Auguste Aman

Modeste N'zi

Reected Backward Doubly Stochastic Dierential Equations and Application

Keywords: Backward doubly SDEs, Stochastic PDE, Obstacle problem. MSC: 60H15; 60H20

This paper is devoted to study the class of reected backward doubly stochastic dierential equation (RBDSDE, for short). We rst prove existence and uniqueness result under Lipschitz condition on the coecient (drift) via the penalization method. As application we derive the obstacle problem of stochastic PDE.

Introduction

The rst approach of linear backward stochastic dierential equations (BSDE, for short) is been initiated by Bismut (see [START_REF] Bismut | Conjuquate convex function in optimalstochastic control[END_REF][START_REF] Bismut | An introductory approach to duality in stochastic control[END_REF]). But since the fundamental paper on this theory was published by Pardoux and Peng [START_REF] Pardoux | Adapted solution of backward stochastic dierential equation[END_REF] in 1990, BSDEs have been taken with great interest in more and more researches. For instance, the theory of BSDE has been found to be powerful tool for mathematical economics and particular, for mathematical nance ( [START_REF] El Karoui | Backward stochastic dierential equations in nance[END_REF] and it reference). Another main reason is due to their connection with the probabilistic interpretation of partial dierential equation (PDE for short)(see [START_REF] Pardoux | Backward stochastic dierential equations and quasilinear parabolic partial dierential equations[END_REF][START_REF] Peng | Probabilistic interpretation for systems of quasilinear parabolic partial dierential equations[END_REF]), variational inequalities and obstacle problems ( [START_REF] Karoui | Reected solution of backward SDE's, and related obstacle problem for PDE's[END_REF]) stochastic games [START_REF] Hamadène | Zero-sum stochastic dierential games and BSDEs[END_REF], and so one. On the other hand in order to give a probabilistic interpretation to the stochastic PDE and generalized a well-know Feynman-Kac formula, Pardoux and Peng [START_REF] Pardoux | Backward doubly stochastic dierential equations and systems of quasilinear SPDEs[END_REF] introduced a new kind of equation. They called it backward doubly stochastic dierential equations (BDSDE for short). It has two dierent direction of stochastic integral involve with two independents standard Brownian motion: the standard (forward) dW t and a backward stochastic integral dB t . Since BDSDE have studied by many authors. Indeed, Buckdahn and Ma [START_REF] Buckdahn | Stochastic viscosity solutions for nonlinear stochastic partial differential equations[END_REF][START_REF] Buckdahn | Stochastic viscosity solutions for nonlinear stochastic partial dierential equations[END_REF][START_REF] Buckdahn | Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs[END_REF] introduced the 1 notion of stochastic viscosity solution to stochastic PDE under the Lipschitz coecient of the BDSDE that coincides with the well-know viscosity solution of PDE if the coecient in the integral which respect dB t is null in the SPDE. But the BDSDE introduced by buckdahn and Ma is in fact a time reversal of that considered by Pardoux and Peng. However they have similar nature and called by the same name. Moreover in view of the work of Buckdahn and Ma [START_REF] Buckdahn | Stochastic viscosity solutions for nonlinear stochastic partial differential equations[END_REF][START_REF] Buckdahn | Stochastic viscosity solutions for nonlinear stochastic partial dierential equations[END_REF] Aman [START_REF] Aman | Representation theorems for backward doubly stochastic dierential equations[END_REF][START_REF] Aman | Paths regularity for solutions of backward doubly stochastic dierential equations[END_REF] derive respectively a representation and path regularity result that generalized the one give by Ma and Zhang [START_REF] Ma | Representation theorems for Backward stochastic dierential Equations[END_REF]15] to the BSDE. We note also note that a comparison theorem of BDSDE has been given by Gu and al [START_REF] Shi | Comparison theorem of backward doubly stochastic dierential equations and application[END_REF]. This result will play a important role in the proof of our main result. In fact, the goal of this paper is to study the related obstacle problem for SPDE and extend the notion of stochastic viscosity solution to the reected SPDE via the reected BDSDE. The prove used penalization method (see El Karoui et al for the case of BSDE). The rest of the paper is organized as follows. Next section is devoted to claim and prove our main result. In the last section we apply the main result to study the related obstacle problem for SPDE.

2 Reected Backward doubly stochastic dierential equation

Preliminaries

Notation. The scalar product of the space IR d , (d ≥ 2) will be denote by < . > and the Euclidian norm associated by . .

For the remaining of the paper, let us x a positive real number T > 0. First of all {W t , 0 ≤ t ≤ T } and {B t , 0 ≤ t ≤ T } are two mutually independent standard Brownian motion with values respectively in IR d and IR , dened respectively on the two probability space (Ω 1 , F 1 , IP 1 ) and (Ω 2 , F 2 , IP 2 ). Let F B = {F B } t≥0 denote the natural ltration generated by B, augmented by IP 1 -null set of F 1 ; and let

F B = F B ∞ .
On the other hand we consider the following family of σ-elds:

F W t,T = σ{W s -W T , t ≤ s ≤ T } ∨ N 2 ,
where

N 2 denotes all the IP 2 -null set in F 2 . Denote F W T = {F W t,T } 0≤t≤T .
Next we consider the product space (Ω, F, IP) where

Ω = Ω 1 × Ω 2 , F = F 1 ⊗ F 2 and IP = IP 1 × IP 2 ;
For each t ∈ [0, T ], we dene

F t = F B t ⊗ F W t,T .
Let us remark that the collection F = {F t , t ∈ [0, T ]} is neither increasing nor decreasing and it does not constitute a ltration.

