Backward Stochastic Variational Inequalities on Random Interval

arXiv:1112.5792v1 [math.PR] 25 Dec 2011

Backward Stochastic Variational Inequalities on Random Interval Lucian Maticiuc a,∗,1 , Aurel R˘a¸scanu a,b,2 a

b

Faculty of Mathematics, “Alexandru Ioan Cuza” University, Carol I Blvd., no. 9, Iasi, 700506, Romania “Octav Mayer” Mathematics Institute of the Romanian Academy, Iasi branch, Carol I Blvd., no. 8, Iasi, 700506, Romania

December 30, 2011

Abstract The aim of this paper is to study, in the infinite dimensional framework, the existence and uniqueness for the solution of the following multivalued generalized backward stochastic differential equation, considered on a random, possibly infinite, time interval: ( −dYt + ∂y Ψ (t, Yt ) dQt ∋ Φ (t, Yt , Zt ) dQt − Zt dWt , t ∈ [0, τ ] , Yτ = η,

where τ is a stopping time, Q is a progresivelly measurable increasing continuous stochastic process and ∂y Ψ is the subdifferential of the convex lower semicontinuous function y 7−→ Ψ (t, y). Our results generalize those of E. Pardoux and A. R˘ a¸scanu (Stochastics 67, 1999) to the case in which the function Φ satisfies a local boundeness condition (instead of sublinear growth condition with respect to y) and also the results from Ph. Briand et al. (Stochastic Process. Appl. 108, 2003) by considering the multivalued equation. As applications, we obtain from our main result applied for suitable convex functions, the existence for some backward stochastic partial differential equations with Dirichlet or Neumann boundary conditions.

AMS Classification subjects: 60H10, 93E03, 47J20, 49J40. Keywords or phrases: Backward stochastic differential equations; Subdifferential operators; Stochastic variational inequalities; Stochastic partial differential equations. ∗

Corresponding author. The work of this author is supported by POSDRU/89/1.5/S/49944 project. 2 The work of this author is supported by the IDEI Project, “Deterministic and stochastic systems with state constraints”, code PN-II-ID-PCE-2011-3-0843. E-mail addresses: [email protected] (Lucian Maticiuc), [email protected] (Aurel R˘ a¸scanu) 1

1

1

Introduction

In this paper we are interested in the following generalized backward stochastic variational inequality (BSVI for short) considered in the Hilbert space framework:  Z τ Z τ Z τ   Yt + Zs dWs , a.s., [F (s, Ys , Zs ) ds + G (s, Ys ) dAs ] − dKs = η + (1) t∧τ t∧τ t∧τ   dK ∈ ∂ϕ (Y ) dt + ∂ψ (Y ) dA , ∀ t ≥ 0, t t t t

where {Wt : t ≥ 0} is a cylindrical Wiener process, ∂ϕ, ∂ψ are the subdifferentials of a convex lower semicontinuous functions ϕ, ψ, {At : t ≥ 0} is a progressively measurable increasing continuous stochastic process, and τ is a stopping time. In fact we will define and prove the existence of the solution for an equivalent form of (1):  Z ∞ Z ∞ Z ∞   Yt + Zs dWs , a.s., Φ (s, Ys , Zs ) dQs − dKs = η + (2) t t t   dK ∈ ∂ Ψ (t, Y ) dQ , ∀ t ≥ 0, t y t t

with Q, Φ and Ψ adequately defined. The study of the backward stochastic differential equations (BSDEs for short) in the finite dimensional case (equation of type (1) with A and ϕ equal to 0) was initiated by E. Pardoux and S. Peng in [13] (see also [14]). The authors have proved the existence and the uniqueness of the solution for the BSDE on fixed time interval, under the assumption of Lipschitz continuity of F with respect to y and z and square integrability of η and F (t, 0, 0). The case of BSDEs on random time interval (possibly infinite), under weaker assumptions on the data, have been treated by R.W.R. Darling and E. Pardoux in [6], where it is obtained, as application, the existence of a continuous viscosity solution to the elliptic partial differential equations (PDEs) with Dirichlet boundary conditions. The existence and uniqueness results for BSDEs with infinite time horizon, considered in the Hilbert spaces, were studied in [8] by M. Fuhrman and G. Tessitore, where F is still suppose to be Lipschitz continuous with respect to y and z. The more general case of scalar BSDEs with one-sided reflection and associated optimal control problems was considered by N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, M.C. Quenez in [7] and with two-sided reflection associated with stochastic game problem by J. Cvitanic and I. Karatzas [4]. When the obstacles are fixed, the reflected BSDE become a particular case of BSVI of type (1), by taking Ψ as convex indicator of the interval defined by obstacles. We must mention that the solution of a BSVI belongs to the domain of the operator ∂Ψ and it is reflected at the boundary of this. The standard work on BSVI in the finite dimensional case is that of E. Pardoux and A. R˘ a¸scanu [15], where it is proved the existence and uniqueness of the solution (Y, Z, K) for the BSVI (1) with A ≡ 0, under the following assumptions on F : monotonicity with respect to y (in the sense that hy ′ − y, F (t, y ′ , z) − F (t, y, z)i ≤ α|y ′ − y|2 ), lipschitzianity with respect to z and a sublinear growth for F (t, y, 0). Moreover, it is shown that, unlike the forward case, the process K is absolute continuous with respect to dt. In [16] the same authors extend these results to the Hilbert spaces framework. 2

Our paper generalizes the existence and uniqueness results from [16] by considering random time interval [0, τ ] and the Lebesgue-Stieltsjes integral terms, and by assuming a weaker boundedness condition for the generator Φ (instead of the sublinear growth), i.e. E

Z

T 0

Φ# ρ (s)ds

p

< ∞, where Φ# ρ (t) := sup|y|≤ρ |Φ(t, y, 0)| .

(3)

We mention that, since τ is a stopping time, the presence of the process A is justified by the possible applications of equation (1) in proving probabilistic interpretation for the solution of elliptic multivalued partial differential equations with Neumann boundary conditions on a domain from Rd . The stochastic approach of the existence problem for finite dimensional multivalued parabolic PDEs, was considered by L. Maticiuc and A. R˘ a¸scanu in [10]. On the other hand, infinite dimensional generalization for parabolic PDEs (namely nonlinear Kolmogorov equations) was treated by M. Fuhrman and G. Tessitore in [9] and the corresponding nonlinear elliptic equations, i.e. nonlinear stationary Kolmogorov equations, in [8]. The next step and the subject of a forthcoming paper is the study of the infinite dimensional (parabolic and elliptic) Kolmogorov equations with state constraints. Concerning the assumption (3), we recall that, in the case of finite dimensional BSDE, E. Pardoux in [12] has used a similar condition, in order to prove the existence of a solution in L2 . His result was generalized by Ph. Briand, B. Deylon, Y. Hu, E. Pardoux, L. Stoica in [3], where it is proved the existence in Lp of the solution for BSDEs considered both with fixed and random terminal time. We mention that the assumptions from our paper are, broadly speaking, similar to those of [3]. The article is organized as follows: in the next Section a brief summary of infinite dimensional stochastic integral and the assumptions are given. Section 3 is devoted to the proof of the existence and uniqueness of a strong solution for (2). In the fourth section it is introduced a new type of solution (called variational weak solution) and it is also proved the existence and uniqueness result. In Section 4 are obtained, as applications, the existence of the solution for various type of backward stochastic partial differential equations with boundary conditions. The Appendix contains, following [17], some results useful throughout the paper.

2 2.1

Preliminaries Infinite dimensional framework

In the beginning of this subsection we give a brief exposition of the stochastic integral with respect to a Wiener process defined on a Hilbert space. For a deeper discussion concerning the notion of cylindrical Wiener process and the construction of the stochastic integral we refer reader to [5]. We consider a complete probability space (Ω, F, P), the set NP = {A ∈ F : P (A) = 0}, a right continuous and complete filtration {Ft }t≥0 , and two real separable Hilbert spaces H, H1 . p Let us denote by SH [0, T ], p ≥ 0, the complete metric space of continuous progressively

3

measurable stochastic process (p.m.s.p.) X : Ω × [0, T ] → H with the metric given by   1∧1/p  ˜ t |p  < ∞, if p > 0, E sup |X − X  t  t∈[0,T ] ˜ ρp (X, X) =     ˜ t | < ∞, if p = 0,  E 1 ∧ sup |Xt − X  t∈[0,T ]

p the space of p.m.s.p. X : Ω × [0, ∞) → H such that, for all T > 0, the restriction and by SH p p . [0, T ]. To shorten notation, we continue to write S p for SH X|[0,T ] ∈ SH p Remark that SH [0, T ] is a Banach space for p ≥ 1. By Mp (Ω × [0, T ] ; H), p ≥ 1, we denote the Banach space of the continuous stochastic processes M such that E (|M (t)|p ) < ∞, ∀t ∈ [0, T ], M (0) = 0 a.s., and EFs (Mt ) = Ms , a.s. for all 0 ≤ s ≤ t ≤ T . The norm is defined by kM kMp = [E (|M (T )|p )]1/p . If p > 1, then p Mp (Ω × [0, T ] ; H) is a closed linear subspace of SH [0, T ]. 0 Let W = {Wt (a) : t ≥ 0, a ∈ H1 } ⊂ L (Ω, F, P) be a Gaussian family of real-valued random variables with zero mean and the covariance function given by

E [Wt (a)Ws (b)] = (t ∧ s) ha, biH1 ,

s, t ≥ 0, a, b ∈ H1 .

We call W a H1 -Wiener process if, for all t ≥ 0, (i)

FtW := σ{Ws (a) : s ∈ [0, t], a ∈ H1 } ∨ NP ⊂ Ft ,

(ii) Wt+h (a) − Wt (a) is independent of Ft , for all h > 0, a ∈ H1 . Let {gi }i∈N ∗ be an orthonormal and complete basis in H1 . We introduce the separable Hilbert space L2 (H1 ; H) of Hilbert–Schmidt operators from H1 to H, i.e. the space of linear operators Z : H1 → H such that |Z|2L2 (H1 ;H) =

∞ X i=1

|Zei |2H = Tr (Z ∗ Z) < ∞.

