NON-LINEAR TIME-ADVANCED BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH JUMPS

ISSN 0806–2439 October 2011

NON-LINEAR TIME-ADVANCED BACKWARD STOCHASTIC 2

PARTIAL DIFFERENTIAL EQUATIONS WITH JUMPS 3

OLIVIER MENOUKEU-PAMEN 4

Abstract. We prove an existence and uniqueness result for non-linear time-advanced backward stochastic partial differential equations with jumps (ABSPDEJs). We then apply our results to study a time-advanced backward type of stochastic generalized porous medium equations with jumps.

1. Introduction 5

The notion of backward stochastic differential equations (BSDEs) has received a lot of 6

attention in the past two decades owing to a range of applications in stochastic optimal 7

control theory, stochastic differential games, econometrics, mathematical finance, and non 8

linear partial differential equation. See [8, 9, 15, 19, 32]. Since the work by Pardoux and 9

Peng [24], there has been significant literature dedicated to the case of BSDE. See e.g., 10

[1, 3, 9, 30].

11

Recently, Peng and Yang [25] introduced the notion of anticipated (or time-advanced) 12

backward stochastic differential equations (ABSDEs). They proved existence and unique- 13

ness of adapted solutions to ABSDEs under Lipschitz continuity of the drift. ABSDEs 14

appear for example as adjoint processes when dealing with the maximum principle for 15

stochastic control for a system with delay. See [6, 25, 20, 29]. The results in [25] were 16

extended to the Poisson jumps case by Øksendal et al. [20] with an additional moving 17

average type of delay in the drift coefficient.

18

In the present paper, we consider an infinite-dimensional version of the previous work.

19

More exactly, we are interested in studying a class of time-advanced backward stochas- 20

tic partial differential equations with jumps (ABSPDEJs) which includes the following 21

Date: First Version: July, 2011. This Version: October 2011.

2010Mathematics Subject Classification. 60G51, 60H05, 60H15, 91G10.

Key words and phrases. Time-advanced BSDE, L´evy process, stochastic evolution equation.

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

1

ABSPDEs:

22

dY(t, x) = −A(t, Y(t, x))dt−Eh b(t, x)

F_ti

dt+Z(t, x) dB(t) +

Z

X

Q(t, x, ζ)Ne(dt,dζ), (t, x)∈[0, T]×Rⁿ (1.1) 23

Y(t, x) =G(t, x), Z(t, x) = G₁(t, x), Q(t, x, ζ) = G₂(t, x, ζ), t∈[T, T +δ], (1.2) where b(t, x) = b(t, x, Y(t, x), Y(t + δ, x), Y_t(x), Z(t, x), Z(t + δ, x), Z_t, Q(t, x,·), Q(t + 24

δ,·), Q_t(·)), dY(t, x) denotes the Itˆo differential with respect to t and where A is a fam- 25

ily of (nonlinear) operators satisfying some conditions (see Assumptiion A1 ), b satisfies 26

Lipschitz continuous conditions (see Assumption A2) and Y_t is defined in (2.3).

27

We assume that G(t, ω) is a continuous H-valued F_t-measurable process, G₁(t, ω) is a 28

continuous L₂(K, H)-valued F_t-measurable process, G(t, ω) is a continuous L²(ν)-valued 29

F_t-measurable process and for all (t, x)∈[T, T +δ]×Rⁿ we have 30

E

hZ T+δ T

Z

Rⁿ

kG(t, x)k²dxdt i

<∞, E

hZ T+δ T

Z

Rⁿ

kG1(t, x)k²dxdt i

<∞, EhZ T+δ

T

Z

Rⁿ

Z

R0

kG₂(t, x, ζ)k²ν(dζ) dxdti

<∞.

The functionb : [0, T]×Rⁿ×R×R×R×R×Ω−→Rand the terminal value functions 31

G, G₁, G₂ are given.

32

We remark that, in the infinite dimensional case, the existence and uniqueness of adapted 33

solutions of BSDEs has also been studied by several authors in the case with no delay. See 34

[10, 11, 12, 13, 17, 18, 31].

35

In the case with delay, Øksendal et al [21] derived an existence and uniqueness result in 36

finite dimension when the operatorAis linear. We aim at giving conditions on the operator 37

A, function b and the terminal value functions, which contain as a special case, the corre- 38

sponding results in the finite dimensional ABSDEs case (A = 0 orA linear). We establish 39

an existence and uniqueness result of a “strong”solution (Y(t, x), Z(t, x), Q(t, x, ζ)) of the 40

ABSPDEJs in an appropriate set, that is the probability space, the noise and the Poisson 41

random measure are given.

42

We shall in the present paper prove the existence and uniqueness of solutions of Equa- 43

tions (1.1)-(1.2) in infinite dimension and whenAis a non-linear operator. We shall employ 44

the Galerkin approximation method (see e.g., [2, 5, 7, 14, 22, 23, 26]), which consists of 45

looking at the ABSPDEJs (1.1)-(1.2) as a special case of time-advanced backward stochas- 46

tic evolution for Hilbert space valued processes.

47

The second motivation of our paper is to apply our results to study time-advanced 48

backward-type stochastic generalized porous medium equations with jumps (BSPMEJs).

49

Let us consider first the deterministic homogeneous Dirichlet problem of the generalized 50

porous medium equation (or Filtration equation) in complete form and with delay.

51

∂_tY = ∆Φ(Y) +b(t, Y(t), Y^t) in [0, T]× O, (1.3)

Y(t, x) = ϕ(t, x) in [−δ,0]× O, (1.4)

Y(t, x) = 0 in [0, T]×∂O, (1.5)

where O is a bounded open subset in Rⁿ, ∆ is the usual Laplace operator, Φ : R → R 52

is continuous, monotone increasing function which satisfies some properties which will be 53

given in Section 4. For Y(·) : R+ → R, Y^t will denote the function defined by Y^t(s) = 54

Y(t−s) fors∈[0, δ]. If we assume for instance that the previous equation represents the 55

gas flow through a porous medium thenY represents the density andbrepresents the mass 56

force source in the medium.