Further, we assume that random variables ξ(ω 1 ), ω 1 ∈ Ω 1 and ζ(ω 2 ), ω 2 ∈ Ω 2 are considered as random variables on Ω via the following identication:

ξ(ω 1 , ω 2 ) = ξ(ω 1 ); ζ(ω 1 , ω 2 ) = ζ(ω 2 ).
In this paper we will always use this collection.

For any d ≥ 2, let M 2 (0, T, IR d or IR ) of (classes of dP⊗dt a.e. equal) d-dimensional jointly measurable stochastic processes ϕ t ; t ∈ [0, T ], which, satisfy :

(i) ϕ 2 M 2 = IE T 0 |ϕ t | 2 dt < ∞; (ii) ϕ is F t -measurable , for any t ∈ [0, T ].
Similarly, we denote by S 2 ([0, T ], IR) the set of one dimensional continuous stochastic processes, which verify:

(iii) ϕ 2 S 2 = IE( sup 0≤t≤T |ϕ t | 2 ) < ∞; (iv) ϕ is F t -measurable , for any t ∈ [0, T ].
As mentioned in introduction this paper studies a reected BDSDE that is an equation of the following type:

Y t = ξ + t 0 f (s, Y s , Z s )ds + t 0 g(s, Y s , Z s ) dB s - t 0 Z s ↓ dW s + K t , 0 ≤ t ≤ T.
(2.1)

f and g are called generators and ξ the terminal condition.

Denition 2.1 By solution of the reected BDSDE we mean a triplet

(Y, Z, K) = {Y t , Z, K t } t∈[0,T ] ∈ S 2 ([0, T ]; IR) × M 2 (0, T ; IR d ) × S 2 ([0, T ]
; IR) such that for a given barrier process S = {S t } t≥0 , the following holds IP-a.s

(i) Y t ≥ S t , 0 ≤ t ≤ T ,
(ii) K is an increasing process such that K 0 = 0 and T 0 (Y t -S t ) dK t = 0;

(iii) processes (Y, Z, K) verify equation (2.1).

In the sequel we will use the notation reected BDSDE (ξ, f, g, S) to say that we consider the reected BDSDE whose generator is f and g, with terminal condition ξ and S = {S t } t≥0 a barrier process.

Existence and Uniqueness result

In this subsection, we consider reected BDSDE (ξ, f, g, S) when generators, terminal condition and barrier process satisfy those assumptions:

(A1) f : Ω × [0, T ] × IR × IR d → IR; g : Ω × [0, T ] × IR × IR d → IR be jointly measurable such that for any (y, z) ∈ IR × IR d f (., y, z) ∈ M 2 (0, T, IR), g(., y, z) ∈ M 2 (0, T, IR )
Moreover, we assume that there exist constants C > 0 and 0 < α < 1 such that for any

(ω, t) ∈ Ω × [0, T ]; (y 1 , z 1 ), (y 2 , z 2 ) ∈ IR × IR d , H1    |f (t, y 1 , z 1 ) -f (t, y 2 , z 2 )| 2 ≤ C(|y 1 -y 2 | 2 + z 1 -z 2 2 ) g(t, y 1 , z 1 ) -g(t, y 2 , z 2 ) 2 ≤ C|y 1 -y 2 | 2 + α z 1 -z 2 2 (A2) ξ is a F 0 -measurable variable such that IE(|ξ| 2 ) < +∞ (A3) (S t
) t≥0 is a continuous progressively measurable real-valued process satisfying:

(i) IE sup 0≤t≤T (S + t ) 2 < +∞ (ii)
S T ≤ ξ P a.s.

Our objectif is to construct a solution of reected BDSDE (ξ, f, g, S). As we explained in introduction, our method (penalization method) relies heavily on a priori estimate and the comparaison theorem.

For each n ∈ IN * we set

f n (s, y, z) = f (s, y, z) -n(S s -y) -
and considered the following BDSDE

Y n t = ξ + t 0 f n (s, Y n s , Z n s )ds + t 0 g(s, Y n s , Z n s ) dB s - t 0 Z n s ↓ dW s (2.2)
that is the penalized version of reected BDSDE (ξ, f, g, S). Then it is known from Pardoux and Peng in [START_REF] Pardoux | Backward doubly stochastic dierential equations and systems of quasilinear SPDEs[END_REF], that for

t ∈ [0, T ] the BDSDE (2.2) has a unique solution (Y n , Z n ) ∈ S 2 ([0, T ]; IR)× M 2 (0, T ; IR d ). Lemma 2.2 Let us consider (Y n , Z n ) ∈ S 2 ([0, T ]; IR)×M 2 (0, T ; IR d ) solution of BDSDE (2.2).
Then there exists

C > 0 such that for each n ∈ IN IE sup 0≤t≤T |Y n t | 2 + T 0 + Z n s 2 ds + |K n T | 2 < C
where

K n t = n t 0 (Y n s -S s ) -ds, 0 ≤ t ≤ T.
Proof.