It will cause no confusion  if we use |Z| to designate the norm in L2 (H1 ; H). The sequence W i = Wti := Wt (gi ) : t ∈ [0, T ]}, i ∈ N∗ defines is a family of real-valued Wiener processes mutually independent on (Ω, F, P). If H1 is finite dimensional space then we have the representation X gi Wti , t ≥ 0, Wt = i

but, in general case, this series does not converge in H1 , but rather in a larger space H2 such that H1 ⊂ H2 with the injection J : H1 →H2 being a Hilbert-Schmidt operator. Moreover, W ∈ M2 (Ω × [0, T ] ; H2 ). For 0 < T ≤ ∞, we will denote by ΛpL2 (H1 ,H) (0, T ), p ≥ 0, the space Lpad (Ω×(0, T ) ; L2 (H1 , H)), i.e. the complete metric space of progressively measurable stochastic processes Z : Ω × (0, T ) → L2 (H1 , H) with metric of convergence   Z T
p/2 1∧1/p  2  ˜  < ∞, if p > 0, |Z − Z | ds E s s  0 ˜ dp (Z, Z) = Z T  
1/2     E 1∧ < ∞, if p = 0. |Zs − Z˜s |2 ds 0

4

The space ΛpL2 (H1 ,H) (0, T ) is a Banach space for p ≥ 1 with norm kZkΛp = dp (Z, 0). From now on, for simplicity of notation, we write Λp (0, T ) instead of ΛpL2 (H1 ,H) (0, T ) (when no confusion can arise). Let us denote by Λp the space of measurable stochastic processes X : Ω × [0, ∞) → H such that, for all T > 0, the restriction X|[0,T ] ∈ Λp (0, T ). For any Z ∈ Λ2 let the stochastic integral, I (Z) (t) =

Z

t

Zs dWs := 0

∞ Z X i=1

t

0

Zs (gi ) dWs (gi ) , t ∈ [0, T ] ,

where {gi }i is an orthonormal basis in H1 . Note that the introduced stochastic integral doesn’t depend on the choice of the orthonormal basis on H1 . By the standard localization procedure we can extend this integral as a linear continuous operator I : Λp (0, T ) → S p [0, T ] , p ≥ 0, and it has the following properties: Proposition 1 Let Z ∈ Λp (0, T ). Then (i) E I (Z) (t) = 0, ∀t ∈ [0, T ] , if p ≥ 1,

(ii) E |I (Z) (T )|2 = kZk2Λ2 , if p ≥ 2, (iii)

1 kZkpΛp ≤ E supt∈[0,T ] |I (Z) (t)|p ≤ cp kZkpΛp , if p > 0, cp (Burkholder-Davis-Gundy inequality),

(iv) I (Z) ∈ Mp (Ω × [0, T ] ; H) , if p ≥ 1. From now on we shall consider that the original filtration {Ft }t≥0 is replaced by the filtration {FtW }t≥0 generated by the Wiener process. The following Hilbert space version of the martingale representation theorem, extended to a random interval, holds: Proposition 2 Let τ : Ω → [0, ∞] be a stopping time, p > 1 and η : Ω → H be a Fτ measurable random variable such that E |η|p < ∞. Then 1. there exists a unique stochastic process ζ ∈ Λp (0, ∞) such that ζt = 1[0,τ ] (t) ζt , ∀t ≥ 0 and Z τ ζs dWs , η = Eη+ 0

or equivalently, 2. there exists a unique pair (ξ, ζ) ∈ S p × Λp (0, ∞) such that Z τ ζs dWs , t ≥ 0, a.s. ξt = η − t∧τ

5

2.2

Assumptions

Let us consider the following BSVI:  Z Z τ Z τ   Yt + [F (s, Ys , Zs ) ds + G (s, Ys ) dAs ] − dKs = η +

Zs dWs , a.s.,

t∧τ

t∧τ

t∧τ

τ

(4)

  dK ∈ ∂ϕ (Y ) dt + ∂ψ (Y ) dA , ∀ t ≥ 0. t t t t

The next assumptions will be needed throughout the paper: (A1 ) The random variable τ : Ω → [0, ∞] is a stopping time; (A2 ) The random variable η : Ω → H is Fτ -measurable such that E |η|p < ∞ and the stochastic process (ξ, ζ) ∈ S p × Λp (0, ∞) is the unique pair associated to η such that we have the martingale representation formula (see Proposition 2)  Z ∞   ξt = η − ζs dWs , t ≥ 0, a.s., (5) t   ξ = EFt η = EFt∧τ η and ζ = 1 t t [0,τ ] (t) ζt ;

(A3 ) The process {At : t ≥ 0} is a progressively measurable increasing continuous stochastic process such that A0 = 0, Qt (ω) = t + At (ω) ,

and {αt : t ≥ 0} is a real positive p.m.s.p. (given by Radon-Nikodym’s representation theorem) such that α ∈ [0, 1] and dt = αt dQt

and

dAt = (1 − αt ) dQt ;

(A4 ) The functions F : Ω × [0, ∞) × H × L2 (H1 , H) → H and G : Ω × [0, ∞) × H → H are such that ( F (·, ·, y, z) , G (·, ·, y) are p.m.s.p., for all (y, z) ∈ H × L2 (H1 , H) , F (ω, t, ·, ·) , G (ω, t, ·) are continuous functions a.e. ,

and P-a.s., Z

where

T 0

Fρ#

(s) ds +

Z

T 0

G# ρ (s) dAs < ∞,

Fρ# (ω, s) := sup|y|≤ρ |F (ω, s, y, 0)| ,

∀ ρ, T ≥ 0,

G# ρ (ω, s) := sup|y|≤ρ |G (ω, s, y)| ;

(A5 ) Assume that there exist two p.m.s.p. µ, ν : Ω × [0, ∞) → R such that Z

0

T

|µt |2 dt < ∞ and

Z

0

T

|νt |2 dAt < ∞, for all T > 0, P-a.s.,

6

(6)

(7)

and there exists ℓ ≥ 0, such that, for all y, y ′ ∈ H, z, z ′ ∈ L2 (H1 , H) , hy ′ − y, F (t, y ′ , z) − F (t, y, z)i ≤ 1[0,τ ] (t) µt |y ′ − y|2 , hy ′ − y, G(t, y ′ ) − G(t, y)i ≤ 1[0,τ ] (t) νt |y ′ − y|2 ,

(8)

|F (t, y, z ′ ) − F (t, y, z)| ≤ 1[0,τ ] (t) ℓ |z ′ − z| ;

Let us introduce the function Φ (ω, t, y, z) := 1[0,τ (ω)] (t) [αt (ω) F (ω, t, y, z) + (1 − αt (ω)) G (ω, t, y)] ,

(9)

in which case (8) become hy ′ − y, Φ(t, y ′ , z) − Φ(t, y, z)i ≤ 1[0,τ ] (t) [µt αt + νt (1 − αt )] |y ′ − y|2 ,

|Φ(t, y, z ′ ) − Φ(t, y, z)| ≤ 1[0,τ ] (t) ℓαt |z ′ − z| .

From now on, p ≥ 2 and, for a > 1, let Z t   a  1[0,τ ] (s) µs + ℓ2 αs + νs (1 − αs ) dQs Vt = 2 0 Z t   a 2
1[0,τ ] (s) µs + ℓ ds + νs dAs . = 2 0

(10)

We can give now some a priori estimates concerning the solution of (4).

˜ ∈ S 0 [0, T ] × Λ0 (0, T ). Under the assumptions (A4 -A5 ) the Lemma 3 Let (Y, Z) , (Y˜ , Z) following inequalities hold, in the sense of signed measures on [0, ∞), hYs , Φ (s, Ys , Zs ) dQs i ≤ |Ys | |Φ (s, 0, 0)| dQs + |Ys |2 dVs + and hYs − Y˜s , Φ (s, Ys , Zs ) − Φ(s, Y˜s , Z˜s )idQs ≤ |Ys − Y˜s |2 dVs +

1 |Zs |2 ds 2a

1 |Zs − Z˜s |2 ds. 2a

(11)

(12)

Proof. The inequalities can be obtained by standard calculus (applying the monotonicity and Lipschitz property of function Φ). (A6 ) ϕ, ψ : H → [0, +∞] are proper convex lower semicontinuous (l.s.c.) functions such that ϕ (0) = ψ (0) = 0 (consequently 0 ∈ ∂ϕ (0) ∩ ∂ψ (0)) and Ψ (ω, t, y) = 1[0,τ (ω)] (t) [αt (ω) ϕ (y) + (1 − αt (ω)) ψ (y)] ; We recall now that the multivalued subdifferential operator ∂ϕ is the maximal monotone operator ∂ϕ (y) := {ˆ y ∈ H : hˆ y , v − yi + ϕ (y) ≤ ϕ (v) , ∀ v ∈ H} . We define

Dom (ϕ) = {y ∈ H : ϕ (y) < ∞} , Dom (∂ϕ) = {y ∈ H : ∂ϕ (y) 6= ∅} ⊂ Dom (ϕ) , 7

and by (y, yˆ) ∈ ∂ϕ we understand that y ∈Dom(∂ϕ) and yˆ ∈ ∂ϕ (y). Recall that Dom (ϕ) = Dom (∂ϕ), int (Dom (ϕ)) = int (Dom (∂ϕ)) . If K is a H-valued bounded variation stochastic process, RA is a real increasing stochastic T process and Y is a H-valued stochastic process such that 0 ϕ (Yt ) dAt < ∞, a.s. ∀ T ≥ 0, then the notation dKt ∈ ∂ϕ (Yt ) dAt , P-a.e. means that Z s Z s Z s ϕ (y (r)) dAr , ∀y ∈ C ([0, T ] ; H) , ∀ 0 ≤ t ≤ s. ϕ (Yr ) dAr ≤ hy (r) − Yr , dKr i + t

t

t

(13)

Let ε > 0 and the Moreau-Yosida regularization of ϕ :   1 2 |y − v| + ϕ (v) : v ∈ H , ϕε (y) := inf 2ε

(14)

which is a C 1 convex function. We mention some properties (see H. Br´ezis [2], and E. Pardoux, A. R˘ a¸scanu [15] for the last one): for all x, y ∈ H ε |∇ϕε (x)|2 + ϕ (x − ε∇ϕε (x)) , 2 (b) ∇ϕε (x) = ∂ϕε (x) ∈ ∂ϕ (x − ε∇ϕε (x)) ,

(a) ϕε (x) =

1 |x − y| , ε (d) h∇ϕε (x) − ∇ϕε (y), x − yi ≥ 0, (c) |∇ϕε (x) − ∇ϕε (y)| ≤

(15)

(e) h∇ϕε (x) − ∇ϕδ (y), x − yi ≥ −(ε + δ) h∇ϕε (x), ∇ϕδ (y)i . We introduce the compatibility conditions between ϕ, ψ and F, G (which have previously been used in [10]): (A7 ) For all ε > 0, t ≥ 0, y ∈ H, z ∈ L2 (H1 , H) (i) h∇ϕε (y) , ∇ψε (y)i ≥ 0, 

(ii) ∇ϕε (y) , G (t, y) + νt− y ≤ |∇ψε (y)| G (t, y) + νt− y , P-a.s., 

− (iii) ∇ψε (y) , F (t, y, z) + µ− t y ≤ |∇ϕε (y)| F (t, y, z) + µt y , P-a.s.,

where µ− = − min {µ, 0} and ν − = − min {ν, 0}. Example 4 Let H = R.

A. Clearly, since ∇ϕε and ∇ψε are increasing monotone, we see that, if

y G (t, y) + νt− y ≤ 0 and y F (t, y, z) + µ− t y ≤ 0, ∀ t, y, z, then compatibility assumptions (16) are satisfied. 8

(16)

B. If ϕ, ψ : R → (−∞, +∞] are the convex indicator functions ( ( 0, if y ∈ [a, b] , 0, if y ∈ [c, d] , ϕ (y) = and ψ (y) = +∞, if y ∈ / [a, b] , +∞, if y ∈ / [c, d] , where a, b, c, d ∈ R are such that 0 ∈ [a, b] ∩ [c, d] (see the assumption (A6 )),

then

∇ϕε (y) =

 1 (y − b)+ − (a − y)+ ε

and

∇ψε (y) =

 1 (y − d)+ − (c − y)+ . ε

Since (A7 − i) is fulfilled, the compatibility assumptions become

G (t, y) + νt− y ≥ 0, for y ≤ a, G (t, y) + νt− y ≤ 0, for y ≥ b,

and, respectively,


F (t, y, z) + µ− t y ≥ 0, for y ≤ c,

The last assumption is the following:


F (t, y, z) + µ− t y ≤ 0, for y ≥ d.

