ≤e^(λ+1)t(E[|φ|²_H] +E[

Z T t

|b(s)|²_Hds])

Hence,

Z _T

0

E[|Y_sⁿ|²_H]ds≤C(E[|φ|²_H] +E[

Z _T

0

|b(s)|²_Hds]), whereC is an appropriate constant. This further implies that

sup

n

{ Z T

0

E[|Y_sⁿ|²_H]ds}<∞ sup

n

{ Z T

0

E[||Y_sⁿ||²_V]ds}<∞ (4.6) sup

n

{ Z T

0

E[|Z¯_sⁿ|²_L

2(K,H)]ds}<∞ (4.7)

sup

n

{ Z T

0

E[|Qⁿ_s|²_L2(ν)]ds}<∞ (4.8) For a separable Hilbert space L, we denote by M²([0, T], L) the Hilbert space of progres- sively measurable, square integrable, L-valued processes equipped with the inner product

< a, b >M=E[RT

0 < at, bt>Ldt]. By the weak compactness of a Hilbert space, it follows from (4.6),(4.7)and (4.8) that a subsequence {n_k, k ≥1} can be selected so thatY_.^nk, k≥ 1 con- verges weakly to some limitY inM²([0, T], V), ¯Z_.^nk, k≥1 converges weakly to some limitZ inM²([0, T], L2(K, H)) andQⁿ_.^k, k≥1 converges weakly to some limitQinM²([0, T], L²(ν)).

Let us prove that ( a version of )(Y, Z, Q) is a solution to the backward stochastic evolution equation (3.3) and (3.4). Forn≥i≥1, we have that

dhY_tⁿ, eii=hP_nAY_tⁿ, eiidt+hb_n(t), eiidt+hZ¯_tⁿdBt, eii+ Z

X

hQⁿ_t(x), eiiNe(dt, dx)

=hAY_tⁿ, eiidt+hb_n(t), eiidt+hZ¯_tⁿdBt, eii+ Z

X

hQⁿ_t(x), eiiNe(dt, dx) (4.9)

Let h(t) be an absolutely continuous function from [0, T] to R with h⁰(·) ∈ L²([0, T]) and h(0) = 0. By the Itˆo formula,

hY_Tⁿ, eiih(T) = Z T

0

h(t)hAY_tⁿ, eiidt+ Z T

0

h(t)hb_n(t), eiidt +

Z T 0

h(t)dh Z t

0

Z¯_sⁿdB_s, e_ii+ Z T

0

Z

X

h(t)hQⁿ_t(x), e_iiNe(dt, dx) + Z T

0

hY_tⁿ, e_iih⁰(t)dt. (4.10) Replacingn byn_k in (4.10) and lettingk→ ∞ to obtain

hφ, e_iih(T)

= Z T

0

h(t)hAY_t, e_iidt+ Z T

0

h(t)hb(t), e_iidt+ Z T

0

Z

X

h(t)hQ_t(x), e_iiNe(dt, dx) +

Z _T

0

h(t)dh Z _t

0

Z_sdB_s, e_ii+ Z _T

0

hY_t, e_iih⁰(t)dt.4.11 (4.12) From (4.10) to (4.11), we have used the fact that the linear mappingGfromM²([0, T], L₂(K, H)) intoL²(Ω) defined by

G(Z) = Z T

0

h(t)dh Z t

0

Z_sdB_s, e_ii=

∞

X

j=1

Z T 0

h(t)hZ_t(k_j), e_iidβ_t^j

is continuous and also the linear mappingF from M²([0, T], L²(ν)) intoL²(Ω) defined by F(Q) =

Z T 0

Z

X

h(t)hQ_t(x), e_iiNe(dt, dx)

is continuous. So, the convergence of (4.10) to (4.11) takes place weakly in L²(Ω). Fix t∈(0, T) and choose , for n≥1,

hn(s) =







1, s≥t+_2n¹ ,

1−_n¹(t+ _2n¹ −s), t−_2n¹ ≤s≤t+_2n¹ ,

0, s≤t− _2n¹

Withh(·) replaced byhn(·) in (4.11), it follows that hφ, e_ii=

Z T 0

h_n(s)hAY_s, e_iids+ Z T

0

h_n(s)hb(s), e_iids+ Z T

0

Z

X

h(t)hQ_t(x), e_iiNe(dt, dx) +

Z T 0

h_n(s)dh Z s

0

Z_udB_u, e_ii+1 n

Z t+_2n¹ t−_2n¹

hY_s, e_iids. (4.12)

Sending nto infinity in (4.12) we get that hφ, e_ii=

Z T t

hAY_s, e_iids+ Z T

t

hb(s), e_iids

+ Z T

t

Z

X

hQ_s(x), eiiNe(ds, dx) + Z T

t

dh Z s

0

ZudBu, eii+hY_t, eii.

for almost allt∈[0, T] ( with respect to Lebesgue measure).