From Itô's formula, it follows that

|Y n t | 2 + t 0 |Z n s | 2 ds = |ξ| 2 + 2 t 0 Y n s f (s, Y n s , Z n s )ds + 2 t 0 Y n s g(s, Y n s , Z n s ), dB s (2.
3)

+ t 0 g(s, Y n s , Z n s ) 2 ds + 2 t 0 Y n s dK n s -2 t 0 Y n s Z n s , ↓ dW s .
By using (H1) we have

2Y n s f (s, Y n s , Z n s ) ≤ β|Y n s | + 1 β |f (s, Y n s , Z n s )| 2 ≤ (β + C β )|Y n s | 2 + C β Z n s 2 + 1 β |f (s, 0, 0)| 2 (2.4) and g(s, Y n s , Z n s ) 2 ≤ (1 + β ) g(s, Y n s , Z n s ) -g(s, 0, 0) 2 + ( 1 β + 1) g(s, 0, 0) 2 (2.5) ≤ C(1 + β )|Y n s | 2 + α(1 + β ) Z n s 2 + ( 1 β + 1) g(s, 0, 0) 2 .

Moreover since

t 0 (Y n s -S s )dK n s ≤ 0 we obtain t 0 Y n s dK n s = t 0 (Y n s -S s )dK n s + t 0 S s dK n s ≤ t 0 S s dK n s .
(2.6)

If we choose β and β such that

C β + α(1 + β ) ≤ 1 it follows from (2.3) -(2.6) that IE |Y n t | 2 + T 0 Z n s 2 ds ≤ CIE |ξ| 2 + T 0 |Y n s | 2 ds + T 0 |f (s, 0, 0)| 2 ds + T 0 g(s, 0, 0) 2 ds + T 0 S s dK n s . Gronwall's lemma applied to Y n gives IE |Y n t | 2 ≤ CIE |ξ| 2 + T 0 |f (s, 0, 0)| 2 ds + T 0 g(s, 0, 0) 2 ds + T 0 S s dK n s .
(2.7)

Then we have

IE T t Z n s 2 ds ≤ C IE |ξ| 2 + T 0 |f (s, 0, 0)| 2 ds + T 0 g(s, 0, 0) 2 ds + T 0 S s dK n s .
(2.8)

We now give an estimation of IE(|K n T | 2 . From the equation

K n T = Y T -ξ - T 0 f (s, Y n s , Z n s )ds - T 0 g(s, Y n s , Z n s ) dB s + T 0 Z n s ↓ dW s (2.9)
and inequalities (2.7) and (2.8), we show the following inequalities

IE |K n T | 2 ≤ CIE |ξ| 2 + T 0 |f (s, 0, 0)| 2 ds + T 0 g(s, 0, 0) 2 ds + T 0 S s dK n s . ≤ CIE |ξ| 2 + T 0 |f (s, 0, 0)| 2 ds + T 0 g(s, 0, 0) 2 ds + γ sup 0≤t≤T (S + t ) 2 + 1 γ IE(|K n T | 2 ).
Choosing γ such C γ < 1 we get

IE |K n T | 2 ≤ CIE |ξ| 2 + T 0 |f (s, 0, 0)| 2 ds + T 0 g(s, 0, 0) 2 ds + sup 0≤t≤T (S + t ) 2 .
(2.10)

Combining (2.7), (2.8) and (2.10), it follows that

IE |Y n t | 2 + T 0 Z n s 2 ds + |K n T | 2 ≤ CIE |ξ| 2 + T 0 |f (s, 0, 0)| 2 ds + T 0 g(s, 0, 0) 2 ds + sup 0≤t≤T (S + t ) 2 . (2.11)
Finally applied Burkhölder-Gundy-Davis inequality we have

IE sup 0≤t≤T |Y n t | 2 + T 0 Z n s 2 ds + |K n T | 2 ≤ CIE |ξ| 2 + T 0 |f (s, 0, 0)| 2 ds + T 0 g(s, 0, 0) 2 ds + sup 0≤t≤T (S + t ) 2 (2.
12) which end the prove of this Lemma

The main result of this section is the following. Theorem 2.3 Assume that (A1), (A2), (A3) hold. Moreover if the obstacle process (S t ) is the Itô process in the form

dS t = m t dt + n t dB t + v t dW t , with IE T 0 |m s | 2 + |v s | 2 + |n s | 2 ds < +∞.
Then there exists a unique triple (Y, Z, K) solution of the reected BDSDE (ξ, f, g, S) . Further,

lim n-→+∞ IE sup 0≤t≤τ |K n t -K t | 2 = 0.
Proof.