57

LetY be solution to the problem (1.3)-(1.5). Define Ye as the time reversal of T i.e., Ye(t, x) =Y(T −t, x) fort ≤T.

Then Ye(t, x) solves the following time-advanced backward generalized porous medium 58

equation.

59

∂_tYe = ∆Φ(Ye) +b(t,Ye(t),Ye_t) in [0, T]× O, (1.6) Ye(t, x) = ϕ(t, x) in [T, Te +δ]× O, (1.7)

Ye(t, x) = 0 in [0, T]×∂O, (1.8)

We shall study the following type of time-advanced BSPMEJs with a given terminal con- 60

dition.

61

dY(t, x) = −∆Φ(Y(t, x))dt−Eh b(t, x)

F_ti

dt+Z(t, x) dB(t) +

Z

X

Q(t, x, ζ)Ne(dt,dζ), (t, x)∈[0, T]× O (1.9) 62

Y(t, x) =G(t, x), Z(t, x) = G1(t, x), Q(t, x, ζ) = G2(t, x, ζ), (t, x)∈[T, T +δ]× O, (1.10) where b(t, x) = b(t, x, Y(t, x), Y(t + δ, x), Y_t(x), Z(t, x), Z(t + δ, x), Z_t, Q(t, x,·), Q(t + 63

δ,·), Qt(·)).

64

Let us mention that in the stochastic framework, since the solution must be adapted to 65

the filtration generated by Brownian motion and the Poisson random measure, we need to 66

condition the anticipated terms with respect to the filtration.

67

In [5, 33], the authors studied the existence and uniqueness of a strong solution for a 68

class of stochastic functional differential equations driven by Brownian motion. The time- 69

advanced backward stochastic generalized porous medium equation can be seen as the 70

inverse problem to determine the stochastic coefficients from the terminal values.

71

The paper is organized as follows: In Section 2, we give the framework needed to establish 72

our results. Section 3 contains the main results of the paper, and in Section 4, we apply 73

the results to study time-advanced BSPMEJs.

74

2. Framework 75

In this section, we introduce the setting in which we shall prove our main result for 76

time-advanced backward stochastic partial differential equations with jumps . 77

LetV, H, K, be three real separable Hilbert spaces such thatV is continuously, densely 78

embedded inH, with 79

V ,→H ≡H^∗ ,→V^∗. (2.1)

Here, V^∗ is the topological dual of V. We assume in particular that V and V^∗ are 80

uniformly convex. Denote by k · k_V, k · k_V^∗ and | · |the norm inV, V^∗ and H respectively, 81

byh·i the duality product between V, V^∗ and and by (·) the scalar product in H.

82

Let A(t,·) : V −→ V^∗ be a family of (nonlinear) operators, defined a.e.t and p ≥ 2.

83

Assume the following conditions:

84

Assumption A1.

85 86

(a1) Coercivity: there exist α > 0, λ ∈ R and an F_t-adapted process f ∈ L¹([0, T]× Ω, dt⊗dP) such that:

2hA(t, u), ui ≤λ|u|²_H −αkuk^p_V +f(t) for all u∈V, a.e. t

(a2) Boundedness: there existsγ >0 and anF_t-adapted processg ∈L^p−1p ([0, T]×Ω, dt⊗ dP) such that

kA(t, u)k_V^∗ ≤γkuk^p−1_V +g(t) for all u∈V, a.e. t (a3) Measurability:

t ∈(0, T)7−→A(t, u), is Lebesgue-measurable for all u∈V, a.e. t (a4) Weak monotonicity: there exists λ >0 such that

2hA(t, u)−A(t, v), u−vi ≤λ|u−v|²_H for all u, v ∈V, a.e. t (a5) Hemicontinuity: The map

θ ∈R→ hA(t, u+θv), wi ∈R is continuous foru, v, w ∈V, a.e. t

87

Let us mention that, in general, the operator A is not necessarily bounded from H to 88

H.

89

Let (Ω,F,P) be a complete probability space and {B(t), t≥ 0} be a cylindrical Brow- 90

nian motion with covariance space K on the probability space (Ω,F,P) i.e., for any 91

k ∈K, hB(t), kiis a real valued Brownian motion with Eh

hB(t), ki²i

=t|k|²_K. 92

Let (X,B(X)) be a measurable space, where X is a topological vector space. Denote 93

by η(t) a L´evy process on X. Let ν(dζ) the L´evy measure of η and L²(ν) be the L²- 94

space of square integrable H-valued measurable functions associated with ν. Define p(t) 95

by p(t) = ∆η(t) = η(t)−η(t−). It follows from the property of the Poisson process that 96

p= (p(t), t ∈ D_p) is a stationary Poisson point process on X with characteristic measure 97

ν. Let N(dt,dζ) = N_p(dt,dζ) be the Poisson counting measure associated with the L´evy 98

process. Then N(dt,dζ) has the compensator E[N(dt,dζ)] = ν(dζ) dt on (X,B(X)).

99

Denote by Ne(dt,dζ) = N(dt,dζ)−ν(dζ) dt the compensated Poisson random measure.

100

Let {F_t}_t∈[0,T], be the σ-algebras generated by {B(s), N(s, A), A ∈ B(X), s ≤ t}.

101

Then for a cylindrical Brownian motion, it is known that the following representation 102

holds:

103

B(t) =

∞

X

i=0

β_i(t)k_i, (2.2)

where {k_i}i≥1 is an orthonormal basis of K and {β_i(t), t ≥ 0} are independent standard 104

Brownian motions.

105

LetLbe a Hilbert space, δ >0, p≥0 and T >0. We denote byM_L^p =M^p(0, T+δ;L), 106

the reflexive Banach space of L-valued processes (Y(t))t∈[0,T+δ] measurable and satisfying:

107

(1) Y(t) is F_t-measurable a.e. in t where F_t=F_T, t∈[T, T +δ].

108

(2) Eh RT+δ

0 kY(t)k^p_Ldti

<∞.