Asiis arbitrary, this implies that φ=

Z T t

AYsds+ Z T

t

b(s)ds+ Z T

t

ZsdBs+ Z T

t

Z

X

Qs(x)Ne(ds, dx) +Yt. (4.13) for almost allt∈[0, T] ( with respect to Lebesgue measure).

Fort∈[0, T], define Yˆ_t=φ−

Z _T

t

AY_sds− Z _T

t

b(s)ds− Z _T

t

Z_sdB_s− Z _T

t

Z

X

Q_s(x)Ne(ds, dx).

Then we see that ( ˆY_t, Z_t, Q_t) also satisfies (ii) in the Theorem 4.1 with Y replaced by ˆY for allt∈[0, T]. Hence, ( ˆYt, Zt, Qt) is a solution to the equations (3.3) and (3.4).

Uniqueness:

Let (Y_t, Z_t, Q_t) and ( ¯Y_t,Z¯_t,Q¯_t) be two solutions of the equation (3.3). Then Z T

t

A(Ys−Y¯s)ds+ Z T

t

(Zs−Z¯s)dBs+ (Yt−Y¯t) + Z T

t

Z

X

(Qs(x)−Q¯s(x))Ne(ds, dx) = 0 Applying Itˆo’s formula, we get

0 =|Y_t−Y¯t|²_H + 2 Z T

t

hY_s−Y¯s, dMsi +

Z T t

Z

X

[|Y_s−−Y¯s−+Q_s(x)−Q¯_s(x)|²_H− |Y_s−−Y¯s−|²_H]Ne(ds, dx) + 2

Z T t

hA(Y_s−Y¯s), Ys−Y¯shds+ Z T

t

|Z_s−Z¯s|²_L

2(K,H)ds +

Z T t

Z

X

|Q_s(x)−Q¯_s(x)|²_Hdsν(dx) (4.14)

whereM_t=Rt

0(Z_s−Z¯_s)dB_s. By (3.2),we get that E[|Y_t−Y¯_t|²_H] =−2

Z T t

E[hA(Y_s−Y¯_s), Y_s−Y¯_si]ds−E[

Z T t

|Z_s−Z¯_s|²_L

2(K,H)ds]

−E[

Z T t

Z

X

|Q_s(x)−Q¯s(x)|²_Hdsν(dx)]

≤ −α Z T

t

E[||Y_s−Y¯_s||²_V]ds+λ Z T

t

E[|Y_s−Y¯_s|²_H]ds

≤λ Z _T

t

E[|Y_s−Y¯_s|²_H]ds.

By a Gronwall type inequality, it follows thatE[|Y_t−Y¯_t|²_H] = 0 , which proves the uniqueness.

Body MathLemma 4.3 Assume that E[|φ|²_H]<∞,and that b(t, y, z, q, ω) =b(t, z, q, ω) is independent of y. Then there exists a unique H×L₂(K, H)×L²(ν)-valued progressively measurable process (Y_t, Z_t, Q_t) such that

(i) E[RT

0 |Y_t|²_Hdt]<∞, E[RT 0 |Z_t|²_L

2(K,H)dt]<∞, E[RT

0 |Q_t|²_L2(ν)dt]<∞.

(ii) φ=Yt+RT

t AYsds+RT

t b(s, Zs, Qs)ds+RT

t ZsdBs+RT t

R

XQs(x)Ne(ds, dx); 0≤t≤T.

Proof.