This proof will be spilt in two steps. In the sequel, C denotes a positive constant which may vary from one line to another.

Existence

According it denition we have f n (t, y, z) ≤ f n+1 (t, y, z). Thanks to the comparison theorem (see Th.3.1 in Yufeng Shi et al [START_REF] Shi | Comparison theorem of backward doubly stochastic dierential equations and application[END_REF]) we deduce that the sequence (Y n ) n>0 is non-decreasing. Hence there exists a progressively measurable process Y such that Y n t Y t a.s. Then, according to Fatou's Lemma and Lemma 2.2, we deduce that

IE sup 0≤t≤T |Y t | 2 < +∞.
Moreover, the Lebegue's dominated convergence theorem ensures that

IE T 0 |Y n s -Y s | 2 ds -→ 0, as n → ∞.
(2.13)

On the other hand, for any n ≥ p, it follows by Itô's formula that

|Y n t -Y p t | 2 + t 0 |Z n s -Z p s | 2 ds = 2 T t (Y n s -Y p s )[f (s, Y n s , Z n s ) -f (s, Y p s , Z p s )]ds + t 0 g(s, Y n s , Z n s ) -g(s, Y p s , Z p s ) 2 ds +2 t 0 (Y n s -Y p s ) (g(s, Y n s , Z n s ) -g(s, Y p s , Z p s )), dB s -2 t 0 (Y n s -Y p s ) (Z n s -Z p s ), ↓ dW s +2 t 0 (Y n s -Y p s )(dK n s -dK p s ).
(2.14)

Following the same computation as in Lemma 2.1, taking the expectation and in virtue of assumptions (H1) we obtain

E |Y n s -Y p s | 2 + T 0 Z n s -Z p s 2 ds ≤ cIE T 0 |Y n s -Y p s | 2 ds + T 0 (Y n s -Y p s )(dK n s -dK p s ) . (2.15)
But it clearly seen that

IE T t (Y n s -Y p s )(dK n s -dK p s ) ≤ IE sup 0≤s≤T (Y n s -S s ) -K n T + sup 0≤s≤T (Y p s -S s ) -K p T
and it follows from (2.15), the Hölder inequality and Lemma 2.1 that

E |Y n s -Y p s | 2 + T 0 Z n s -Z p s 2 ds ≤ cIE T 0 |Y n s -Y p s | 2 ds + sup 0≤s≤T | (Y n s -S s ) -| 2 + sup 0≤s≤T | (Y p s -S s ) -| 2 .
(2.16)

If we admit for a moment the following Lemma 2.4

IE sup 0≤t≤T (Y n t -S t ) -2 -→ 0, as n -→ ∞;
then we can conclude now. Indeed by passing of limit in (2.16) and using Lemma 2.2, we deduce that

IE |Y n s -Y p s | 2 + T 0 |Z n s -Z p s | 2 ds -→ 0, as n, p -→ ∞.
(2.17)

Taking sup 0≤s≤T (.) in (2.14) and using Bürkhölder-Davis-Gundy's inequality, it holds that

IE sup 0≤s≤T |Y n s -Y p s | 2 -→ 0, as n, p -→ ∞, (2.18)
and then from (2.9), we derive easily (Y s -S s )dK s = 0, a.s.

IE sup 0≤s≤T |K n s -K p s | 2 -→ 0, as n, p → ∞. ( 2
(Y, Z, K) ∈ S 2 ([0, T ]; IR) × M 2 (0, T ; IR d ) × S 2 ([0, T ]; IR) such that IE sup 0≤s≤T |Y n s -Y s | 2 + T 0 |Z n s -Z s | 2 ds + sup 0≤s≤T |K n s -K s | 2 → 0, as n → ∞. ( 2
Finally passing to the limit in (2.2) we proved that (Y, Z, K) solves the reected BDSDE associated to data (ξ, f, g, S). Now to end let us return to the proof of Lemma 2.2. Let ( Y t n , Z n t ), 0 ≤ t ≤ T be the unique solution of the BDSDE

Proof of Lemma

Y n t = ξ + t 0 f (s, Y n s , Z n s )ds + t 0 g(s, Y n s , Z n s ) dB s +n t 0 (S s -Y n s )ds - t 0 Z n s ↓ dW s , 0 ≤ t ≤ T
From the comparison theorem for BDSDE (see [START_REF] Shi | Comparison theorem of backward doubly stochastic dierential equations and application[END_REF]), we have for every

n ∈ IN * , Y n t ≤ Y n t , a.s.
On the other hand, let ν be a stopping time such that 0 ≤ ν ≤ T . Hence it follows from Itô's formula applying to e -n(ν-t) Y t that

Y n ν = e -n(ν) ξ + ν 0 e -n(ν-s) f (s, Y n s , Z n s )ds + ν 0 e -n(ν-s) g(s, Y n s , Z n s )dB s + n ν 0 e -n(ν-s) S s ds - ν 0 e -n(ν-s) Z n s dW s (2.21)
Taking the conditional expectation of both side of (2.21) we get