109

Forp= 2, M²(0, T +δ;L) is a Hilbert space equipped with the following scalar product hY, Zi_M²

L =EhZ T+δ 0

hY(t), Z(t)i_Ldti

The space S² =S²(0, T +δ;L) is defined in a similar way with (2), replaced by 110

kYk_S² =Eh sup

0≤t≤T+δ

kY(t)k²_Li¹₂

<∞.

Let L₂(K, H) denote the space of Hilbert-Schmidt operators acting from K into H.

111

Then L₂(K, H) is a separable Hilbert space with the inner product hY, Zi_L2(K,H) = 112

∞

P

i=0

hY ki, ZkiiH. Let k · k_L2(K,H) represent the corresponding (Hilbert-Schmidt) norm.

113

Given a stochastic process Y(t)∈M^p(0, T +δ;V)∩S²(0, T +δ;H), we denoted by Yt, 114

the M^p(0, δ;V)∩S²(0, δ;H)-valued stochastic process by setting 115

Y_t(s)(ω) = Y(t+s)(ω); s∈[0, δ] (2.3)

As mentioned in the introduction, the purpose of the paper is to establish existence and 116

uniqueness results for a class of time-advanced BSPDEJ of the form 117







dY(t) =−Eh

b(t, Y(t), Y(t+δ), Y_t, Z(t), Z(t+δ), Z_t, Q(t,·), Q(t+δ,·), Q_t(·)) F_ti

dt

−A(t, Y(t))dt+Z(t) dB(t) + R

XQ(t, ζ)Ne(dt,dζ), t ∈[0, T] Y(t) =G(t), Z(t) = G₁(t), Q(t, ζ) =G₂(t, ζ), t∈[T, T +δ],

(2.4) where A is a nonlinear operator, G(t, ω) is a continuous H-valued F_t-measurable process, 118

G₁(t, ω) is a continuous L₂(K, H)-valued F_t-measurable process, G₂(t, ω) is a continuous 119

L²(ν)-valuedF_t-measurable process andb : [0, T]×H³×(L₂(K, H))³×(L²(ν))³×Ω−→H 120

satisfies the following conditions:

121

Assumption A2.

122 123

(b1) E

RT

0 |b(t,0,0,0,0,0,0,0,0,0)|²_Hdt

<∞.

124

(b2) t∈[0, T]7−→b(t, y, y1, y2, z, z1, z2, q, q1, q2, ω) is Lebesgue-measurable.

125

(b3) Lipschitz condition: There exists a C such that 126

|b(t, y, y₁, y₂, z, z₁, z₂, q, q₁, q₂)−b(t,y,¯ y¯₁,y¯₂,z,¯ z¯₁,z¯₂,q,¯ q¯₁,q¯₂)|_H

≤C

|y−y|¯_H +|y₁−y¯₁|_H +|y₂−y¯₂|_L²_(0,δ;H)+kz−zk¯ _L²_(K,H)+kz₁−z¯₁k_L²_(K,H) +kz₂−z¯₂k_L²_(0,δ;L²_(K,H))+kq−qk¯ _L²_(ν)+kq₁−q¯₁k_L²_(ν)+kq₂−q¯₂k_L²_(0,δ;L²_(ν))

3. Main results 127

In this section, we present the main results of the paper.

128

3.1. Existence and uniqueness.

129

Theorem 3.1. Assume that the terminal values G ∈ S²(T, T + δ;H) ∩ M^p(T, T + 130

δ;V), G₁ ∈ M²(T, T + δ;L₂(K, H)) and G₂ ∈ M²(T, T + δ;L²(ν)). Moreover, as- 131

sume that the conditions of Assumptions A1-A2 are fulfilled. Then there exists a unique 132

H×L₂(K, H)×L²(ν)-valued progressively measurable process (Y(t), Z(t), Q(t))solution of 133

equation (2.4) in M²(0, T+δ;H)∩M^p(0, T +δ;V)×M²(0, T +δ;L₂(K, H))×M²(0, T + 134

δ;L²(ν)).

135

We shall first prove the uniqueness of the solution when such a solution exists. Then we 136

shall prove the existence in several lemmas.

137

Proof of the uniqueness. Let (Y(t), Z(t), Q(t)) and ( ¯Y(t),Z(t),¯ Q(t)) in¯ M²(T, T +δ;H)∩ 138

M^p(T, T +δ;V)×M²(T, T +δ;L₂(K, H))×M²(T, T +δ;L²(ν)) be two solutions of the 139

time-advanced BSPDEJ (2.4).

140

By the Itˆo formula, we get 141

0 =|G(T)−G(T¯ )|²_H =|Y(T)−Y¯(T)|²_H

=|Y(t)−Y¯(t)|²_H −2 Z T

t

hA(s, Y(s))−A(s,Y¯(s)), Y(s)−Y¯(s)ids

−2 Z T

t

Eh

b(s)−¯b(s) F_si

, Y(s)−Y¯(s) ds+

Z T t

kZ(s)−Z¯(s)k²_L2(K,H)ds + 2

Z T t

Y(s)−Y¯(s),(Z(s)−Z(s)) dB¯ (s) +

Z T t

Z

X

|Q(s, ζ)−Q(s, ζ)|¯ ²_HN(ds,dζ) + 2

Z T t

Y(s)−Y¯(s),(Q(s, ζ)−Q(s, ζ))¯ Ne(ds,dζ)

, (3.1)

where we have used the short hand notation 142

b(t) = b(t, Y(t), Y(t+δ), Y_t, Z(t), Z(t+δ), Z_t, Q(t,·), Q(t+δ,·), Q_t(·))

This implies that 143

|Y(t)−Y¯(t)|²_H + Z T

t

kZ(s)−Z(s)k¯ ²_L2(K,H)ds+ Z T

t

Z

X

|Q(s, ζ)−Q(s, ζ)|¯ ²_HN(ds,dζ)

= +2 Z T

t

hA(s, Y(s))−A(s,Y¯(s)), Y(s)−Y¯(s)ids+ 2 Z T

t

E

h

b(s)−¯b(s) Fs

i

, Y(s)−Y¯(s)

ds

−2 Z T

t

Y(s)−Y¯(s),(Z(s)−Z(s)) dB(s)¯

−2 Z T

t

Y(s)−Y¯(s),(Q(s, ζ)−Q(s, ζ))¯ Ne(ds,dζ) .