 (A8 ) There exist the p.m.s.p. µ ˜, ν˜ : Ω × [0, ∞) → R with µ ˜ ≥ max µ, 12 µ and ν˜ ≥ 
RT max ν, 12 ν , such that 0 |˜ µt |2 dt + |˜ νt |2 dAt < ∞, ∀ T > 0, P-a.s. and, using the notation Z t h i a
1[0,τ ] (s) µ ˜s + ℓ2 ds + ν˜s dAs , V˜t = (17) 2 0 we suppose the following compatibility conditions between η and ϕ, ψ, F and G :   ˜ (i) E e2 sups∈[0,τ ] Vs (ϕ(η) + ψ(η)) < ∞,

˜ (ii) E ep sups∈[0,τ ] Vs |η|p + E QpT < ∞, ∀ T > 0, Z τ Z τ p/2 p ˜s ˜s 2 V V e Ψ (s, ξs ) dQs e |Φ (s, ξs , ζs )| dQs (iii) E +E < ∞, 0

(18)

0

and the locally boundedness conditions: (iv) E

Z

0

T

p
˜ ˜ ˜s y ds eVs sup|y|≤ρ F s, e−Vs y, 0 − µ

p T ˜
˜ +E eVs sup|y|≤ρ G s, e−Vs y − ν˜s y dAs < ∞, ∀ T, ρ > 0, 0 Z τ
2 ˜ ˜ (v) E e2Vs sup|y|≤ρ F s, e−Vs y, 0 ds 0 Z τ ˜
2 ˜ +E e2Vs sup|y|≤ρ G s, e−Vs y dAs < ∞, ∀ ρ > 0. Z

0

9

(19)

3

Main result: the existence of the strong solution

Using the definition of Q, Φ and Ψ (given in the previous assumptions) we can rewrite (4) in the form  Z ∞ Z ∞ Z ∞   Yt + dKs = η + Φ (s, Ys , Zs ) dQs − Zs dWs , a.s., (20) t t t   dK ∈ ∂ Ψ (t, Y ) dQ , ∀ t ≥ 0. t y t t

Definition 5 We call (Yt , Zt , Ut )t≥0 a solution of (20) if (Y, Z) ∈ S 0 ×Λ0 , (Yt , Zt ) = (ξt , ζt ) = (η, 0) for t > τ and   Ut = 1[0,τ ] (t) αt Ut1 + (1 − αt ) Ut2 , where U 1 and U 2 are p.m.s.p.,

such that

(i) (ii)

Z

T

0 Ut1

(|Φ (s, Ys , Zs )| + |Us |) dQs < ∞, P-a.s., for all T ≥ 0, ∈ ∂ϕ (Yt ) , dP ⊗ dt- a.e. ,

(iii) Ut2 ∈ ∂ψ (Yt ) , dP ⊗ dAt - a.e., Z ∞ prob. 2 2V T e2Vs |Zs − ζs |2 ds −−−−→ 0, as T → ∞, |YT − ξT | + (iv) e

(21)

T

and, for t ∈ [0, T ], (v) Yt +

Z

T

Us dQs = YT + t

Z

T t

Φ (s, Ys , Zs ) dQs −

Z

T

Zs dWs , a.s.

(22)

t

Remark 6 If there exists a constant C such that supt∈[0,τ ] |Vt (ω)| ≤ C, P − a.s. ω ∈ Ω, then the condition (21-iv) is equivalent to Z ∞ prob. 2 |Zs |2 ds −−−−→ 0, as T → ∞. (23) |YT − η| + T

We can now formulate the main result. In order to obtain the absolute continuity with respect to dQt of the process K (as in Definition 5) it is necessary to impose a supplementary assumption: (A9 ) There exists R0 > 0 such that, for all t ≥ 0, 2 Z τ ˜ 2 Ft 2 sups≥t V˜s Ft E (e |η| ) + E eVs |Φ (s, 0, 0)| dQs ≤ R0 , a.s.

(24)

t∧τ

Theorem 7 Let the assumptions (A1 −A9 ) be satisfied. Then the backward stochastic variational inequality (20) has a unique solution (Y, Z, U ) such that ˜

E supt∈[0,T ] epVt |Yt |p < ∞, for all T ≥ 0, 10

(25)

and Yt +

Z

T

Us dQs = YT + t

Z

T

Φ (s, Ys , Zs ) dQs −

t

Z

T

Zs dWs , a.s.

(26)

t

Moreover, for all 2 ≤ q ≤ p, there exists a constant C = C (a, q) > 0 such that, for all t ≥ 0, P-a.s. Z T q/2 ˜ ˜t q F q V t e2Vs |Zs |2 ds (a) e |Yt | + E t q i Z ∞ h ˜ ˜ , eVs |Φ (s, 0, 0) |dQs ≤ C EFt eq sups≥t Vs |η|q + t

(b)

˜ eqVt

q

|Yt − ξt | +

≤ C EFt

hZ

EFt

∞

Z

∞

˜ e2Vs

t

2

|Zs − ζs | ds

˜

e2Vs Ψ (s, ξs ) dQs

t

q/2

+

q/2

Z

∞

t

(27) ˜

eVs |Φ (s, ξs , ζs )| dQs

   ˜  ˜ E e2Vt (ϕ(Yt ) + ψ(Yt )) ≤ E e2V∞ (ϕ(η) + ψ(η)) , Z ∞ h p/2 i ˜T ˜ p p V |YT − ξT | + (d) lim E e = 0, e2Vs |Zs − ζs |2 ds (c)

T →∞

q i

,

T

and (e) E

Z

τ

h

0

˜

e2Vs |Us1 |2 ds + |Us2 |2 dAs


i

< ∞.

(28)

¯ are two solutions, in the sense of Definition 5, that satisfy (25), then Proof. If (Y, Z), (Y¯ , Z) ˜ E supt∈[0,T ] epVs |Ys − Y¯s |p < ∞,

From (12), satisfied by the process Ys − Y¯s , we conclude that ¯s idQs ≤ |Ys − Y¯s |2 dV˜s + 1 |Zs − Z˜s |2 ds, hYs − Y¯s , Φ (s, Ys , Zs ) − Φ(s, Y¯s , Z¯s ) − Us + U 2a since

and

 ¯s ∈ ∂y Ψ(s, Y¯s ), ¯s ≥ 0, for Us ∈ ∂y Ψ(s, Ys ) and U Ys − Y¯s , Us − U dVs ≤ dV˜s on [0, τ ] .

Applying Proposition 16, it follows that there exists C = C (a, p) > 0 such that pV˜s

E sups∈[0,T ] e

|Ys − Y¯s | + E p

Z

0

T

2V˜s

e

|Zs − Z¯s | ds 2

p/2

  ˜ ≤ C E epVT |YT − Y¯T |p −−−−→ 0,

and the uniqueness is proved. The proof of the existence will be split into several steps. A. Approximating problem. 11

T →∞

Let n ∈ N∗ and ε = 1/n. We consider the approximating stochastic equation Z ∞ Z ∞ n n n 1[0,n] (s) Φ(s, Ysn , Zsn )dQs 1[0,n] (s) ∇y Ψ (s, Ys )dQs = η + Yt + t t Z ∞ Zsn dWs , P-a.s., ∀ t ≥ 0, −

(29)

t

or equivalent, P-a.s.,  Z n Z n Z n  n , Z n )dQ − n (s, Y n )dQ = EFn η n+  Zsn dWs , Φ(s, Y ˜ + ∇ Ψ Y  s s y s s s   t t t t ∀ t ∈ [0, n] ,

     (Y n , Z n ) = (ξ , ζ ), ∀ t > n, t t t t

with

(30)

  Ψn (ω, s, y) = 1[0,τ (ω)] (s) αs (ω) ϕ1/n (y) + (1 − αs (ω)) ψ1/n (y) .

We notice that

Φ1 (t, y, z) := 1[0,n] (t) Φ (t, y, z) − ∇y Ψn (t, y)

satisfies the inequalities




hy ′ − y, Φ1 (t, y ′ , z) − Φ1 (t, y, z)i ≤ 1[0,n∧τ ] (t) [(µt − n) αt + (νt − n) (1 − αt )] |y ′ − y|2 ≤ 1[0,n∧τ ] (t) [˜ µt αt + ν˜t (1 − αt )] |y ′ − y|2 ,

|Φ1 (t, y, z ′ ) − Φ1 (t, y, z)| ≤ 1[0,n∧τ ] (t) ℓαt |z ′ − z| ,

since µs ≤ µ ˜s and νs ≤ ν˜s on [0, ∞). The corresponding process Vt1 (see definitions (10) and (17)) is given by Z t h i a
1[0,n∧τ ] (s) µ Vt1 = ˜s + ℓ2 ds + ν˜s dAs . 2 0

Obviously, Vt1 = V˜t∧n , ∀t ≥ 0. Applying Proposition 18 from the Appendix, with Φ replaced with Φ1 , we deduce that equation (30) has a unique solution (Y n , Z n ) such that p/2 Z n n 2 2V˜s n p pV˜t < ∞, e |Zs | ds E supt∈[0,n] e |Yt | + E 0

and, using (77), can be prove that pV˜t

E supt∈[0,T ] e

|Ytn |p

B. Boundedness of Y n and Z n .

+E

Z

T

2V˜s

e

0

|Zsn |2 ds

p/2

< ∞, for all T ≥ 0.

Since ϕn , ψ n are convex functions and it is assumed that ϕ (0) = ψ (0) = 0, we see that h∇y Ψn (t, y) , yi ≥ 0, ∀y ∈ H, and therefore (11) becomes

 1 Ytn , Φ1 (t, Ytn , Ztn )dQt ≤ 1[0,n] (t) |Ytn ||Φ (t, 0, 0) |dQt + |Ytn |2 dVt1 + |Ztn |2 dt. 2a 12

Equation (29) can be written, for any T ≥ 0, in the form Ytn = YTn +

T

Z

t

Z

Φ1 (s, Ysn , Zsn )dQs −

T t

Zsn dWs , P-a.s., ∀ t ∈ [0, T ] .

Applying Proposition 16 we deduce that, for all q ∈ [2, p], there exists a constant C = C (a, q) > 0 such that such that, P-a.s., for all 0 ≤ t ≤ T ≤ n, EFt

˜ sups∈[t,T ] eqVs |Ysn |q

≤C ≤C ≤C

EFt EFt EFt







+

EFt

Z

T

˜

e2Vs |Zsn |2 ds

t

˜ eqVT |YTn |q

+

q sups∈[t,T ] V˜s

e

q sups∈[t,T ] V˜s

e

Z |ξT

T

1[0,n]

t

|q

|η|q

+

+

˜ (s) eVs |Φ (s, 0, 0) |dQ

Z

Z

q/2

∞

1[0,n]

t ∞

1[0,n]

t

s

q 

˜ (s) eVs |Φ (s, 0, 0) |dQ

˜ (s) eVs |Φ (s, 0, 0) |dQ

s

s

q 

q 

,

since by Jensen’s inequality we have |ξT |q = |EFT η|q ≤ EFT |η|q . Using (77), it can be proved that the above inequality holds also for all 0 ≤ t ∨ n ≤ T . Passing to limit as T → ∞ we infer, using Beppo Levi’s Theorem, that P-a.s. EFt

˜ sups≥t eqVs |Ysn |q

+

EFt

Z

t

h

∞

˜ e2Vs |Zsn |2 ds

q sups∈[t,τ ] V˜s

≤ C EFt (e

|η|q ) +

q/2

Z

∞

t

1[0,n]

˜ (s) eVs |Φ (s, 0, 0) |dQ

s

q i

(31) .