SetZ_t⁰= 0, Q⁰_t = 0. Denote by (Y_tⁿ, Z_tⁿ, Qⁿ_t) the unique solution of the backward stochas- tic evolution equation:

dY_tⁿ = AY_tⁿdt+b(t, Z_tⁿ⁻¹, Qⁿ⁻¹_t )dt+Z_tⁿdBt+ Z

X

Qⁿ_t(x)Ne(dt, dx)4.15 (4.13)

Y_Tⁿ = φ(ω).4.16 (4.14)

The existence of such a solution (Y_tⁿ, Z_tⁿ, Qⁿ_t) has been proved in Lemma 4.2. PuttingM_tⁿ= Rt

0 Z_sⁿdB_s, and by Itˆo’s formula we get that 0 =|Y_Tⁿ⁺¹−Y_Tⁿ|²_H

=|Y_tⁿ⁺¹−Y_tⁿ|²_H + 2 Z T

t

hA(Y_sⁿ⁺¹−Y_sⁿ), Y_sⁿ⁺¹−Y_sⁿids + 2

Z T t

hb(t, Z_sⁿ, Qⁿ_s)−b(t, Z_sⁿ⁻¹, Qⁿ⁻¹_s ), Y_sⁿ⁺¹−Y_sⁿids +

Z T t

Z

X

[|Y_s−ⁿ⁺¹−Y_s−ⁿ +Qⁿ⁺¹_s −Qⁿ_s|²_H − |Y_s−ⁿ⁺¹−Y_s−ⁿ|²_H]Ne(ds, dx) +

Z T t

Z

X

|Qⁿ⁺¹_s −Qⁿ_s|²_H]dsν(dx) + 2

Z T t

hY_sⁿ⁺¹−Y_sⁿ, d(M_sⁿ⁺¹−M_sⁿ)i+ Z T

t

|Z_sⁿ⁺¹−Z_sⁿ|²_L

2(K,H)ds

In virtue of (3.2), forε >0, E[|Y_tⁿ⁺¹−Y_tⁿ|²_H] +E[

Z _T

t

|Z_sⁿ⁺¹−Z_sⁿ|²_L

2(K,H)ds] +E[

Z _T

t

Z

X

|Qⁿ⁺¹_s −Qⁿ_s|²_H]dsν(dx)]

=−2E[

Z T t

hA(Y_sⁿ⁺¹−Y_sⁿ), Y_sⁿ⁺¹−Y_sⁿids]

−2E[

Z T t

hb(t, Z_sⁿ, Qⁿ_s)−b(t, Z_sⁿ⁻¹, Qⁿ⁻¹_s ), Y_sⁿ⁺¹−Y_sⁿids]

≤λE[

Z T t

|Y_sⁿ⁺¹−Y_sⁿ|²_Hds]−αE[

Z T t

||Y_sⁿ⁺¹−Y_sⁿ||²_Vds]

+εE[

Z T t

|b(t, Z_sⁿ, Qⁿ_s)−b(t, Z_sⁿ⁻¹, Qⁿ⁻¹_s )|²_Hds] +1 εE[

Z T t

|Y_sⁿ⁺¹−Y_sⁿ|²_Hds] (4.17)

Choose ε < _4c¹, where cis the Lipschitz constant in (3.5). It follows from (4.17) that E[|Y_tⁿ⁺¹−Y_tⁿ|²_H] +E[

Z T t

|Z_sⁿ⁺¹−Z_sⁿ|²_L

2(K,H)ds]

+αE[

Z T t

||Y_sⁿ⁺¹−Y_sⁿ||²_Vds] +E[

Z T t

Z

X

|Qⁿ⁺¹_s −Qⁿ_s|²_H]dsν(dx)]

≤ (λ+ 1 ε)E[

Z T t

|Y_sⁿ⁺¹−Y_sⁿ|²_Hds] + ¹₂E[

Z T t

|Z_sⁿ−Z_sⁿ⁻¹|²_L

2(K,H)ds]

+1 2E[

Z T t

Z

X

|Qⁿ_s −Qⁿ⁻¹_s |²_H]dsν(dx)]4.18 (4.15) Letβ =λ+¹_ε. Then we have from (4.18) that

− d dt(e^βtE[

Z T t

|Y_sⁿ⁺¹−Y_sⁿ|²_Hds]

+e^βtE[

Z T t

|Z_sⁿ⁺¹−Z_sⁿ|²_L

2(K,H)ds]

+e^βtE[

Z T t

Z

X

|Qⁿ⁺¹_s −Qⁿ_s|²_H]dsν(dx)]

+αe^βtE[

Z T t

||Y_sⁿ⁺¹−Y_sⁿ||²_Vds]

≤1 2e^βtE[

Z T t

|Z_sⁿ−Z_sⁿ⁻¹|²_L

2(K,H)ds]

+1 2e^βtE[

Z T t

Z

X

|Qⁿ_s −Qⁿ⁻¹_s |²_H]dsν(dx)] (4.19) From here, following a similar proof as in [PP1] we will show that (Yⁿ, Zⁿ, Qⁿ) converges to

some limit (Y, Z, Q) in the product space ofM²([0, T], V),M²([0, T], L₂(K, H)) andM²([0, T], L²(ν)).