Y n ν = IE e -n(ν) ξ + ν 0 e -n(ν-s) f (s, Y n s , Z n s )ds + ν 0 e -n(ν-s) g(s, Y n s , Z n s )dB s +n ν 0 e -n(ν-s) S s ds | F ν (2.22)
It easily seen that

e -n(ν) ξ + n ν 0 e -n(s-ν) S s ds -→ ξ1 {ν=0} + S ν 1 {ν>0}
(2.23) a.s.and in L 2 (Ω), the conditional expectation converge also in L 2 (Ω). Moreover, since

ν 0 e -n(s-ν) f (s, Y n s , Z n s )ds ≤ 1 √ 2n ν 0 |f (s, Y n s , Z n s )| 2 ds 1 2
and

IE ν 0 e -n(ν-s) g(s, Y n s , Z n s ) dB s = 0 Y n ν = IE ν 0 e -n(ν-s) f (s, Y n s , Z n s )ds + ν 0 e -n(ν-s) g(s, Y n s , Z n s )dB s | F ν -→ 0 in L(Ω), as n -→ ∞. Consequently, Y n ν -→ ξ1 {ν=0} + S ν 1 {ν>0} in L 2 (Ω) and further Y ν ≥ S ν a.s.
(2.24) From (2.24) and the section theorem in Dellacherie and Meyer [START_REF] Dellacherie | Probabilities and Potential[END_REF], ( page 220), we get that

Y t ≥ S t a.s.
Hence (Y n t -S t ) -↓ 0, a.s. and according Dini's theorem, the convergence is uniform in t.

Finally, since

(Y n t -S t ) -≤ (S t -Y 0 t ) + ≤ |S t | + Y 0 t , the dominated convergence theorem en- sures that lim n-→+∞ IE( sup 0≤t≤T | (Y n t -S t ) -| 2 ) = 0.

Uniqueness

Let us dne

{(∆Y t , ∆Z t , ∆K t ) , 0 ≤ t ≤ T } = {(Y t -Y t , Z t -Z t , K t -K t ), 0 ≤ t ≤ T }
where {(Y t , Z t , K t ) , 0 ≤ t ≤ τ } and {(Y t , Z t , K t ), 0 ≤ t ≤ T } denote two solutions of the reected BDSDE associated to the data (ξ, f, g, S). Let us rst note that T t ∆Y s ∆dK s ≤ 0.

(2.25)

Moreover, Itô formula yields that for every 0 ≤ t ≤ T

|∆Y t | 2 + t 0 Z s 2 ds = 2 t 0 Y s (f (s, Y s , Z s ) -f (s, Y s , Z s ))ds + t 0 g(s, Y s , Z s ) -g(s, Y s , Z s ) 2 ds + t 0 Y s , (g(s, Y s , Z s ) -g(s, Y s , Z s )) dB s -2 t 0 Y s , Z s dW s + 2 t 0 Y s dK s .
Then, by using similar computation as the proof of existence and using (2.25) one have

IE |∆Y t | 2 + T 0 Z s 2 ds ≤ CIE T 0 | Y s |ds
from which, we deduce that ∆Y t = 0 and further ∆Z t = 0. On the other hand since

∆K t = ∆Y t - t 0 f (s, Y s , Z s ) -f (s, Y s , Z s )ds - t 0 (g(s, Y s , Z s ) -g(s, Y s , Z s ))dB s + t 0 Z s dW s ,
we have ∆K t = 0 which end the proof of the theorem.

3 Reected BDSDE and obstacle problem for a non linear parabolic stochastic PDE

Preliminaries and denitions

According notations of subsection 2.1, let recall F B = {F B t } 0≤t≤T and M B 0,T denote all the F Bstopping times τ such 0 ≤ τ ≤ T , a.s. For generic Euclidean spaces E and E 1 we introduce the

eld u : Ω 1 × [0, T ] × IR → IR which satises (f, g, h, l)            min {u(t, x) -h(t, x), du(t, x) + [Lu(t, x) + f (t, x, u(t, x), σ * (x)D x u(t, x))]dt +g(t, x, u(t, x)) dB s } = 0, (t, x) ∈ [0, T ] × IR d u(0, x) = l(x), x ∈ IR d where L = 1 2 d i,j (σ(x)σ * (x)) i,j ∂ 2 ∂x i ∂x j + d i=1 b i (x) ∂ ∂x i .
More precisely, we shall consider the solution of SPDE associated to the data (f, g, f, h, l) in the two stochastic viscosity sense, inspired respectively by the work of Buckdahn and Ma [START_REF] Buckdahn | Stochastic viscosity solutions for nonlinear stochastic partial differential equations[END_REF][START_REF] Buckdahn | Stochastic viscosity solutions for nonlinear stochastic partial dierential equations[END_REF] and [START_REF] Buckdahn | Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs[END_REF]. It will convenient for the sequel to dene this two notion of stochastic viscosity for SPDE (f, g, h, l). Let us remark that the second stochastic viscosity notion use stochastic g-superjet and g-subjet notion. So Before give that denition, it is reasonable to recall the notion of stochastic g-superjet or subjet.