In particular, for q = 2, there exists another constant C ≥ 1 such that, for all t ≥ 0, Z ∞ h 2 i ˜ ˜t 2 sups∈[t,τ ] V˜s 2 n 2 F 2 V t |η| ) + 1[0,n] (s) eVs |Φ (s, 0, 0) |dQs e |Yt | ≤ C E (e t

≤

2CR02

=

R0′ ,

(32)

P-a.s.,

where R0 is given by (24). C. Other boundedness results on Y n and Z n . Since for all u ∈ H, hu − y, ∇y Ψn (t, y)i ≤ Ψn (t, u) − Ψn (t, y) , we can deduce (see inequality (12)) that, as signed measures on [0, n] , 

n Yt − ξt , Φ1 (s, Ysn , Zsn ) dQt ≤ [Ψn (t, ξt ) − Ψn (t, Ytn )] dQt +|Ytn − ξt | |Φ(t, ξt , ζt )|dQt + |Ytn |2 dVt1 +

1 n |Z − ζt |2 dt. 2a t

But 0 ≤ Ψn (t, ξt ) ≤ Ψ(t, ξt ) = 1[0,τ (ω)] (t) [αt (ω) ϕ(ξt ) + (1 − αt (ω)) ψ(ξt )] , 13

(33)

therefore (33) becomes

 Ψn (t, Ytn )dQt + Ytn − ξt , Φ1 (s, Ysn , Zsn ) dQt

≤ Ψ(t, ξt )dQt + |Ytn − ξt | |Φ(t, ξt , ζt )|dQt + |Ytn |2 dVt1 +

1 n |Z − ζt |2 dt. 2a t

From (30) we see that (Y n , Z n ) satisfies the equation Z n Z n n n n (Zsn − ζs )dWs , ∀ t ∈ [0, n] , Φ1 (s, Ys , Zs )dQs − Yt − ξt = t

t

Rn

since ξt = ξn − t ζs dWs , ∀ t ∈ [0, n] . Applying now Proposition 16, there exists a constant C = C (a, p) > 0 such that, P-a.s., for all t ∈ [0, n] , p/2 p/2 Z n Z n ˜ ˜s ˜s 2 n 2 V p F n p V F t t e2Vs Ψn (s, Ysn )dQs +E e |Zs − ζs | ds E sups∈[t,n] e |Ys − ξs | + E t t  Z n p/2 Z n p  ˜s ˜s 2 V V F t + e Ψ(s, ξs )dQs e |Φ(s, ξs , ζs )|dQs ≤CE . t

t

Therefore

Z ∞ p/2 ˜ ˜ e2Vs |Zsn − ζs |2 ds EFt sups≥t epVs |Ysn − ξs |p + EFt t Z ∞ p/2 Z ∞ p  ˜s ˜s 2 V V F t e Ψ(s, ξs )dQs + e |Φ(s, ξs , ζs )|dQs . ≤CE t

since

(Ysn , Zsn )

(34)

t

= (ξs , ζs ) for s > n.

D. Boundedness of ∇ϕ1/n (Ytn ) and ∇ψ1/n (Ytn ). In order to obtain the boundedness for |∇ϕ1/n (Ysn )|2 it is essential to use the following stochastic subdifferential inequality (see Proposition 11 in [10]), written first for ϕ1/n : Z s Z s ˜ ˜r ˜t ˜s 2 V n n 2 V n 2 V e2Vr ∇ϕ1/n (Yrn )dYrn , ϕ1/n (Yr )d(e ) + e ϕ1/n (Ys ) ≥ e ϕ1/n (Yt ) + t

t

∀ 0 ≤ t ≤ s ≤ n.

Hence 2V˜s

e

ϕ1/n (Ysn )

2V˜t

≥e

ϕ1/n (Ytn ) +

Z s Z s ˜ n 2V˜r ˜ e2Vr ∇ϕ1/n (Yrn )dYrn . 2 e ϕ1/n (Yr )dVr + t

t

It follows that, P-a.s. for all 0 ≤ t ≤ s ≤ n, Z s  ˜

˜t n 2 V e2Vr ∇ϕ1/n (Yrn ), 1[0,n] (r) ∇y Ψn (r, Yrn ) dQr e ϕ1/n (Yt ) + t Z s  ˜

˜s n 2 V e2Vr ∇ϕ1/n (Yrn ), 1[0,n] (r) Φ(r, Yrn , Zrn )dQr ≤ e ϕ1/n (Ys ) + t Z s Z s

 ˜ ˜ − e2Vr ∇ϕ1/n (Y n ), Z n dWr − 2 e2Vr ϕ1/n (Y n )dV˜r t

r

r

t

14

r

(and similar inequality for ψ1/n ). Since ϕ1/n (0) + ψ1/n (0) = 0 ≤ ϕ1/n (y) + ψ1/n (y) ≤ ϕ(y) + ψ(y), ∀y ∈ H,

(35)

we infer that Z

s

h ˜ 1[0,n∧τ ] (r) e2Vr αr |∇ϕ1/n (Yrn )|2 t i 

+ ∇ϕ1/n (Yrn ), ∇ψ1/n (Yrn ) + (1 − αr ) |∇ψ1/n (Yrn )|2 dQr Z s  ˜

˜s n n 2 V e2Vr ∇ϕ1/n (Yrn ) + ∇ψ1/n (Yrn ), 1[0,n] (r) Φ(r, Yrn , Zrn ) dQr ≤ e [ϕ(Ys ) + ψ(Ys )] + t Z s Z s   ˜  ˜r

n n n 2 V ∇ϕ1/n (Yr ) + ∇ψ1/n (Yr ), Zr dWr − 2 e2Vr ϕ1/n (Yrn ) + ψ1/n (Yrn ) dV˜r . − e

˜ e2Vt



ϕ1/n (Ytn )

+

ψ1/n (Ytn )



+

t

t

(36)

Using definition (9), the compatibility assumptions (16) gives us h∇ϕε (y), Φ(t, y, z)i = 1[0,τ ] (t) h∇ϕε (y), αt F (t, y, z) + (1 − αt ) G(t, y)i

and respectively

(37)

≤ 1[0,τ ] (t) αt |F (t, y, z)||∇ϕε (y)| + (1 − αt ) |G(t, y)||∇ψε (y)|
+ (1 − αt ) νt− |y||∇ψε (y)| − (1 − αt ) νt− h∇ϕε (y), yi

h∇ψε (y), Φ(t, y, z)i = 1[0,τ ] (t) h∇ψε (y), αt F (t, y, z) + (1 − αt ) G(t, y)i

(38)

≤ 1[0,τ ] (t) αt |F (t, y, z)||∇ϕε (y)| + (1 − αt ) |G(t, y)||∇ψε (y)|
− + αt µ− t |y||∇ϕε (y)| − αt µt h∇ψε (y), yi

From (16-i), (32), (36-38) and the inequality 2ab ≤ α1 a2 + αb2 with α ∈ {2, 4}, we obtain Z i h  1 s ˜t  ˜ n n 2 V ϕ1/n (Yt ) + ψ1/n (Yt ) + e 1[0,n∧τ ] (r) e2Vr |∇ϕ1/n (Yrn )|2 dr + |∇ψ1/n (Yrn )|2 dAr 2 t Z s i h ˜ ˜ 1[0,n∧τ ] (r) 4e2Vr |F (r, Yrn , Zrn )|2 dr + |G(r, Yrn )|2 dAr ≤ e2Vs [ϕ(Ysn ) + ψ(Ysn )] + t Z s i h ˜ 2 dr + |ν − |2 dA | + 1[0,n∧τ ] (r) e2Vr |Yrn |2 |µ− r r r t Z s

 

 ˜  n n − n n − 1[0,n∧τ ] (r) e2Vr µ− r ∇ψ1/n (Yr ), Yr dr + νr ∇ϕ1/n (Yr ), Yr dAr t Z s Z s   ˜  ˜r

n n n 2 V ∇ϕ1/n (Yr ) + ∇ψ1/n (Yr ), Zr dWr − 2 e2Vr ϕ1/n (Yrn ) + ψ1/n (Yrn ) dV˜r . − e t

t

Using (35), the definition of V˜ , the assumption dVs ≤ 2dV˜s , on [0, τ ], and the inequality

 0 ≤ ϕ1/n (y) ≤ ∇ϕ1/n (y), y , ∀y ∈ H, 15

(39)

we have that



   n n − n n n n ˜ −µ− r ∇ψ1/n (Yr ), Yr dr − νr ∇ϕ1/n (Yr ), Yr dAr − 2 ϕ1/n (Yr ) + ψ1/n (Yr ) dVr
≤ 2 ϕ1/n (Yrn ) + ψ1/n (Yrn ) ((˜ µr )− dr + (˜ νr )− dAr )



 n n −νr− ∇ϕ1/n (Yrn ), Yrn dAr − µ− r ∇ψ1/n (Yr ), Yr dr



 ≤ 2 ∇ϕ1/n (Yrn ), Yrn (˜ µr )− dr + 2 ∇ψ1/n (Yrn ), Yrn (˜ νr )− dAr



  n n − ≤ ∇ϕ1/n (Yrn ), Yrn µ− r dr + 2 ∇ψ1/n (Yr ), Yr νr dAr , since µ = µ+ − µ− and 2˜ µ ≥ µ ⇔ 2 (˜ µ)− ≤ µ− (and similar for ν). Hence Z s

 

 ˜  n n − n n − 1[0,n∧τ ] (r) e2Vr µ− r ∇ψ1/n (Yr ), Yr dr + νr ∇ϕ1/n (Yr ), Yr dAr t Z s  ˜  −2 e2Vr ϕ1/n (Yrn ) + ψ1/n (Yrn ) dV˜r t Z s h i 1 ˜ 1 1[0,n∧τ ] (r) e2Vr |∇ϕ1/n (Yrn )|2 dr + |∇ψ1/n (Yrn )|2 dAr ≤ 4 4 t Z s i h ˜ 2 − 2 + 1[0,n∧τ ] (r) e2Vr |Yrn |2 |µ− r | dr + |νr | dAr .

(40)

t

Moreover, we see that E because E

Z

s

˜ e4Vr

t

Ft

Z

s t

 ˜

e2Vr ∇ϕ1/n (Yrn ) + ∇ψ1/n (Yrn ), Zrn dWr = 0,

|∇ϕ1/n (Yrn )|

"

+


2 |∇ψ1/n (Yrn )| |Zrn |2 dr

1/2  Z s  ˜r ˜r 2 n 2 V n V ≤ E supr∈[t,s] 2n e |Yr | e |Zr | dr t

1/2

#

" Z 1/2 # Z s 1/2 s √ p ′ ˜ ˜ < ∞. e2Vr |ζ˜r |2 dr +E e2Vr |Zrn − ζ˜r |2 dr ≤ 2 2n R0 E t

t

For s = n, Jensen’s inequality yields h i h i h i ˜ ˜ ˜ E e2Vs ϕ(Ysn ) + ψ(Ysn ) = E e2Vn (ϕ(ξn ) + ψ(ξn )) ≤ E e2Vn (ϕ(η) + ψ(η)) ,

and using (40), the inequality (39) becomes Z n h h
i 1 ˜ ˜ E e2Vt ϕ1/n (Ytn ) + ψ1/n (Ytn ) + E 1[0,n∧τ ] (r) e2Vr |∇ϕ1/n (Yrn )|2 dr 4 t i i h ˜ +|∇ψ1/n (Yrn )|2 dAr ≤ E e2Vn (ϕ(η) + ψ(η)) Z n
˜ 1[0,n∧τ ] (r) e2Vr |F (r, Yrn , Zrn )|2 dr + |G(r, Yrn )|2 dAr +4E t Z n i h ˜ 2 − |2 dA . | dr + |ν 1[0,n∧τ ] (r) e2Vr |Yrn |2 |µ− +2E r r r t

16

(41)

But the right hand side in the above inequality is bounded since ˜

˜

˜

e2Vr |G(r, Yrn )|2 ≤ sup|y|≤√R′ e2Vr |G(r, e−Vr y)|2 , 0

2V˜r

e

|F (r, Yrn , Zrn )|2

≤ 3 sup|y|≤

Therefore and E

Z

√

˜

˜

˜

˜

e2Vr |F (r, e−Vr y, 0)|2 + 3ℓ2 e2Vr |Zrn − ζr |2 + 3ℓ2 e2Vr |ζr |2 .

R′0

h
i ˜ E e2Vt ϕ1/n (Ytn ) + ψ1/n (Ytn ) ≤ C, for all t ≥ 0

∞

i h ˜ ˜ 1[0,n∧τ ] (r) e2Vr |∇ϕ1/n (Yrn )|2 dr + e2Vr |∇ψ1/n (Yrn )|2 dAr ≤ C.