Using weak monotonicity (a(2)), Lipschitz condition (b(3)) and taking the expectation, 144

we get 145

Eh

|Y(t)−Y¯(t)|²_Hi

+EhZ T t

kZ(s)−Z¯(s)k²_L2(K,H)dsi +EhZ T

t

Z

X

|Q(s, ζ)−Q(s, ζ¯ )|²_Hν(dζ)dsi

≤λEhZ T t

|Y(s)−Y¯(s)|²_Hdsi + 1

εEhZ T t

|Y(s)−Y¯(s)|²_Hdsi +CεE

hZ T t

|Y(s)−Y¯(s)|²_Hds i

+CεE hZ T

t

|Y1(s)−Y¯1(s)|²_Hds i

+C_εEhZ T t

kY₂(s)−Y¯₂(s)k²_L2(0,δ;H)dsi

+C_εEhZ T t

kZ(s)−Z¯(s)k²_L

2(K,H)dsi +C_εEhZ T

t

kZ₁(s)−Z¯₁(s)k²_L2(K,H)dsi

+C_εEhZ T t

kZ₂(s)−Z¯₂(s)k²_L2(0,δ;L2(K,H))dsi +C_εEhZ T

t

Z

X

|Q(s, ζ)−Q(s, ζ)|¯ ²_Hν(dζ)dsi

+C_εEhZ T t

Z

X

|Q₁(s, ζ)−Q¯₁(s, ζ)|²_Hν(dζ)dsi +CεE

hZ T t

Z

X

kQ2(s, ζ)−Q¯2(s, ζ)k²_L2(0,δ;H)ν(dζ)ds i

. (3.2)

Note also that since Y(t) = ¯Y(t) =G(t) for t∈[T, T +δ], we have 146

EhZ T t

|Y₁(s)−Y¯₁(s)|²_Hdsi

=EhZ T t

|Y(s+δ)−Y¯(s+δ)|²_Hdsi

≤EhZ T t

|Y(s)−Y¯(s)|²_Hdsi

, (3.3)

and by interchanging the order of integration, we get 147

EhZ T t

|Y₂(s)−Y¯₂(s)|²_L2(0,δ;H)dsi

=EhZ T t

Z δ 0

|Y(s+r)−Y¯(s+r)|²_Hdr dsi

≤EhZ T+δ t

|Y(u)−Y¯(u)|²_Hdu Z u

u−δ

dsi

≤δEhZ T t

|Y(s)−Y¯(s)|²_Hdsi

. (3.4)

The same inequalities also hold for Z−Z¯ and Q−Q. Using (3.3) and (3.4), it follows¯ 148

from (3.2) that 149

Eh

|Y(t)−Y¯(t)|²_Hi

+EhZ T t

kZ(s)−Z(s)k¯ ²_L2(K,H)dsi +E

hZ T t

Z

X

|Q(s, ζ)−Q(s, ζ)|¯ ²_Hν(dζ)ds i

≤(λ+1

ε +C_ε,δ)EhZ T t

|Y(s)−Y¯(s)|²_Hdsi

+C_ε,δ¹ EhZ T t

kY₂(s)−Y¯₂(s)k²_L2(0,δ;H)dsi +C_ε,δ² EhZ T

t

Z

X

|Q(s, ζ)−Q(s, ζ)|¯ ²_Hν(dζ)dsi

. (3.5)

Now choosing ε small enough such that C_ε,δ¹ <1 and C_ε,δ² <1, we get 150

E h

|Y(t)−Y¯(t)|²_Hi

≤Cε,δ,λE hZ T

t

|Y(s)−Y¯(s)|²_Hds i

, (3.6)

where C_ε,δ,λ =λ+1

ε +C_ε,δ. Hence, Gronwall’s lemma obviously implies uniqueness.

151

152

Proof of the existence. We shall first give the following result on existence and uniqueness 153

of a stochastic evolution equation in finite dimension.

154

Proposition 3.2. Assume thatV =H =V^∗ =R^d and b= 0, and the operator A in (2.4) 155

satisfies Assumption A1 with λ = 0 in (a4). Then for G ∈ S²(T, T +δ;H)∩M^p(T, T + 156

δ;V), G₁ ∈M²(T, T +δ;L₂(K, H)) and G₂ ∈M²(T, T +δ;L²(ν)), there exists a unique 157

H×L₂(K, H)×L²(ν)-valued progressively measurable process (Y(t), Z(t), Q(t))solution of 158

equation (2.4) in M²(0, T+δ;H)∩M^p(0, T +δ;V)×M²(0, T +δ;L₂(K, H))×M²(0, T + 159

δ;L²(ν)).

160

Proof. The result follows by combining the results in [4, 21, 33].

161

We shall prove the following lemmas.

162

Lemma 3.1. Required conditions of Theorem 3.1. Moreover, assume that 163

b(t, y, y₁, y₂, z, z₁, z₂, q, q₁, q₂, ω) = b(t, ω) is independent of y, y₁, y₂, z, z₁, z₂, q, q₁, q₂ and 164

that Eh RT

0 |b(t)|²_Hdti

< ∞. Then there exists a unique H ×L₂(K, H) × L²(ν)-valued 165

progressively measurable process (Y(t), Z(t), Q(t)) solution of equation (2.4) in M²(0, T + 166

δ;H)∩M^p(0, T +δ;V)×M²(0, T +δ;L2(K, H))×M²(0, T +δ;L²(ν)).

167

Proof of Lemma 3.1.

168 169

The uniqueness has already been shown.