Denition 3.1 Let τ ∈ M B 0,T and ξ ∈ L 0 (F B τ ; IR d ).
We say that a sequence of random variables

(τ k , ξ k ) is a (τ, ξ)-approximating sequence if (τ k , ξ k ) ∈ M B 0,T × L 2 (F B τ ; IR d ), ∀ k such that (i) ξ k -→ ξ in probability (ii) either τ k ↑ τ a.s. and τ k < τ on {τ > 0}.
If {ξ k } is a sequence of random variable that converge to 0 in probability, then we shall denote θ k = o(ξ k ), k = 1, 2, ..., ... to be any sequence of random variables such that

[o(ξ k )/ξ k ]I {ξ k =0} -→ 0 as k -→ ∞ in probability. Denition 3.2 Assume g ∈ C 0,1 ([0, T ] × R d+1 ; IR ). Let (τ, ξ) ∈ M B 0,T × L 2 (F τ , IR), and u ∈ C(F B ; [0, T ] × R d ). A triplet (a, p, X) is called a stochastic g-superjet of u at (τ, ξ) if the following hold (i) (a, b, c, p, q, X) is an IR × IR × IR × IR n × IR n × S(n)-valued F τ -measurable random vector; (ii) denoting    b = g(τ, ξ, u(τ, ξ)) c = (g∂ u g)(τ, ξ, u(τ, ξ)) q = ∂ x g(τ, ξ, u(τ, ξ)) + ∂ u g(τ, ξ, u(τ, ξ))p
then for any (τ, ξ)-approximation sequence (τ k , ξ k ) it holds that

u(τ k , ξ k ) ≤ u(τ, ξ) + a(τ k -τ ) + b(B k -B τ ) + c 2 (B k -B τ ) 2 + < p, ξ k -ξ > + < q, ξ k -ξ > (B k -B τ ) + 1 2 < X(ξ k -ξ), ξ k -ξ > +o(|τ k -τ |) + o(|ξ k -ξ| 2 ). (3.1)
We denote the set of all stochastic g-superjet of u at (τ, ξ) by J 1,2+ u(τ, ξ). Similarly, we say that the triple (a, p, X) is a stochastic g-subjet of u at (τ, ξ) if (i) holds and the inequality in (3.1) is reversed; and we denote the set of stochastic g-subjet by J 1,2-u(τ, ξ). Now we will be able to give the two denitions of stochastic viscosity solution of the reected SPDE (f, g, h, l). For this end let us recall some notations in [START_REF] Buckdahn | Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs[END_REF]. Indeed, we consider process

η ∈ C(F B , [0, T ] × IR d × IR)
Further the mapping y → η(s, x, y) denes a dieomorphism for all t, x a.s. (see Protter, 1990 [?]). Denote the y-inverse of η(s, x, y) by ε(s, x, y). Then since ε(s, x, η(s, x, y)) = y, one can show that (see Buckdhan and Ma [START_REF] Buckdahn | Stochastic viscosity solutions for nonlinear stochastic partial differential equations[END_REF][START_REF] Buckdahn | Stochastic viscosity solutions for nonlinear stochastic partial dierential equations[END_REF])

ε(t, x, y) = y - t 0
D y ε(s, x, y)g(s, x, y) • dB s .

(3.3)

To simplify the notation in the sequel we denote

A f,g (ϕ(t, x)) = Lϕ(t, x) + f (t, x, ϕ(t, x), σ * D x ϕ(t, x)) - 1 2 (g, D y g)(t, x, ϕ(t, x)). Denition 3.3 (a) A random eld u ∈ C(F B , [0, T ] × IR d
) is called a stochastic viscosity subsolution of the SPDE (f, g, h, l) if u(0, x) ≤ l(x), x ∈ IR d and if for any stopping time

τ ∈ M B 0,T , any state variable ξ ∈ L 0 (F B τ ; IR d ), any random eld ϕ ∈ C 1,2 (F B τ , [0, T ] × IR)
with the property that for almost all ω ∈ {0 < τ < T }, u(t, x) -η(t, x, ϕ(t, x)) ≤ 0 = u(τ (ω), ξ(ω)) -η(τ (ω), ξ(ω), ϕ(τ (ω), ξ(ω))) is fullled for all (t, x) in some neighborhood V(ω, τ (ω), ξ(ω)) of (τ (ω), ξ(ω)), the following condition is satised:

min (u(τ, ξ) -h(τ, ξ), A f,g (ψ(τ, ξ)) -D y ψ(τ, ξ)D t ϕ(τ, ξ)) ≤ 0 hold almost surely. (b) A random eld u ∈ C(F B , [0, T ] × IR d
) is called a stochastic viscosity supersolution of the SPDE (f, g, h, l) if u(T, x) ≥ l(x), x ∈ IR d and if for any stopping time τ ∈ M B 0,t , any state variable ξ ∈ L 0 (F B τ ; R d ), any random eld ϕ ∈ C 1,2 (F B τ , [0, T ]×IR) with the property that for almost all ω ∈ {0 < τ < T }, u(t, x) -η(t, x, ϕ(t, x)) ≥ 0 = u(τ (ω), ξ(ω)) -η(τ (ω), ξ(ω), ϕ(τ (ω), ξ(ω))) is fullled for all (t, x) in some neighborhood V(ω, τ (ω), ξ(ω)) of (τ (ω), ξ(ω)), the following condition is satised:

min (u(τ, ξ) -h(τ, ξ), A f,g (ψ(τ, ξ)) -D y ψ(τ, ξ)D t ϕ(τ, ξ)) ≥ 0 hold almost surely. (c) u ∈ C(F B , [0, T ] × IR d
) is said to be a stochastic solution of SPDE(f, g, h, l) if it both a stochastic viscosity subsolution and a stochastic viscosity supersolution.

Denition 3.4 (a) A random eld u ∈ C(F B , [0, T ] × IR d ) is called a stochastic viscosity subsolution of SPDE (f, g, h, l) if u(0, x) ≤ l(x), ∀ x ∈ IR d and for any (τ, ξ) ∈ M B 0,T × L 2 (F B τ ; IR d ) and any (a, p, X) ∈ J 1,2+ u(τ, ξ) it holds that min u(τ, ξ) -h(τ, ξ), -a - 1 2 T ra(σσ * X) -b(x), p -f (τ, ξ, u(τ, ξ), pσ(τ, ξ)) + 1 2
(g∂ u g)(τ, ξ, u(τ, ξ)) ≤ 0. (3.4)

In the other words at any (τ, ξ)

∈ M B 0,T × L 2 (F B τ ; IR d ) where u(τ, ξ) > h(τ, ξ), -a - 1 2 T ra(σσ * X) -b(x), p -f (τ, ξ, u(τ, ξ), pσ(τ, ξ)) + 1 2 (g∂ u g)(τ, ξ, u(τ, ξ)) ≤ 0.
(b) u is said to be a stochastic viscosity supersolution if u(0, x) ≥ l(x), ∀ x ∈ IR d , and for any (τ, ξ) ∈ M B 0,T × L 2 (F B τ ; IR d ) and any (a, p, X) ∈ J 1,2-u(τ, ξ), (3.4) holds with the inequality being reversed. In the other words, at each

(τ, ξ) ∈ M B 0,T × L 2 (F B τ ; IR d ), we have both u(τ, ξ) ≥ h(τ, ξ) and -a - 1 2 T ra(σσ * X) -b(x), p -f (τ, ξ, u(τ, ξ), pσ(τ, ξ)) + 1 2 (g∂ u g)(τ, ξ, u(τ, ξ)) ≥ 0.
(c) If u is both a stochastic viscosity subsolution and supersolution, we say that u is a stochastic viscosity solution of SPDE (f, g, h, l).

Remark 3.5 Observe that if f, h are deterministic and g ≡ 0 the Denition 3.3 coincide with the deterministic case (see [START_REF] Karoui | Reected solution of backward SDE's, and related obstacle problem for PDE's[END_REF]).

Existence of stochastic viscosity solutions

In this subsection we apply the result of the previous section to prove the existence of stochastic viscosity solutions with respect Denition 3.4 to obstacle problem of quasi-linear SPDE. Let consider u a random eld dened on

Ω 1 × [0, T ] × IR d by u(t, x) = Y t,x t , (t, x) ∈ [0, T ] × IR d , (3.5)
which is

F B t -measurable for each t ∈ [0, T ]; in the other words, u ∈ C(F B , [0, T ] × IR d ).
Theorem 3.6 u dened by (3.5), is the stochastic viscosity solution with respect Denition 3.4

of the obstacle problem of SPDE (f, g, l, h).

Proof. For each

(t, x) ∈ [0, T ] × IR n , n ≥ 1, let { n Y t,x s , n Z t,x s , 0 ≤ s ≤ t} denote the solution of the BDSDE n Y t,x s = l(X t,x T ) = s 0 f (r, X t,x r , n Y t,x r , n Z t,x r )dr + n s 0 ( n Y t,x r -h(r, X t,x r )) -dr + s 0 g(r, X t,x r , n Y t,x r )dB r - s 0 n Z t,x r dW r .
It is know from Buckdahn and Ma [START_REF] Buckdahn | Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs[END_REF] that

u n (t, x) = n Y t,x
t , (t, x) ∈ [0, T ] × IR n , is the stochastic viscosity solution of the parabolic SPDE:

          
du n (t, x) + [Lu n (t, x) + f n (t, x, u n (t, x), σ * D x u n (t, x))]dt +g n (t, x, u n (t, x)) dB t = 0, (t, x) ∈ [0, T ] × IR d u n (0, x) = l(x), x ∈ IR d , where f n (t, x, y, z) = f (t, x, y, z) + n(y -h(t, x)) -. But from the proof of theorem 2.1, for each (t, x) ∈ [0, T ] × IR d we have u n (t, x) ↑ u(t, x) a.s. as n → ∞. Since u n and u are continuous, it follows from Dini's theorem that the above convergence is uniform on the compacts. We now show that u is a stochastic viscosity subsolution of obstacle problem of SPDE (f, g, l, h). Let (τ, ξ) ∈ M B 0,T × L 2 (F B τ ; IR d ) at which u(τ, ξ) > h(τ, ξ), and let (a, p, X) ∈ J 1,2+ u(τ, ξ). The stochastic verson of Lemma 6.1 in Crandall and al [START_REF] Crandall | User's guide to the viscosity solutions of second order partial dierential equations[END_REF], proved there exists sequences