0

From (42) and (15−a) we see that, for all t ≥ 0,  h 2 i 2 ˜ ≤ 2C/n, E e2Vt 1/n∇ϕ1/n (Yrn ) + 1/n∇ψ1/n (Yrn ) and

h

i
˜ ≤ C. E e2Vt ϕ Ytn − 1/n∇ϕ1/n (Yrn ) + ψ Ytn − 1/n∇ψ1/n (Yrn )

(42)

(43)

(44)

(45)

E. Cauchy sequences and convergence. From (34) we have ˜ E sups≥n epVs |Ysn+l

≤CE

Z

∞

|p

− ξs + E

˜ e2Vs Ψ(s, ξ

Z

n

s )dQs

n

∞

˜ e2Vs |Zsn+l

p/2

+

Z

∞

n

− ζs

|2 ds

p/2

˜ eVs |Φ(s, ξ

s , ζs )|dQs

p 

(46) −→ 0, n → ∞.

By uniqueness it follows that, for all t ∈ [0, n], Z n Z n n,l n n+l n+l (Zsn+l − Zsn )dWs , a.s., dKs − − Yt = Yn − ξn + Yt t

t

where   dKsn,l = Φ(s, Ysn+l , Zsn+l ) − Φ(s, Ysn , Zsn ) − [∇y Ψn+l (Ysn+l ) − ∇y Ψn (Ysn )] dQs .

By (15-d) with ε = 1/ (n + l) and δ = 1/n 


− Ysn+l − Ysn , ∇y Ψn+l (s, Ysn+l ) − ∇y Ψn (s, Ysn ) dQs 

 

 ≤ (ε + δ)1[0,τ ] (s) ∇ϕε (Ysn+l ), ∇ϕδ (Ysn ) ds + ∇ψε (Ysn+l ), ∇ψδ (Ysn ) dAs ,

and using (12) we have on [0, n]

h  ε+δ
Ysn+l − Ysn , dKsn,l ≤ 1[0,τ ] (s) |∇ϕε (Ysn )|2 + |∇ϕδ (Ysn+l )|2 ds 2 i
1 2 n |Zsn+l − Zsn |2 ds. + |∇ψε (Ys )| + |∇ψδ (Ysn+l )|2 dAs + |Ysn+l − Ysn |2 dVs1 + 2a 17

Proposition 16 yields once again ˜ E sups∈[0,n] e2Vs |Ysn+l

+(ε + δ) C E

Z

n∧τ

0

+(ε + δ) C E

Z

n∧τ

0

−

Ysn |2

+E

Z

0

n

˜

˜

e2Vs |Zsn+l − Zsn |2 ds ≤ C Ee2Vn |Ynn+l − ξn |2


˜ e2Vs |∇ϕε (Ysn )|2 + |∇ϕδ (Ysn+l )|2 ds

(47)


˜ e2Vs |∇ψε (Ysn )|2 + |∇ψδ (Ysn+l )|2 dAs .

The estimates (43) and (46) give us ˜ E sups∈[0,n] e2Vs |Ysn+l

−

Ysn |2

+E

Z

n

0

˜

e2Vs |Zsn+l − Zsn |2 ds

˜

≤ E sups≥n e2Vs |Ysn+l − ξs |2 +

C −→ 0, as n → ∞. n

Hence ˜

E sups≥0 e2Vs |Ysn+l − Ysn |2

˜

˜

≤ E sups∈[0,n] e2Vs |Ysn+l − Ysn |2 + E sups≥n e2Vs |Ysn+l − ξs |2 −→ 0, as n → ∞ and E

Z

0

∞

˜

e2Vs |Zsn+l − Zsn |2 ds Z Z n ˜s n+l n 2 2 V e |Zs − Zs | ds + E ≤E 0

∞ n

˜

e2Vs |Zsn+l − ζs |2 ds −→ 0, as n → ∞.

F. Passage to the limit. Consequently there exists (Y, Z) ∈ S 2 × Λ2 such that Z ∞ ˜ 2V˜s n 2 e2Vs |Zsn − Zs |2 ds −→ 0, as n → ∞. E sups≥0 e |Ys − Ys | + E 0

We have that (Yt , Zt ) = (η, 0) for t > τ , since Ytn = ξt = η and Ztn = ζt = 0 for t > τ . Applying Fatou’s Lemma to (31) and (34) we obtain (27−a, b) and taking the limit along a subsequence in (32), we deduce that ˜

e2Vt |Yt |2 ≤ R0′ , P-a.s., for all t ≥ 0. From (43) there exist two p.m.s.p. U 1 and U 2 , such that along a subsequence still indexed by n, ˜

1[0,τ ∧n] e2V ∇ϕ1/n (Y n ) ⇀ 1[0,τ ] U 1 , ˜

1[0,τ ∧n] e2V ∇ψ1/n (Y n ) ⇀ 1[0,τ ] U 2 ,

weakly in L2 (Ω × R+ , dP ⊗ dt; H) ,

weakly in L2 (Ω × R+ , dP ⊗ dAt ; H) .

18

Applying Fatou’s Lemma to (43) we obtain (28) and from (44) we deduce that for all t ≥ 0 fixed, there exists a subsequence indexed also by n, such that 1 a.s. ∇ϕ1/n (Ytn ) −−−→ 0 and n

1 a.s. ∇ψ1/n (Ytn ) −−−→ 0. n

We now apply Fatou’s Lemma to (45) and we conclude (27−d). From (29) we have for all 0 ≤ t ≤ T ≤ n, P-a.s. Ytn

+

Z

T

t

∇y Ψ

n

(s, Ysn )dQs

YTn

=

+

Z

T

Φ(s, Ysn , Zsn )dQs

t

Z

−

T t

Zsn dWs ,

and passing to the limit we conclude that Yt +

Z

T

Us dQs = YT +

Z

T

Φ(s, Ys , Zs )dQs −

t

t

Z

T

Zs dWs , a.s.

(48)

t

with   Us = 1[0,τ ] (s) αs Us1 + (1 − αs ) Us2 , for s ≥ 0.

(49)

Since (15−b), we see that, for all E ∈ F, 0 ≤ s ≤ t and X ∈ S 2 [0, T ] , E

Z

s

t

˜ 1E e2Vr ∇ϕ1/n (Yrn ), Xr



−

Yrn



dr + E

Z

s

t

˜

1E e2Vr ϕ(Yrn − 1/n∇ϕ1/n (Ysn ))dr ≤E

Z

t

˜

1E e2Vr ϕ(Xr )dr.

s

Passing to lim inf for n → ∞ in the above inequality we obtain Us1 ∈ ∂ϕ(Ys ), dP ⊗ ds-a.e.

(50)

Us2 ∈ ∂ψ(Ys ), dP ⊗ dAs -a.e.

(51)

and, with similar arguments,

Summarizing the above conclusions we see that (Y, Z, U ) is solution of the BSVI (20) under the assumption (A1 −A9 ).

4

Variational weak formulation

The existence and the uniqueness of a solution (Y, Z) are still true without the additional assumption (A9 ), but in this case we can’t prove the absolute continuity of K (or even the bounded variation property). Therefore we shall give a new definition for the solution of the BSVI (20). Let us define the space M of the semimartingales M ∈ S 1 of the form Z t Z t Rr dWr , Nr dQr + Mt = γ − 0

0

19

where N and R are two p.m.s.p. such that

Z

T 0

|Nr | dQr +

Z

T 0

|Rr |2 dr < ∞ a.s., ∀ T > 0 and

γ ∈ L0 (Ω; F0 , P; H). For a intuitive introduction, let M ∈ M and (Y, Z) be a solution of (20), in the sense of Definition 5. By Itˆo’s formula we deduce the inequality 1 1 |Mt − Yt |2 + 2 2

Z

Z T 1 2 Ψ (r, Mr ) dQr Ψ (r, Yr ) dQr ≤ |MT − YT | + |Rr − Zr | dr + 2 t t Z T Z T hMr − Yr , (Rr − Zr ) dWr i, hMr − Yr , Nr − Φ (r, Yr , Zr )idQr − +

T t

Z

2

T

t

t

since dKt = Ut dQt ∈ ∂y Ψ (t, Yt ) dQt (see the inequality (13)). We propose the following weak formulation of the Definition 5 Definition 8 We call (Yt , Zt )t≥0 a variational weak solution of (20) if (Y, Z) ∈ S 2 × Λ2 , (Yt , Zt ) = (ξt , ζt ) = (η, 0) for t > τ and (i)

Z

T

(|Φ (s, Ys , Zs )| + |Ψ (s, Ys )|) dQs < ∞, P-a.s., for all T ≥ 0, Z s Z 1 1 s 1 2 2 |Mt − Yt | + Ψ (r, Yr ) dQr ≤ |Ms − Ys |2 |Rr − Zr | dr + 2 2 t 2 t Z s Z s + Ψ (r, Mr ) dQr + hMr − Yr , Nr − Φ (r, Yr , Zr )idQr t t Z s − hMr − Yr , (Rr − Zr ) dWr i , 0

(ii)

(52)

t

L2 (Ω

(iii) Ee2VT

∀ 0 ≤ t ≤ s, ∀ (N, R) ∈ × [0, ∞); H) × Λ2 , ∀ M ∈ M, Z ∞ e2Vs |Zs − ζs |2 ds −→ 0, as T → ∞. |YT − ξT |2 + E T

Theorem 9 Let the assumptions (A1 −A8 ) be satisfied. Then the backward stochastic variational inequality (20) has a unique solution (Y, Z) in the sense of Definition 8 such that ˜

E supt∈[0,T ] epVs |Ys |p < ∞, for all T ≥ 0.

(53)

Moreover, the inequalities (27) hold. Proof. Firstly we shall approximate the data η and Φ by η n respectively Φn which satisfy (24). Let
η n (ω) = η (ω) 1[0,n] |η (ω)| + | sups∈[0,τ ] V˜s | ,
Φn (t, y, z) = Φ (t, y, z) − Φ (t, 0, 0) + Φ (t, 0, 0) 1[0,n] t + |Φ (t, 0, 0)| + |V˜t | . 20

Obviously, p sups∈[0,τ ] V˜s

E e

n

|η − η|

p


+E

Z

τ

V˜s

e 0

n

|Φ (s, 0, 0) − Φ (s, 0, 0)| dQs

p

−→ 0, as n → ∞.

Applying Theorem 7 we deduce that there exists a unique solution (Y n , Z n , U n ) of the BSVI (20) corresponding to η n and Φn :  Z ∞ Z ∞ Z ∞  n n n n n  Ytn + Zsn dWs , a.s., Φ (s, Ys , Zs ) dQs − Us dQs = η + (54) t t t   U n ∈ ∂ Ψ (t, Y n ) , ∀ t ≥ 0. y t t

This solution satisfies inequalities (27-28) with Y, Z, U, Φ, η, ξ, ζ replaced respectively with Y n , Z n , U n , Φn , η n , ξ n , ζ n . Since |η n | ≤ |η| and |Φn (t, 0, 0)| ≤ |Φ (t, 0, 0)|, 2 i Z ∞ h ˜ 2 sups∈[t,τ ] V˜s 2 n 2 Ft 2V˜t , ∀t ≥ 0, P-a.s. |η| ) + eVs |Φ (s, 0, 0) |dQs e |Yt | ≤ C E (e t

Using (12), we see that

hYsn − Ysm , Φn (s, Ysn , Zsn ) − Φm (s, Ysm , Zsm ) − (Usn − Usm )idQs

≤ hYsn − Ysm , Φ (s, Ysn , Zsn ) − Φ(s, Ysm , Zsm )idQs
+hYsn − Ysm , Φ (s, 0, 0)i 1[0,n] − 1[0,m] (t + |Φ (t, 0, 0)| + |V˜t |)dQs
≤ |Ysn − Ysm | |Φ (s, 0, 0)| 1[0,n] − 1[0,m] (t + |Φ (t, 0, 0)| + |V˜t |) dQs

(55)

1 +|Ysn − Ysm |2 dV˜s + |Zsn − Zsm |2 ds, 2a

since hYsn − Ysm , Usn − Usm i ≥ 0, for Usn ∈ ∂y Ψ(s, Ysn ) and Usm ∈ ∂y Ψ(s, Ysm ), and dVs ≤ dV˜s on [0, τ ]. Applying Proposition 16 for the equation satisfied by Y n − Y m on [0, T ], it follows that p/2 Z T ˜s ˜s n m 2 2 V n m p p V e |Zs − Zs | ds E sups∈[0,T ] e |Ys − Ys | + E 0

≤CE

Z

T

0

p
˜ eVs |Φ (s, 0, 0)| 1[0,n] − 1[0,m] (t + |Φ (t, 0, 0)| + |V˜t |) dQs   ˜ +C EepVT |YTn − ξTn |p + |ξTn − ξTm |p + |ξTm − YTm |p .