170 171 172

Existence : 173

Let D(A) = {v; v ∈ V, Av ∈ H}. Then the subspace D(A) is dense in H. We fix an orthonormal basis {e₁, . . . , e_n, . . .} of H where e_i ∈D(A) for all i ≥1. Let V_n =H_n =V_n^∗ be the vector space generated by {e₁, . . . , e_n}. Let P_n ∈ L(H, H_n =V_n) be the orthogonal projection from H into V_n. Then, P_n can be extended to an operator Pe_n from V^∗ onto V_n^∗ =Vn as follows:

Pe_nu=

n

X

i=1

hu, e_iie_i, u∈V^∗.

PutA_n=Pe_nA. ThenA_n is an operator from V_n into V_n =V_n^∗ satisfying Assumption A1.

174

Denote by K_n the subspace generated by {k₁, . . . , k_n} with k_n given as in (2.2). Let P¯_n ∈ L(K, K_n) be the projection from K onto K_n. Let Bⁿ(t) be the K_n-valued Wiener process defined by Bⁿ(t) = ¯P_nB(t). Define

F_tⁿ=σ{Bⁿ(s), N(s, A), A∈ B(X), s≤t}

completed by the probability measure P. 175

Define

b_n(t) =Eh P_nb(t)

F_tⁿi and for t∈[T, T +δ],

Gⁿ(t) =E h

PnG(t) F_Tⁿi

, Gⁿ₁(t) = E h

PnG1(t) F_Tⁿi

, Gⁿ₂(t, ζ) =E h

PnG2(t, ζ) F_Tⁿi

. Now, we consider the following time-advanced BSDEJ on V_n

176







dYⁿ(t) =A_n(t, Yⁿ(t))dt+b_n(t)dt+Zⁿ(t)dBⁿ(t) +R

XQⁿ(t, ζ)Ne(dt,dζ), t ∈[0, T]

Yⁿ(t) =Gⁿ(t), Zⁿ(t) = Gⁿ₁(t), Qⁿ(t, ζ) = Gⁿ₂(t, ζ), t∈[T, T +δ].

(3.7) A_n is an operator satisfying Assumption A1 on the finite dimensional spaceV_n ontoV_n. 177

It is also easy to check that Bⁿ, Gⁿ, Gⁿ₁ and G²_n satisfy the assumptions of Proposition 178

3.2 by replacing V, H, V^∗ by V_n, H_n, V_n^∗. We can then conclude that Equation (3.7) has a 179

unique F_tⁿ-adapted solution (Yⁿ(t), Zⁿ(t), Qⁿ(t,·))∈V_n×L₂(K_n, V_n)×L²(ν).

180

We also have that for each n and t 181

|Gⁿ(T)|H ≤ |G(T)|H, lim

n→∞E

|Gⁿ(T)−G(T)|²_H

= 0 (3.8)

182

|Gⁿ(t)|_H ≤ |G(t)|_H, lim

n→∞E

Z T+δ T

|Gⁿ(t)−G(t)|²_Hdt

= 0 (3.9)

183

kGⁿ₁(t)k_L2(H,K) ≤ kG₁(t)k_L2(H,K), lim

n→∞E

Z T+δ T

kGⁿ₁(t)−G₁(t)k²_L2(H,K)dt

= 0 (3.10) 184

kGⁿ₂(t)k_L²_(ν) ≤ kG₂(t)k_L²_(ν), lim

n→∞E

Z T+δ T

Z

X

|Gⁿ₂(t, ζ)−G₂(t, ζ)k_Hν(dζ)dt

= 0 (3.11)

185

|bⁿ(t)|_H ≤ |b(t)|_H, lim

n→∞E Z T

0

|bⁿ(t)−b(t)|²_Hdt

= 0 (3.12)

In what follows, we shall split the proof into three steps. In the first step, we shall show 186

that the sequence (Yⁿ(t), Zⁿ(t), Qⁿ(t,·)) is bounded inM²(0, T+δ;H)∩M^p(0, T+δ;V)× 187

M²(0, T+δ;L₂(K, H))×M²(0, T+δ;L²(ν)). In step 2, we shall show that the weak limit 188

as a version that satisfies the following time-advanced BSPDE 189

dY(t) =−b(t)dt−X(t)dt+Z(t) dB(t) +R

X Q(t, ζ)Ne(dt,dζ), t∈[0, T] Y(t) =G(t), Z(t) = G₁(t), Q(t, ζ) = G₂(t, ζ), t∈[T, T +δ].

In the last step, we shall prove that X(t) =A(t, Y(t)) in M²(0, T;V^∗).

190

Step 1. Let show that the sequence (Yⁿ(t), Zⁿ(t), Qⁿ(t,·)) is bounded inM²(0, T+δ;H)∩ 191

M^p(0, T +δ;V)×M²(0, T +δ;L₂(K, H))×M²(0, T +δ;L²(ν)). By the Itˆo formula, we 192

have 193

Eh

|Yⁿ(t)|²_Hi

=Eh

|Gⁿ(T)|²_Hi +Eh

2 Z T

t

hPe_nA(s, Yⁿ(s)), Yⁿ(s)idsi + 2EhZ T

t

b_n(s), Y(s) dsi

−EhZ T t

kZⁿ(s)k²_L2(Kn,Vn)dsi

−E hZ T

t

Z

X

|Qⁿ(s, ζ)|²_Hν(dζ)dt i

, (3.13)

where kZⁿ(s)k²_L

2(Kn,Vn) =

n

P

i,j=1

Z_(i,j)ⁿ (s)2

denotes the Hilbert-Schmidt norm. Using the 194

coercivity argument, we obtain 195

Eh

|Yⁿ(t)|²_Hi

≤Eh

|G(T)|²_Hi

−αEhZ T t

kYⁿ(s)k^p_Vdsi

+λEhZ T t

|Yⁿ(s)|²_Hdsi

(3.14) +EhZ T

t

f(s)dsi +1

εEhZ T t

|Yⁿ(s)|²_Hdsi

+C_εEhZ T t

|b(s)|²_Hdsi

−EhZ T t

kZ¯ⁿ(s)k²_L2(K,H)dsi

−EhZ T t

Z

X

|Qⁿ(s, ζ)|²_Hν(dζ)dti , where ¯Zⁿ(s) = ¯P_nZⁿ(s) with ¯P_n been the projection from K intoK_n. Therefore, 196