n k → +∞ (τ k , ξ k ) → (τ, ξ) (a k , p k , X k ) ∈ J 1,2+ u n k (τ k , ξ k )
such that (a k , p k , X k ) → (a, p, X).

But for any k, since (a k , p k , X k ) ∈ J 1,2+ u n k (τ k, ξ k ) and u n k is a stochastic viscosity solution of SPDE (f n , g, l) we have (3.6)

On the other hand since u(τ, ξ) > h(τ, ξ) and the uniform convergence of u n , it follows that for k large enough u(τ k , ξ k ) > h(τ k , ξ k ), hence by passing to the limit as k → ∞ in (3.6) we get -a -1 2 T ra(σσ * X) -b, p -f (τ, ξ, u(τ, ξ), pσ(τ, ξ)) + 1 2 (g∂ u g)(τ, ξ, u(τ, ξ)) ≤ 0, and we have proved that u given by (3.5) is a stochastic viscosity subsolution of SPDE (f, g, h, l).

Using the similarly argument with replace J 1,2+ u n k (τ k, ξ k ) by J 1,2-u n k (τ k, ξ k ) it not dicult to prove that u given by (3.5) is a stochastic viscosity supersolution of SPDE (f, g, h, l). So we conclude that u is a stochastic viscosity of SPDE (f, g, h, l) and nish the proof.

Remark 3.7 If the function g takes a simpler form: g(t, x, u) = g(t, x)u then according of Buckdahn and Ma work's [START_REF] Buckdahn | Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs[END_REF], it follows easily that the two previous denitions of stochastic viscosity solution of refected SPDE (f, g, h, l) coincide.

2 t 0 (

 20 as the solution to the equation η(t, x, y) = y + 1 gD y g)(s, x, η(s, x, y))ds + t 0 g(s, x, η(s, x, y))ds..

-a k - 1 2 T

 2 ra(σσ * X k ) -b, p -f n k (τ k , ξ k , u n k (τ k , ξ k ), p k σ(τ k , ξ k )) + 1 2 (g∂ u g)(τ k , ξ k , u(τ k , ξ k )) ≤ 0.

  .[START_REF] Pardoux | Backward doubly stochastic dierential equations and systems of quasilinear SPDEs[END_REF] According(2.17) and(2.19) we deduce that the sequence of processes (Y n , Z n , K n ) is a Cauchy sequence in the Banach space S 2 ([0, T ]; IR) × M 2 (0, T ; IR d ) × S 2 ([0, T ]; IR). Consequently there exists a triplet

  .20) In view of Lemma 2.2 we have Y t ≥ S t a.s. Now, to conclude the proof of the existence, it remains to show (ii) of Denition 2.1. For instance let us remark that by (2.13) and (2.19) we get But since Y t ≥ S t a.s. and

	T	T
	0 (Y T (Y n s -S s )dK n s -→ 0 T
	(Y n s -S s )dK n s = -n	|(Y n
	0	0

s -S s )dK s in L 1 (Ω), as n -→ ∞. s -S s ) -| 2 ds ≤ 0 a.s. we obtain T 0

  3.1

	Since Y n t ≥ Y 0

t , we can w.l.o.g. replace S t by S t ∨Y 0 t , i.e. we may assume that IE( sup 0≤t≤T S 2 t ) < ∞.

following vector spaces of functions:

• for integers k, n the symbol C k,n ([0, T ]×E; E 1 ) stands for the space of all E 1 -valued functions dened on [0, T ] × E which are k-times continuously dierentiable in t and n-times continuously dierentiable in x, and C k,n b ([0, T ] × E; E 1 ) denotes the subspace of C k,n ([0, T ] × E; E 1 ) in which all functions have uniformly bounded partial derivatives.

• For any sub-σ-eld G ⊆ F B and a real number p ≥ 0, L p (G; E) to be all E-valued G-measurable random variable ξ such that IE|ξ| p < ∞. Furthermore, for (t, x, y)

The meaning of D xy and D yy is then self-explanatory. In this section, we consider the continuous coecients f, g as the following:

with the property that for all x ∈ IR d , f (., x, ., .) and g(., x, .) are Lipschitz continuous in x and satisfy the condition (H1) uniformly in x where for some constant K > 0. Moreover h is jointly continuous in (t, x) such that

(H2).

Furthermore, we shall make use the following assumptions; Let us consider the related obstacle problem for stochastic partial dierential equation. Roughly speaking, a solution of a stochastic obstacle problem of SPDE (f, g, h, l) is a random