Therefore passing to the limit for T → ∞, Z ˜s n m p p V E sups≥0 e |Ys − Ys | + E

0

≤CE

Z

∞

0

∞

˜ e2Vs |Zsn

−

Zsm |2 ds

p/2

p
˜ eVs |Φ (s, 0, 0)| 1[0,n] − 1[0,m] (t + |Φ (t, 0, 0)| + |V˜t |) dQs ˜

+C Eep sups≥0 Vs |η n − η m |p −−−−−→ 0. n,m→∞

21

Consequently there exists (Y, Z) ∈ S 0 × Λ0 a solution of the BSVI (20) such that Z ∞ ˜ n 2 2V˜s e2Vs |Zsn − Zs |2 ds −→ 0, as n → ∞ E sups≥0 e |Ys − Ys | + E 0

and (Yt , Zt ) = (η, 0) for t > τ , since Ytn = ξtn = η n and Ztn = ζtn = 0 for t > τ . Let M ∈ M given by Z t Z t Rr dWr . Nr dQr + Mt = γ − 0

0

From the Itˆo’s formula applying to |Mt − Ytn |2 we deduce that, for all 0 ≤ t ≤ s ≤ τ, Z s Z s Z 1 1 s 1 2 2 2 n n n n Ψ (r, Mr ) dQr |Mt − Yt | + Ψ (r, Yr ) dQr ≤ |Ms − Ys | + |Rr − Zr | dr + 2 2 t 2 t t Z s Z s n n n hMr − Yrn , (Rr − Zrn ) dWr i . + hMr − Yr , Nr − Φ (r, Yr , Zr )idQr − t

t

Since on a subsequence (still denoted by n) sup s∈[0,T ]

|Ysn

2

− Ys | +

Z

T 0

|Zsn − Zs |2 ds −→ 0, a.s.,

it follows easily, passing to the lim inf, that the couple (Y, Z) satisfies the inequality (52-ii). In the same manner, the inequalities (27) follow now from the similar properties satisfied by the approximate solution (Y n , Z n ).

In order to prove the uniqueness of the solution let Y 1 , Z 1 and Y 2 , Z 2 be two solutions of (20) corresponding to η 1 and η 2 respectively. Therefore Z 2
1 s 2
1 2 1 Rr − Zr1 2 + Rr − Zr2 2 dr + Mt − Yt + Mt − Yt 2 2 t Z s Z s 2
2


1 1 2 2 1 Ψ r, Yr + Ψ r, Yr dQr ≤ + Ψ (r, Mr ) dQr +2 Ms − Ys + Ms − Ys 2 t t Z s


+ hMr − Yr1 , Nr − Φ r, Yr1 , Zr1 i + hMr − Yr2 , Nr − Φ r, Yr2 , Zr2 i dQr t Z s

hMr − Yr1 , Rr − Zr1 i + hMr − Yr2 , Rr − Zr2 dWr i , ∀ 0 ≤ t ≤ s, ∀ M ∈ M. − t



Φ r, Yr1 , Zr1 + Φ r, Yr2 , Zr2 Z= and Φ (r) = . Let Y = 2 From the convexity of Ψ we see that

2Ψ (r, Yr ) ≤ Ψ r, Yr1 + Ψ r, Yr2 , Y 1 +Y 2 , 2

Z 1 +Z 2 2

and using the identity







y1 + y2 f 1 + f 2  1 1 , + y − y2, f 1 − f 2 = y1 , f 1 + y2, f 2 , 2 2 2 2 22

we obtain

hMr − Yr1 , Nr − Φ r, Yr1 , Zr1 i + hMr − Yr2 , Nr − Φ r, Yr2 , Zr2 i

−2 hMr − Yr , Nr − Φ (r)i − 21 hYr1 − Yr2 , Φ r, Yr1 , Zr1 − Φ r, Yr2 , Zr2 i = 0,

and

hMr − Yr1 , Rr − Zr1 i + hMr − Yr2 , Rr − Zr2 i

−2 hMr − Yr , Rr − Zr i − 12 hYr1 − Yr2 , Zr1 − Zr2 i = 0.

Therefore, since 1 2


m − y 1 2 + m − y 2 2 = m −

y 1 +y 2 2 2

2 + 14 y 1 − y 2 .

we have Z s 1 1 Zr − Zr2 2 dr ≤ 8Bt,s (M ) + Ys1 − Ys2 2 Yt − Yt2 2 + t Z s Z s


1 2 1 1 2 2 hYr − Yr , Φ r, Yr , Zr − Φ r, Yr , Zr idQr − 2 hYr1 − Yr2 , Zr1 − Zr2 dWr i, +2 t

t

∀ 0 ≤ t ≤ s, ∀ M ∈ M,

(56)

where Z s Z s 1 1 2 hMr − Yr , Nr − Φ (r)idQr − |Mt − Yt |2 Ψ (r, Mr )dQr + Bt,s (M ) = |Ms − Ys | + 2 2 t t Z s Z s Z s 1 − hMr − Yr , (Rr − Zr ) dWr i. Ψ (r, Yr )dQr − |Rr − Zr |2 dr − 2 t t t Our next goal will be to prove that there exists M ε ∈ M such that lim Bt,s (M ε ) = 0, a.s., ∀ 0 ≤ t ≤ s. ε→0

Let Q ε

− Qt

Mtε = e



Y0 +

1 Qε

Z

0

t

 Qr e Qε Yr dQr .

(57)

(58)

Rt Clearly, M ε ∈ M since Mtε = M0ε + 0 dMrε . The next result it is necessary in order to obtain the limit in the Stieltjes type integrals: Lemma 10 Let a : [0, T ] → R be a strictly increasing continuous function such that a (0) = 0 and f : [0, T ] → H be a measurable function such that |f (t)| ≤ C a.e. t ∈ [0, T ]. Define, for ε > 0, Z t a(r)−a(t) −a(t) 1 a(ε) + e a(ε) f (r) da (r) . fε (t) = f (0) e a (ε) 0 Then as ε → 0, fε (t) −→ f (t) , a.e. t ∈ [0, T ] . (59) and, if f is continuous, then

sup |fε (t) − f (t)| −→ 0 .

t∈[0,T ]

23

(60)

Remark 11 The same conclusions are true if we consider fε (t) replaced by gε (t) = f (T ) e

a(t)−a(T ) a(ε)

+

1 a (ε)

Z

T

e

a(t)−a(r) a(ε)

t

f (r) da (r) , t ∈ [0, T ] .

Proof of the Lemma. Obviously we have Z 0 Z t a(r)−a(t)
1 u f a−1 (ua (ε) + a (t)) du a(ε) e f (r) da (r) = e a(ε) −a(t) a(ε)

0

=

0

Z

−a(t) a(ε)

eu



f

◦ a−1




(ua (ε) + a (t)) − f

◦ a−1




 (a (t)) du + f (t)

Z

0 −a(t) a(ε)

(61) eu du.

But Z 0 

 u −1 −1 lim sup e f ◦a (ua (ε) + a (t)) − f ◦ a (a (t)) du −a(t) ε→0

a(ε)

Z

≤ lim sup ε→0

≤ 2C

Z

−n

0

−∞



eu f ◦ a−1 ((ua (ε) + a (t)) ∨ 0) − f ◦ a−1 (a (t)) du

eu du +

−∞

≤

2Ce−n

≤

2Ce−n ,

Z

+ lim sup ε→0

0

−n

Z

for all n,

since lim

Z

β

δ→0 α

Therefore there exists



eu f ◦ a−1 ((ua (ε) + a (t)) ∨ 0) − f ◦ a−1 (a (t)) du

0

−n



f ◦ a−1 ((ua (ε) + a (t)) ∨ 0) − f ◦ a−1 (a (t)) du



f ◦ a−1 (s + δu) − f ◦ a−1 (s) du = 0, a.e.

Z 0 
 u −1 lim e f ◦a (ua (ε) + a (t)) − f (t) du = 0, −a(t)

ε→0

a(ε)

and (59) follows. In the case of continuity for f it is sufficient to write Z t a(r)−a(t) −a(t) 1 fε (t) = f (0) e a(ε) + e a(ε) f (r) da (r) a (ε) 0 Z tε a(r)−a(t) Z t a(r)−a(t) −a(t) 1 1 a(ε) a(ε) = f (0) e e e a(ε) f (r) da (r) , + f (r) da (r) + a (ε) 0 a (ε) tε p
where tε := a−1 a (t) − a (ε) −→ t, as ε → 0, and tε < t.

24

Applying the above Lemma we can conclude from (58) that Mtε −→ Yt , ∀ t ∈ [0, T ] . Next we shall prove that, for all t ≤ s, Z Z s ε Ψ (r, Mr )dQr −→

(62)

s

Ψ (r, Yr )dQr .

t

t

From (58) and the convexity of the functions ϕ and ψ we deduce that  Z s Z s Z s Z r −Qr Qu −Qr 1 ε Qε e Qε ϕ (Y0 ) dr + ϕ (Mr ) αr dQr ≤ ϕ (Yu ) dQu dr Qε e t

t

= ϕ (Y0 ) −

Z

s

e

= ϕ (Y0 )

dr +

t

Z t Z 0

−Qr Qε

t

0 Z s

1 Qε e

e

Qu −Qr Qε

−Qr Qε

dr +

t

− and Z

t

s

Z t

ψ (Mrε ) (1

= ψ (Y0 ) −

Z

s

e

= ψ (Y0 )

−Qr Qε

dAr +

r

1 Qε e

e

Qu −Qr Qε

0

Z

−Qr Qε

dAr +

ψ (Yu )

Z

0

Z

t 0

Z

0

1 Qε e

Qu −Qr Qε



ϕ (Yu ) 1[0,r] (u) dQu dr

Z

0

s

ϕ (Yu )

Qu −Qr Qε

s

e

−Qr Qε

Z

s

0

1 Qε e

1[u,t] (r) dr dQu .

ψ (Y0 ) dAr +

Z

t

0

1 Qε e

s

1 Qε e

Qu −Qr Qε

ψ (Yu )

 1[u,s] (r) dr dQu



t

s Z r

(63)

Qu −Qr Qε

Qu −Qr Qε

Z

0

s

s Z r 0

1 Qε e

Qu −Qr Qε



ψ (Yu ) dQu dAr



ψ (Yu ) dQu dAr

 ψ (Yu ) dQu dAr

t

−

0

0

s Z s

 ϕ (Yu ) 1[0,r] (u) dQu dr

1 Qε e

0

0 Z s

Z t

t

− αr ) dQr ≤

t

Z t Z 0

ϕ (Yu )

0

Z

0

t

Z

(64) 1 Qε e

Qu −Qr Qε

 1[u,s] (r) dAr dQu



1[u,t] (r) dAr dQu .