Eh

|Yⁿ(t)|²_Hi

≤(λ+1

ε)EhZ T t

|Yⁿ(s)|²_Hdsi +Eh

|G(T)|²_H +C_ε Z T

t

|b(s)|²_Hds+ Z T

t

|f(s)|dsi

It follows from Gronwall’s lemma that EhZ T

0

|Yⁿ(s)|²_Hdsi

≤CEh

|G(T)|²_H + Z T

0

|b(s)|²_Hds+ Z T

0

|f(s)|dsi

for a suitable constant C. Since the right hand side does not depend on n we can con- 197

clude that (Yⁿ, n ≥ 1) is bounded in M²(0, T +δ;H). This also implies that the se- 198

quence (Yⁿ(t),Z¯ⁿ(t), Qⁿ(t,·)) is bounded inM²(0, T+δ;H)∩M^p(0, T+δ;V)×M²(0, T+ 199

δ;L₂(K, H))×M²(0, T+δ;L²(ν)). Moreover, it follows from boundedness ofA(condition 200

(a2)) that the sequence (A(·, Yⁿ), n ≥ 1) is bounded in M^p0(0, T;V^∗) (where p⁰ is the 201

conjugate of p). Hence, by the weak compactness of Hilbert spaces and separable reflex- 202

ive Banach spaces, there exist a subsequence (Y^nk(·),Z¯^nk(·), Q^nk(·), A(·, Y^nk) k ≥ 1) of 203

(Yⁿ(·), Zⁿ(·), Qⁿ(·)) such that 204

Y^nk →Y weakly in M^p(0, T +δ;V) (3.15) Y^nk(0) →Y₀ weakly inL²(Ω;H) (3.16) Z¯^nk →Z weakly in M²(0, T +δ;L₂(K, H)) (3.17) Q^nk(·)→Qweakly in M²(0, T +δ;L²(ν)) (3.18) A(·, Y^nk)→X weakly inM^p0(0, T;V^∗) (3.19) Step 2. We shall now show that (Y, Z, Q) has a version which is solution of the time- 205

advanced BSDEJ (2.4). We first remark that for n, i≥1, we have 206

d(Yⁿ(t), ei) = hP¯nA(t, Yⁿ(t)), eiidt−(bn(t), ei)dt+ ( ¯Zⁿ(t)dBⁿ(t), ei) +

Z

X

(Qⁿ(s, ζ), e_i)Ne(dt,dζ)

=−hA(t, Yⁿ(t)), e_iidt−(b_n(t), e_i)dt+ ( ¯Zⁿ(t)dB(t), e_i) +

Z

X

(Qⁿ(s, ζ), ei)Ne(dt,dζ) (3.20)

Leth(t) be an absolutely continuous function fromR to Rwith h⁰(·)∈L⁽[0, T]).

207

Ifϕ is a function from [0, T] into R, we define ¯ϕ from [−δ, T +δ] into R as follows:

¯ ϕ(t) =

ϕ(t), if t∈[0, T], 0, otherwise.

The latter and Itˆo formula permit us to rewrite (3.20) as follows 208

(Yⁿ(T), e_i)h(T −t)−(Yⁿ(0), e_i)h(−t)

=− Z T

0

h(s−t)hA(s, Yⁿ(s)), e_iids− Z T

0

h(s−t)(b_n(s), e_i)ds +

Z T 0

h(s−t)d(

Z s 0

Z¯ⁿ(r)dB(r), ei) + Z T

0

Z

X

h(s−t)(Qⁿ(s, ζ), ei)Ne(ds,dζ) +

Z T 0

h⁰(s−t)(Yⁿ(s), e_i)ds, ∀t∈[−δ, T +δ]. (3.21)

We use the fact that the linear maps Z 7→

Z T 0

h(s−t)d(

Z s 0

Z¯ⁿ(r)dBⁿ(r), e_i) =

∞

X

j=1

Z T 0

h(s−t)(Z(s, k_j), e_i)dβ^j(s)

is continuous from M²(0, T +δ;L₂(K, H)) into L²(Ω), and that Z ∈M²(0, T +δ;L²(ν))7→

Z T 0

Z

X

h(s−t)(Qⁿ(s, ζ), ei)Ne(ds,dζ)

is continuous from M²(0, T +δ;L²(ν)) into L²(Ω). We replace n by n_k in (3.21) and take 209

the weak limit in L²(Ω) to obtain 210

(Y(T), e_i)h(T −t)−(Y₀, e_i)h(−t)

=− Z T

0

h(s−t)hA(s, Y(s)), e_iids− Z T

0

h(s−t)(b(s), e_i)ds +

Z T 0

h(s−t)d(

Z s 0

Z(r)dB(r), ei) + Z T

0

Z

X

h(s−t)(Q(s, ζ), ei)Ne(ds,dζ) +

Z T 0

h⁰(s−t)(Y(s), e_i)ds, ∀t∈[−δ, T +δ]. (3.22) Choose for n≥1

211

h_n(u) =







1, u≥ _2n¹

1− ¹_n(s− _2n¹ ), − _2n¹ ≤u≤ _2n¹

0, u≤ −_2n¹ .

We replace h(·) by h_n(·) in (3.22) to get 212

(Y(T), e_i)h_n(T −t)−(Y₀, e_i)h_n(−t)

=− Z T

0

h_n(s−t)hA(s, Y(s)), e_iids− Z T

0

h_n(s−t)(b(s), e_i)ds +

Z T 0

h_n(s−t)d(

Z s 0

Z(r)dB(r), e_i) + Z T

0

Z

X

h_n(s−t)(Q(s, ζ), e_i)Ne(ds,dζ) +

Z t+_2n¹ t−_2n¹

(Y(s), e_i)ds, ∀t ∈[−δ, T +δ]. (3.23)

We apply twice the change of variable and by letting n tend to infinity leads to 213

(Y(T), e_i)h(T −t)−(Y₀, e_i)h(−t)

=− Z T

t

hA(s, Y(s)), e_iids− Z T

t

(b(s), e_i)ds+ Z T

t

d(

Z s 0

Z(r)dB(r), e_i) +

Z T t

Z

X

(Q(s, ζ), e_i)Ne(ds,dζ) + (Y(t), e_i), ∀t ∈[−δ, T +δ], i≥1, (3.24) where h is defined fromR into Rby

h(t) =

1, if t≥0 0, if u <0.