Integrating both sides in (4.19) we get that E[

Z T 0

|Y_sⁿ⁺¹−Y_sⁿ|²_Hds] + Z T

0

E[

Z T t

|Z_sⁿ⁺¹−Z_sⁿ|²_L

2(K,H)ds]e^βtdt +

Z T 0

e^βtE[

Z T t

Z

X

|Qⁿ⁺¹_s −Qⁿ_s|²_H]dsν(dx)]dt+α Z T

0

E[

Z T t

||Y_sⁿ⁺¹−Y_sⁿ||²_Vds]e^βtdt

≤ ¹₂ Z T

0

E[

Z T t

|Z_sⁿ−Z_sⁿ⁻¹|²_L

2(K,H)ds]e^βtdt+1 2

Z T 0

e^βtE[

Z T t

Z

X

|Qⁿ_s −Qⁿ⁻¹_s |²_H]dsν(dx)]dt (4.20) In particular,

Z T 0

E[

Z T t

|Z_sⁿ⁺¹−Z_sⁿ|²_L

2(K,H)ds]e^βtdt+ Z T

0

e^βtE[

Z T t

Z

X

|Qⁿ⁺¹_s −Qⁿ_s|²_H]dsν(dx)]dt

≤ ¹₂ Z T

0

E[

Z T t

|Z_sⁿ−Z_sⁿ⁻¹|²_L

2(K,H)ds]e^βtdt+ Z T

0

e^βtE[

Z T t

Z

X

|Qⁿ_s −Qⁿ⁻¹_s |²_H]dsν(dx)]dt

(4.21)

This implies that Z T

0

E[

Z T t

|Z_sⁿ⁺¹−Z_sⁿ|²_L

2(K,H)ds]e^βtdt+ Z T

0

e^βtE[

Z T t

Z

X

|Qⁿ⁺¹_s −Qⁿ_s|²_H]dsν(dx)]dt

≤(¹₂)ⁿC for some constantC. Thus, it follows from (4.20) that

E[

Z T 0

|Y_sⁿ⁺¹−Y_sⁿ|²_Hds]≤(1

2)ⁿC (4.22)

Hence, we conclude from (4.18) that E[

Z _T

0

|Z_sⁿ⁺¹−Z_sⁿ|²_L

2(K,H)ds] +E[

Z _T

0

Z

X

|Qⁿ⁺¹_s −Qⁿ_s|²_H]dsν(dx)]dt

≤(¹₂)ⁿCβ+¹₂{E[

Z T 0

|Z_sⁿ−Z_sⁿ⁻¹|²_L

2(K,H)ds] +E[

Z T 0

Z

X

|Qⁿ_s −Qⁿ⁻¹_s |²_H]dsν(dx)]} (4.23) Using the above inequality repeatedly gives

E[

Z T 0

|Z_sⁿ⁺¹−Z_sⁿ|²_L

2(K,H)ds] +E[

Z T 0

Z

X

|Qⁿ⁺¹_s −Qⁿ_s|²_H]dsν(dx)]

≤(¹₂)ⁿnCβ (4.24)

Combining (4.18) and (4.23) we have that E[

Z T 0

||Y_sⁿ⁺¹−Y_sⁿ||²_Vds]≤(1

2)ⁿ(n+ 1)Cβ (4.25)

It follows now from (4.24) and (4.25) that the the sequence (Y_tⁿ, Z_tⁿ, Qⁿ_t), n ≥ 1 converges in M²([0, T], V)×M²([0, T], L2(K, H))×M²([0, T], L²(ν)) to some limit (Y_t, Z_t, Q_t). Letting n→ ∞ in (4.14), we see that (Y_t, Z_t, Q_t) satisfies

Yt+ Z T

t

AYsds+ Z T

t

b(s, Zs)ds+ Z T

t

ZsdBs+ Z T

t

Z

X

Qs(x)N(ds, dx) =e φ (4.26) i.e., (Yt, Zt, Qt) is a solution to equation (3.3).

Uniqueness