On the other hand, using Remark 11 and Lebesgue’s dominated convergence theorem, we conclude that Z t Z s 1 QuQ−Qr 1 QuQ−Qr ε αr dQr = lim (65) e e ε αr dQr = αu , a.e. lim ε→0 u Qε ε→0 u Qε and respectively, Z t Z s 1 QuQ−Qr 1 QuQ−Qr ε (1 − αr ) dQr = lim e e ε (1 − αr ) dQr = αu , a.e. lim ε→0 u Qε ε→0 u Qε 25

(66)

From the inequalities (63-64) we obtain Z s Z s Ψ (r, Mrε )dQr Ψ (r, Yr )dQr ≤ t t Z s Z s −Qt −Qt Qt −Qr Qt −Qr 1 1 Q Q Q Qε ε ε ε ≤ ϕ (Y0 ) Qε e α dQ + ψ (Y ) Q e (1 − αr ) dQr e e r r 0 ε Qε Qε t t  Z s Z s Z s Qu −Qr Qu −Qr 1 1 Q Q ε ε dr + ψ (Yu ) dAr dQu + ϕ (Yu ) Qε e Qε e 0

Z t Z − ϕ (Yu ) 0

u t

u

1 Qε e

Qu −Qr Qε

dr + ψ (Yu )

Z

u t

u

1 Qε e

Qu −Qr Qε



dAr dQu ,

and applying the limits (65-66), we deduce Z Z s Z s  ε→0 ε ϕ (Yu ) αu + ψ (Yu ) (1 − αu ) dQu = Ψ (r, Mr )dQr −−−−→

s

Ψ (u, Yu )dQu .

(67)

Now returning to the inequality (56), Z s Z s 1 1

Zr − Zr2 2 dr ≤ 2 Yt − Yt2 2 + hYr1 − Yr2 , Φ r, Yr1 , Zr1 − Φ r, Yr2 , Zr2 idQr t t Z s 1
2 + Ys − Ys2 − 2 hYr1 − Yr2 , Zr1 − Zr2 dWr i, ∀ 0 ≤ t ≤ s.

(68)

t

t

t

Therefore (57) follows immediately, since we have (62) and (67).

t

From (12)



1 hYr1 − Yr2 , Φ r, Yr1 , Zr1 − Φ r, Yr2 , Zr2 idQr ≤ |Yr1 − Yr2 |2 dV˜r + |Zr − Zr2 |2 dr 2a

and therefore inequality (68) becomes Z s Z s 1 1 1 Y − Y 2 2 dV˜r Z − Z 2 2 dr ≤ Y 1 − Y 2 2 + 2 Y − Y 2 2 + (1 − 1/a) r r s r s r t t t t Z s
−2 hYr1 − Yr2 , Zr1 − Zr2 dWr i.

(69)

t

Applying a Gronwall’s type stochastic inequality (see Lemma 12 from the Appendix of [11]) we conclude that, for all 0 ≤ t ≤ s, P-a.s. Z s
˜ 2V˜s 1 2 2 2V˜t 1 2 2 Ys − Ys − 2 e2Vr hYr1 − Yr2 , Zr1 − Zr2 dWr i. e Yt − Yt ≤ e t

Therefore, using also the condition (52−iii) form the definition of weak variational solution, the uniqueness follows.

26

5

Examples

Let D ⊂ Rd be an open bounded subset with boundary Bd (D) sufficiently smooth. In what follows H m (D) and H0m (D) stand for the usual Sobolev spaces. Let (Ω, F, (Ft )t≥0 , P) be a complete probability space, {Ws : 0 ≤ s ≤ t} a real Wiener process and set H = H1 := L2 (D). We notice that the space of Hilbert-Schmidt operators from L2 (D) to L2 (D)) can be identified with L2 (D × D). Let j : R → (∞, ∞] be a proper convex l.s.c. function, for which we assume that j (u) ≥ j (0) = 0, ∀u ∈ R. Our aim is to obtain, via Theorem 7, the existence and uniqueness of the solution for some backward stochastic partial differential equations (SPDE) suggested in [16]. We recall the assumptions (A1 −A5 ), (A8 -19), the condition
E QpT < ∞, ∀ T > 0, and definitions (9) and (10) from Section 2.2.

Example 12 First we consider the backward SPDE with Dirichlet boundary condition   −dY (t, x) + ∂j (Y (t, x)) dQt ∋ ∆Y (t, x) dQt + Φ (t, Y (t, x) , Z (t, x)) dQt       −Z (t, x) dWt , in Ω × [0, τ ] × D,  Y (ω, t, x) = 0 on Ω × [0, τ ] × Bd (D) ,    Z ∞   prob.  2  e2Vs kZ (s) − ζs k2 ds −−−−→ 0,  e2VT kY (T ) − ξT k + T →∞

T

where kf k2 :=

R

D

(70)

|f (x)|2 dx .

Let us apply Theorem 7, with Ψ = ϕ = ψ (in which case the compatibility assumptions (16) are satisfied), where ϕ : L2 (D) → (−∞, ∞] is given by  Z Z 1  2  j (u (x)) dx, if u ∈ H01 (D) , j (u) ∈ L1 (D) , |∇u (x)| dx +  2 ϕ (u) = D D    +∞, otherwise. Proposition 2.8 from [1], Chap. II, shows that the following properties hold:

(a) function ϕ  is proper, convex and l.s.c., (b) ∂ϕ (u) = u∗ ∈ L2 (D) : u∗ (x) ∈ ∂j (u (x)) − ∆u (x) a.e. on D , ∀u ∈ Dom (∂ϕ) ,  (c) Dom (∂ϕ) = u ∈ H01 (D) ∩ H 2 (D) : u (x) ∈ Dom (∂j) a.e. on D .

Moreover, there exists a positive constant C such that

(d) kukH 1 (D)∩H 2 (D) ≤ C ku∗ kL2 (D) , ∀ (u, u∗ ) ∈ ∂ϕ. 0

Let η be a H01 (D)-valued random variable, Fτ -measurable such that (A9 ) is satisfied and   ˜ ˜ E ep sups∈[0,τ ] Vs |η|p < ∞, e2 sups∈[0,τ ] Vs j (η) ∈ L1 (Ω × D) , 27

and the stochastic processes ξ, ζ, associated to η by the martingale representation theorem, such that Z τ Z τ p/2 p 2V˜s V˜s e ϕ (ξs ) dQs e |Φ (s, ξs , ζs )| dQs E +E < ∞, (71) 0

0

where V˜ is defined by (17).

Applying now Theorem 7, we deduce that, under the above assumptions, backward SPDE (70) has a unique solution (Y, Z, U ) ∈ SL0 2 (D) × Λ0L2 (D×D) × Λ0L2 (D) such that (Y (t) , Z (t)) = (ξt , ζt ) = (η, 0), for t ≥ τ , and Z T Z T ∆Y (s, x) dQs U (s, x) dQs = Y (T, x) + (i) Y (t, x) + t

t

+

Z

t

T

Φ (s, Y (s, x) , Z (s, x)) dQs −

Z

T

t

Z (s, x) dWs , in [0, T ] × D, a.s. ,

(ii) Y (t) ∈ H01 (D) ∩ H 2 (D) , dP × dt a.e., (iii) Y (t, x) ∈ Dom (∂j) , dP × dQt × dx a.e., (iv) U (t, x) ∈ ∂j (Y (t, x)) , dP × dQt × dx a.e.,


˜ ˜ (v) e2V Y ∈ L∞ 0, T ; L2 Ω; H01 (D) and e2V j (Y ) ∈ L∞ 0, T ; L1 (Ω × D) , for all T > 0, Z τ ˜ (vi) E e2Vs kY (s)k2H 1 (D)∩H 2 (D) dQs < ∞. 0

0

Remark 13 If we renounce the Assumption (A9 ), then we obtain that the backward SPDE (70) admits a variational weak solution. More precisely, Theorem 9 shows that there exists a unique solution (Y, Z) ∈ SL0 2 (D) × Λ0L2 (D×D) such that (Y (t) , Z (t)) = (ξt , ζt ) = (η, 0), for t ≥ τ , and for all 0 ≤ t ≤ s Z sZ Z 1 1 s 2 2 j (Y (r, x)) dx dQr (i) kM (t) − Y (t)k + |R (r) − Z (r)| dr + 2 2 t t D Z sZ 1 2 ≤ kM (s) − Y (s)k + j (M (r, x)) dx dQr 2 t D Z s + hhM (r) − Y (r) , N (r) − ∆Y (r) − Φ (r, Y (r) , Z (r))iidQr t Z s − hhM (r) − Y (r) , (R (r) − Z (r)) dWr ii , t


∀ (N, R) ∈ L2 Ω × [0, ∞); L2 (D) × Λ2L2 (D×D), ∀ M ∈ M,

(ii) Y (t) ∈ H01 (D) , dP × dt a.e.,

(iii) Y (t, x) ∈ Dom (j) , dP × dQt × dx a.e., where M is defined at the beginning of the Section 4 and Z Z f (x) g (x) dx. |f (x)|2 dx and hhf, gii := kf k2 := D

D

.

28

Example 14 As a second example we consider the backward SPDE with Neumann boundary condition  −dY (t, x) = ∆Y (t, x) dQt + Φ (t, Y (t, x) , Z (t, x)) dQt − Zt dWt , in Ω × [0, τ ] × D,       ∂Y (ω, t, x) ∈ ∂j (Y (ω, t, x)) , on Ω × [0, τ ] × Bd (D) , − ∂n  Z ∞    prob. 2  2V T  e e2Vs kZ (s) − ζs k2 ds −−−−→ 0. kY (T ) − ξT k + T →∞

T

(72)

We apply again Theorem 7, with Ψ = ϕ = ψ, where ϕ : L2 (D) → (−∞, ∞] is given by Z  Z 1 2   j (u (x)) dx, if u ∈ H 1 (D) , k (u) ∈ L1 (Bd (D)) , |∇u (x)| dx +   2 D Bd(D) ϕ (u) =     +∞, otherwise.

Proposition 2.9 from [1], Chap. II, shows that:

(a) function ϕ is proper, convex and l.s.c., (b)

∂ϕ (u) = −∆u (x) , ∀u ∈ Dom (∂ϕ) ,

(c)

Dom (∂ϕ) = {u ∈ H 2 (D) : −

∂u (x) ∈ ∂j (u (x)) a.e. on Bd (D)}, ∂n

∂u where is the outward normal derivative to the boundary. ∂n Moreover, there are some positive constants C1 , C2 such that (d) kukH 2 (D) ≤ C1 ku − ∆ukL2 (D) + C2 , ∀u ∈ Dom (∂ϕ) . Let η be a H 1 (D)-valued random variable, Fτ -measurable such that (A9 ) is satisfied and   ˜ E ep sups∈[0,τ ] Vs |η|p < ∞,

˜

e2 sups∈[0,τ ] Vs j (η) ∈ L1 (Ω × Bd (D)) ,

and the stochastic processes ξ and ζ (from the martingale representation theorem) be such that (71) holds. Applying Theorem 7 we conclude that, under the above assumptions, backward SPDE (72) has a unique solution (Y, Z) ∈ SL0 2 (D) × Λ0L2 (D×D) such that (Y (t) , Z (t)) = (ξt , ζt ) =

29

(η, 0), for t ≥ τ , and (i) Y (t, x) = Y (T, x) +

Z

T

∆Y (s, x) dQs +

Z

T

Φ (s, Y (s, x) , Z (s, x)) dQs

t

t

− (ii) Y (t) ∈ H 2 (D) , dP × dt a.e.,

Z

T

t

Z (s, x) dWs , in [0, T ] × D, a.s.,

∂Y (t, x) ∈ Dom (∂j) , dP × dQt × dx a.e., ∂n


˜ ˜ (iv) e2V Y ∈ L∞ 0, T ; L2 Ω; H 1 (D) and e2V j (Y ) ∈ L∞ 0, T ; L1 (Ω × Bd (D)) , Z τ ˜ (v) E e2Vs kY (s)k2H 2 (D) dQs < ∞.