It then follows that Y(0) =Y₀ and 214

dY(t) = −b(t)dt−X(t)dt+Z(t) dB(t) +R

XQ(t, ζ)Ne(dt,dζ), t∈[0, T]

Y(t) =G(t), Z(t) =G₁(t), Q(t, ζ) =G₂(t, ζ), t∈[T, T +δ]. (3.25)

Step 3. We shall prove thatX(t) = A(t, Y(t)) in M^p0(0, T;V^∗). It follows from (3.13) and 215

the weak monotonicity argument (a4) that for any Θ∈M^p(0, T;V)∩M²(0, T;H) 216

Eh

|Y^nk(t)|²_Hi

+EhZ T t

kZ¯^nk(s)k²_L

2(Kn,Vn)dsi

+EhZ T t

Z

X

|Q^nk(s, ζ)|²_Hν(dζ)dti

=Eh

|G^nk(T)|²_Hi

+ 2EhZ T t

hA(s, Y^nk(s))−A(s,Θ(s)),Θ(s)idsi + 2EhZ T

t

hA(s,Θ(s)), Y^nk(s)idsi

+ 2EhZ T t

hA(s, Y^nk(s))−A(s,Θ(s)), Y^nk(s)−Θ(s)idsi + 2E

hZ T t

bn_k(s), Y(s)

ds i

≤Eh

|G^nk(T)|²_Hi

+ 2EhZ T t

hA(s, Y^nk(s))−A(s,Θ(s)),Θ(s)idsi + 2EhZ T

t

hA(s,Θ(s)), Y^nk(s)idsi

+λEhZ T t

|Y^nk(s)−Θ(s)|²_Hdsi + 2EhZ T

t

b_nk(s), Y^nk(s) dsi

(3.26)

We take the limit as k goes to infinity and set t= 0 to get, 217

lim inf

k→∞ Eh

|Y^nk(0)|²_Hi

+ lim inf

k→∞ EhZ T 0

kZ¯^nk(s)k²_L2(Kn,Vn)dsi

+ lim inf

k→∞ EhZ T 0

Z

X

|Q^nk(s, ζ)|²_Hν(dζ)dti

≤Eh

|G(T)|²_Hi

+ 2EhZ T 0

hX(s)−A(s,Θ(s)),Θ(s)idsi + 2E

hZ T 0

hA(s,Θ(s)), Y(s)idsi +λE

hZ T 0

|Y(s)−Θ(s)|²_Hds i

+ 2EhZ T 0

b(s), Y(s) dsi

(3.27) On the other hand, by (3.25), we have

218

Eh

|Y₀|²_Hi

+EhZ T 0

kZ(s)k¯ ²_L2(K,V)dsi

+EhZ T 0

Z

X

|Q(s, ζ)|²_Hν(dζ)dti

=Eh

|G(T)|²_Hi

+ 2EhZ T 0

hX(s), Y(s)idsi

+ 2EhZ T t

b(s), Y(s) dsi

(3.28) and by using (3.16)-(3.18), we get

219

Eh

|Y₀|²_Hi

≤lim inf

k→∞ Eh

|Y^nk(0)|²_Hi EhZ T

0

kZ(s)k¯ ²_L2(K,V)dsi

≤lim inf

k→∞ EhZ T 0

kZ¯^nk(s)k²_L2(Kn,Vn)dsi E

hZ T 0

Z

X

|Q(s, ζ)|²_Hν(dζ)dt i

≤lim inf

k→∞ E hZ T

0

Z

X

|Q^nk(s, ζ)|²_Hν(dζ)dt i

It then follows that 220

−2EhZ T 0

hX(s)−A(s,Θ(s)), Y(s)−Θ(s)idsi +λE

hZ T 0

|Y(s)−Θ(s)|²_Hds i

≥0 Now, we set Θ =Y −µΘ₁ (forµ >0, Θ₁ ∈M²(0, T +δ;H)∩M²(0, T +δ;V)), we get 221

−2EhZ T 0

hX(s)−A(s, Y −µΘ₁), µΘ₁idsi

+λµ²EhZ T 0

|Θ₁|²_Hdsi

≥0 (3.29)

We divide (3.29) by µ, take the limit when µ → 0 and use the hemicontinuity (a5) to obtain

−2EhZ T 0

hX(s)−A(s, Y),Θ₁idsi

≥0, for all Θ₁ ∈M²(0, T +δ;H)∩M^p(0, T +δ;V).

Hence X =A(·, Y). The proof is complete.

222

223

Lemma 3.2. Assume that the conditions of Theorem 3.1 hold. Moreover, assume that 224

b(t, y, y₁, y₂, z, z₁, z₂, q, q₁, q₂, ω) = b(t, z, z₁, z₂, q, q₁, q₂, ω) is independent of y, y₁, y₂ and 225

that Eh RT

0 |b(tz, z₁, z₂, q, q₁, q₂)|²_Hdti

< ∞. Then there exists a unique H ×L₂(K, H)× 226

L²(ν)-valued progressively measurable process (Y(t), Z(t), Q(t)) solution of equation (2.4) 227

in M²(0, T +δ;H)∩M^p(0, T +δ;V)×M²(0, T +δ;L₂(K, H))×M²(0, T +δ;L²(ν)).

228

Proof of Lemma 3.2.

229 230

The uniqueness has already been shown.