(iii) −

0

Example 15 The third example is the backward stochastic porous media equation   −dY (t, x) = ∆ (∂j) (Y (t, x)) dQt + Φ (t, Y (t, x) , Z (t, x)) dQt − Z (t, x) dWt ,       in Ω × [0, τ ] × D,  ∂j (Y (ω, t, x)) ∋ 0 on Ω × [0, τ ] × Bd (D) ,    Z ∞   prob.  2 2V T  e2Vs kZ (s) − ζs k2 ds −−−−→ 0. kY (T ) − ξT k +  e

(73)

T →∞

T

In Theorem 7, let H = H −1 (D) (the dual of H01 (D)), H1 = Rd and Ψ = ϕ = ψ, where ϕ : H −1 (D) → (−∞, ∞] is given by  Z   j (u (x)) dx, if u ∈ L1 (D) , j (u) ∈ L1 (D) ,  ϕ (u) = D    +∞, otherwise,

and j : R → R+ is suppose, moreover, to be continuous with lim j (r) /r = ∞. r→∞

Proposition 2.10 from [1], Chap. II, shows that: (a) function ϕ is proper, convex and l.s.c.,

(b) ∂ϕ (u) = {u∗ ∈ H −1 (D) : u∗ (x) = −∆v (u (x)) , v ∈ H01 (D) , (c)

v (x) ∈ ∂j (u (x)) a.e. on D}, ∀u ∈ Dom (∂ϕ) ,  Dom (∂ϕ) = u ∈ H −1 (D) ∩ L1 (D) : u (x) ∈ Dom (∂j) a.e. on D .

Let η be a H −1 (D)-valued random variable, Fτ -measurable such that (A9 ) is satisfied and   ˜ ˜ E ep sups∈[0,τ ] Vs |η|p < ∞, η ∈ L1 (Ω × D) , e2 sups∈[0,τ ] Vs j (η) ∈ L1 (Ω × D) , 30

and the stochastic processes ξ and ζ (from the martingale representation theorem) be such that (71) holds. From the Theorem 7 it follows that, under the above assumptions, backward SPDE (70) 0 0 such that (Y (t) , Z (t)) = (ξt , ζt ) = (η, 0), has a unique solution (Y, Z) ∈ SH −1 (D) ×Λ (H −1 (D))d for t ≥ τ , and Z T Z T Φ (s, Y (s, x) , Z (s, x)) dQs ∆U (s, x) dQs = Y (T, x) + (i) Y (t, x) + t

t

−

Z

T

t

Z (s, x) dWs in [0, T ] × D, a.s. ,

(ii) Y (t, x) ∈ Dom (∂j) , dP × dt × dx a.e., (iii) U (t, x) ∈ ∂j (Y (t, x)) , dP × dt × dx a.e.,
˜ (iv) e2V j (Y ) ∈ L∞ 0, T ; L1 (Ω × D) , for all T > 0, Z τ ˜ (v) E e2Vs kU (s)k2H 1 (D) dQs < ∞ . 0

0

6

Appendix

In this section we first present some useful and general estimates on (Y, Z) ∈ S 0 [0, T ] × Λ0 (0, T ) satisfying an identity of type Yt = YT +

Z

T

dKs −

t

Z

t

T

Zs dWs , t ∈ [0, T ] , P-a.s.,

where K ∈ S 0 [0, T ] and t 7−→ Kt (ω) is a bounded variation function, P-a.s. The following results and their proofs are given in monograph E. Pardoux, A. R˘ a¸scanu [17], Annex C (a forthcoming book). Assume there exist: ♦ three progressively measurable increasing continuous stochastic processes D, R, N such that D0 = R0 = N0 = 0, ♦ a progressively measurable bounded variation continuous stochastic process V with V0 = 0, ♦ some constants a, p > 1, such that, as signed measures on [0, T ] :


np dDt + hYt , dKt i ≤ 1p≥2 dRt + |Yt |dNt + |Yt |2 dVt + kZt k2 dt, 2a

(74)

where n p = (p − 1) ∧ 1. Let Y eV [t,T ] := sups∈[t,T ] Ys eVs .

Proposition 16 Assume (74) and that

p E Y eV [0,T ] + E

Z

T

2Vs

e

1p≥2 dRs

0

31

p/2

+E

Z

T

Vs

e dNs 0

p

< ∞.

(75)

Then there exists a positive constant C = C (a, p) such that, P-a.s., for all t ∈ [0, T ] : " p/2 # Z T p/2 Z T p e2Vs kZs k2 ds + e2Vs dDs EFt sups∈[t,T ] eVs Ys + t

t

+E

Ft

Z

T

pVs

e

t

≤CE

Ft

|Ys |

p−2

1Ys 6=0 dDs +

Z

T

pVs

e

t

" Z V p T e YT +

T

2Vs

e

1p≥2 dRs

t

p/2

|Ys |

+

p−2

Z

2

1Ys 6=0 kZs k ds

T

Vs

e dNs t

p #



(76)

.

In particular for all t ∈ [0, T ] : h i
+ |Yt |p ≤ C EFt |YT |p + 1p≥2 RTp + NTp ep||(V· −Vt ) ||[t,T ] , P-a.s. As a simple consequence we can deduce, from the above Proposition, an estimate for the stochastic processes (ξ, ζ) associated to η as in Proposition 2: Corollary 17 Let (Vt )t≥0 be a bounded variation and continuous p.m.s.p. with V0 = 0,
η : Ω → H a random variable such that E ep supr∈[0,T ] Vr |η|p < ∞, and (ξ, ζ) ∈ S 0 ×Λ0 (0, ∞) the unique solution of the following equation (see the martingale representation formula (5)): Z T FT ζr dWr , s ∈ [0, T ] , a.s. ξs = E η − s

Therefore, there exists C = C (p) > 0 such that for all t ∈ [0, T ], E

Ft

pVs

sups∈[t,T ] e

p

|ξs | + E

Ft

Z

T

2Vs

e t

2

|ζs | ds

p/2


≤ C EFt ep supr∈[t,T ] Vr |η|p .

Proof. We see at once that the stochastic pair (ξ, ζ) satisfy the equation Z T ζr dWr , s ∈ [0, T ] , a.s. ξs = ξT −

(77)

(78)

s


For any fixed t ∈ [0, T ] let V¯st s∈[0,T ] be the increasing continuous p.m.s.p. defined by V¯st = Vt , s < t, and V¯st = supr∈[t,s] Vr , s ≥ t. Applying Jensen’s inequality and Proposition 16 for (ξ, ζ) (which satisfies (78) and an inequality of type (74), with K = 0 and R = N = 0), we deduce that for all p > 1, there exists C = C (p) > 0 such that Z T p/2 p F pV F t s t e2Vs |ζs |2 ds E sups∈[t,T ] e |ξs | + E t

¯t

≤ EFt sups∈[t,T ] epVs |ξs |p + EFt ≤C

EFt

¯t epVT

p


|ξT |

32

≤C

EFt

Z

T

¯t

e2Vs |ζs |2 ds

t p supr∈[t,T ] Vr

e

p/2


|η|p .

Let us now discuss the existence and uniqueness of a solution for the backward stochastic equation of the form Z T Z T Zs dBs , a.s., ∀ t ∈ [0, T ] . (79) Φ (s, Ys , Zs ) dQs − Yt = η + t

t

We will need the following basic assumptions: (A′3 ) the process {Qt : t ≥ 0} is a progressively measurable increasing continuous stochastic process such that Q0 = 0, and {αt : t ≥ 0} is a real positive p.m.s.p. (given by RadonNikodym’s representation theorem) such that α ∈ [0, 1] and dt = αt dQt ; (A′4 ) the function Φ : Ω × [0, ∞) × H × L2 (H1 , H) → H is such that ( Φ (·, ·, y, z) is p.m.s.p., ∀ (y, z) ∈ H × L2 (H1 , H) , Φ (ω, t, ·, ·) is continuous function, dP ⊗ dt-a.e. ,

and P-a.s., Z

T

0

where

Φ# ρ (s) dQs < ∞,

∀ ρ ≥ 0,

Φ# ρ (ω, s) := sup|u|≤ρ |Φ (ω, s, u, 0)| ; (A′5 ) there exist a p.m.s.p. µ : Ω × [0, ∞) → R and a function ℓ : [0, ∞) → [0, ∞) such that Z

T 0

|µt |dQt +

Z

T 0

ℓ2 (t) dt < ∞,

P-a.s.

and, for all y, y ′ ∈ H, z, z ′ ∈ L2 (H1 , H) , hy ′ − y, Φ(t, y ′ , z) − Φ(t, y, z)i ≤ µt |y ′ − y|2 ,

|Φ(t, y, z ′ ) − Φ(t, y, z)| ≤ ℓ (t) αt |z ′ − z| . Let a > 1 and Vt =

Z

0

where np = 1 ∧ (p − 1) .

t

µs +


a 2 ℓ (s) αs dQs , 2np

Proposition 18 Let p > 1 and η : Ω → H be a random variable measurable with respect to σ ({Ft : t ≥ 0}). Under the hypotheses (A′3 −A′5 ), if moreover, pVT

E e

|η|

p


+E

Z

T 0


sup|y|≤ρ eVt Φ t, e−Vt y, 0 − µs y dQs 33

p

< ∞, for all ρ ≥ 0,

(80)

there exists a unique pair (Yt , Zt )t≥0 ∈ S 0 × Λ0 solution of the BSDE (79) in the sense that (j) Yt = η +

Z

T t

Φ (s, Ys , Zs ) dQs −

(jj) E supt∈[0,T ] epVt |Yt |p + E

Z

T 0

Z

t

T

Zs dBs , a.s., ∀t ∈ [0, T ] ,

e2Vs |Zs |2 ds

p/2

(81)

< ∞.

Remark 19 If (Vt )t≥0 is a deterministic process then the assumption (80) is equivalent to p

E (|η| ) + E

Z

T 0

Φ# ρ (s)dQs

p

< ∞.

(82)

References [1] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach spaces, Springer Monographs in Mathematics, Springer, New York, 2010. [2] H. Br´ezis, Op´erateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973. [3] Ph. Briand, B. Delyon, Y. Hu, E. Pardoux, L. Stoica, Lp solutions of backward stochastic differential equations, Stochastic Process. Appl. 108 (2003), 109-129. [4] J. Cvitanic, I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab. 24 (1996), 2024-2056. [5] G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, Great Britain, 1992. [6] R.W.R. Darling, E. Pardoux, Backward SDE with random terminal time and applications to semilinear elliptic PDE, Ann. Probab. 25 (1997) 1135-11593. [7] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, M.C. Quenez, Reflected solutions of backward SDE’s and related obstacle problems for PDE’s, Ann. Probab. 25 (1997) 702-737. [8] M. Fuhrman, G. Tessitore, Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces, Ann. Probab. 32 (1B) (2004), 607–660. [9] M. Fuhrman, G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control, Ann. Probab. 30 (3) (2002), 1397-1465. [10] L. Maticiuc, A. R˘ a¸scanu, A stochastic approach to a multivalued Dirichlet-Neumann problem, Stochastic Process. Appl. 120 (2010), 777-800.

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[11] L. Maticiuc, A. R˘ a¸scanu, Viability of moving sets for a nonlinear Neumann problem, Nonlinear Anal. 66 (2007), 1587–1599. [12] E. Pardoux, BSDEs, weak convergence and homogenization of semilinear PDEs, Nonlinear Analysis, Differential Equations and Control (Montreal, QC, 1998), Kluwer Academic Publishers, Dordrecht (1999), 503–549. [13] E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), 55-61. [14] E. Pardoux, S. Peng, Backward SDE’s and quasilinear parabolic PDE’s, Stochastic PDE and Their Applications (B.L. Rozovskii, R.B. Sowers eds.), LNCIS 176, Springer (1992) 200-217. [15] E. Pardoux, A. R˘ a¸scanu, Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Process. Appl. 76 (1998) 191-215. [16] E. Pardoux, A. R˘ a¸scanu, Backward stochastic variational inequalities, Stochastics 67 (3-4) (1999) 159–167. [17] E. Pardoux, A. R˘ a¸scanu, Stochastic differential equations, book submitted, 2011.

35