231 232 233

Existence : 234

Set Z⁰(t) = 0 and Q⁰(t, x) = 0. For n ≥ 1, define (Yⁿ(t), Zⁿ(t), Qⁿ(t, x)) to be the 235

unique solution of the following BSPDEJ 236







dYⁿ(t) =−Eh

b(t, Zⁿ⁻¹(t), Zⁿ⁻¹(t+δ), Z_tⁿ⁻¹, Qⁿ⁻¹(t,·), Qⁿ⁻¹(t+δ,·), Qⁿ⁻¹_t (·)) Ft

i dt

−A(t, Yⁿ(t))dt+Zⁿ(t)dB(t) + R

XQⁿ(t, ζ)Ne(dt,dζ), t∈[0, T] Yⁿ(t) =G(t), Zⁿ(t) =G₁(t), Qⁿ(t, ζ) =G₂(t, ζ), t∈[T, T +δ],

(3.30) The existence of solution of (3.30) is a consequence of Lemma 3.1. We shall show that 237

(Yⁿ(t), Zⁿ(t), Qⁿ(t, x)) is a Cauchy sequence inM²(0, T+δ;H)∩M^p(0, T+δ;V)×M²(0, T+ 238

δ;L₂(K, H))×M²(0, T +δ;L²(ν)).

239

By the Itˆo formula, we have 240

0 =|G(T)−G(T¯ )|²_H =|Yⁿ⁺¹(T)−Yⁿ(T)|²_H

=|Yⁿ⁺¹(t)−Yⁿ(t)|²_H −2 Z T

t

hA(s, Yⁿ⁺¹(s))−A(s, Yⁿ(s)), Yⁿ⁺¹(s)−Yⁿ(s)ids +

Z T t

kZⁿ⁺¹(s)−Zⁿ(s)k²_L2(K,H)ds+ Z T

t

Z

X

|Qⁿ⁺¹(s, ζ)−Qⁿ(s, ζ)|²_HN(ds,dζ)

−2 Z T

t

E

h

bⁿ(s)−bⁿ⁻¹(s) Fs

i

, Yⁿ⁺¹(s)−Yⁿ(s)

ds + 2

Z T t

Yⁿ⁺¹(s)−Yⁿ(s),(Zⁿ⁺¹(s)−Zⁿ(s))dB(s) + 2

Z T t

Yⁿ⁺¹(s)−Yⁿ(s),(Qⁿ⁺¹(s, ζ)−Qⁿ(s, ζ))N(ds,e dζ)

, (3.31)

where we have used the short hand notation 241

bⁿ(t) = b(t, Zⁿ(t), Zⁿ(t+δ), Z_tⁿ, Q(t,·), Qⁿ(t+δ,·), Qⁿ_t(·)) and

bⁿ⁻¹(t) =b(t, Zⁿ⁻¹(t), Zⁿ⁻¹(t+δ), Z_tⁿ⁻¹, Qⁿ⁻¹(t,·), Qⁿ⁻¹(t+δ,·), Qⁿ⁻¹_t (·))

We take the expectation, use the weak monotonicity argument (a4) ofAand the Lipschitz 242

condition onb to get 243

Eh

|Yⁿ⁺¹(t)−Yⁿ(t)|²_Hi

+EhZ T t

kZⁿ⁺¹(s)−Zⁿ(s)k²_L2(K,H)dsi +EhZ T

t

Z

X

|Qⁿ⁺¹(s, ζ)−Qⁿ(s, ζ)|²_Hν(dζ)dsi

≤λE hZ T

t

|Yⁿ⁺¹(s)−Yⁿ(s)|²_Hds i

+ 1 εE

hZ T t

|Yⁿ⁺¹(s)−Yⁿ(s)|²_Hds i

+C_εEhZ T t

kZⁿ(s)−Zⁿ⁻¹(s)k²_L

2(K,H)dsi

+C_εEhZ T t

kZ₁ⁿ(s)−Z₁ⁿ⁻¹(s)k²_L

2(K,H)dsi +C_εEhZ T

t

kZ₂ⁿ(s)−Z₂ⁿ⁻¹(s)k²_L2(0,δ;L2(K,H))dsi

+C_εEhZ T t

Z

X

|Qⁿ(s, ζ)−Qⁿ⁻¹(s, ζ)|²_Hν(dζ)dsi +C_εEhZ T

t

Z

X

|Qⁿ₁(s, ζ)−Qⁿ⁻¹₁ (s, ζ)|²_Hν(dζ)dsi

+C_εEhZ T t

Z

X

|Qⁿ₂(s, ζ)−Qⁿ⁻¹₂ (s, ζ)|²_L2(0,δ;H)ν(dζ)dsi . Since Zⁿ(t) =Zⁿ⁻¹(t) = G₁(t) fort ∈[T, T +δ], we have from (3.3) and (3.4) that

244

EhZ T t

kZ₁ⁿ(s)−Z₁ⁿ⁻¹(s)k²_L2(K,H)dsi

≤EhZ T t

kZⁿ⁺¹(s)−Zⁿ(s)k²_L2(K,H)dsi , EhZ T

t

kZ₂ⁿ(s)−Z₂ⁿ⁻¹(s)k²_L2(0,δ;L2(K,H))dsi

≤δEhZ T t

kZⁿ⁺¹(s)−Zⁿ(s)k²_L2(K,H)dsi . The same inequalities also hold for Qⁿ−Qⁿ⁻¹. Choosing ε small enough (for e.g. such 245

that 2Cε+δCε≤ ¹₂) and using the preceding inequalities, we get 246

Eh

|Yⁿ⁺¹(t)−Yⁿ(t)|²_Hi

+EhZ T t

kZⁿ⁺¹(s)−Zⁿ(s)k²_L2(K,H)dsi +EhZ T

t

Z

X

|Qⁿ⁺¹(s, ζ)−Qⁿ(s, ζ)|²_Hν(dζ)dsi

≤λεE hZ T

t

|Yⁿ⁺¹(s)−Yⁿ(s)|²_Hds i

+1 2E

hZ T t

kZⁿ(s)−Zⁿ⁻¹(s)k²_L2(K,H)ds i

+ 1

2EhZ T t

Z

X

|Qⁿ(s, ζ)−Qⁿ⁻¹(s, ζ)|²_Hν(dζ)dsi

, (3.32)

where λ_ε =λ+ 1

ε. Hence 247