Obstacle problem for SPDE with nonlinear Neumann boundary condition via reflected generalized backward doubly SDEs

Accepted Manuscript Obstacle problem for SPDE with nonlinear Neumann boundary condition via reflected generalized backward doubly SDEs Auguste Aman, Naoul Mrhardy PII: DOI: Reference:

S0167-7152(12)00411-7 10.1016/j.spl.2012.11.004 STAPRO 6474

To appear in:

Statistics and Probability Letters

Received date: 22 December 2011 Revised date: 1 November 2012 Accepted date: 2 November 2012 Please cite this article as: Aman, A., Mrhardy, N., Obstacle problem for SPDE with nonlinear Neumann boundary condition via reflected generalized backward doubly SDEs. Statistics and Probability Letters (2012), doi:10.1016/j.spl.2012.11.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Obstacle problem for SPDE with nonlinear Neumann boundary condition via reflected generalized backward doubly SDEsI Auguste Aman∗ U.F.R de Math´ematique et Informatique, Universit´e F´elix H. Boigny, 22 BP 582 Abidjan-Cocody, Cˆote d’Ivoire

Naoul Mrhardy Facult´e polydisciplinaire de Khouribga, Universit´e Hassan 1er. Al Hay El Jadid-El Biout ”OCP”. Khouribga, Maroc

Abstract This paper is intended to give a representation for stochastic viscosity solution of semi-linear reflected stochastic partial differential equations with nonlinear Neumann boundary condition. We use its connection with reflected generalized backward doubly stochastic differential equations. Keywords: 2000 MSC: 60H15; 60H20 Backward doubly SDEs, Stochastic PDEs, Obstacle problem, stochastic viscosity solutions. 1. Introduction Backward doubly stochastic differential equations (BDSDEs, for short) were introduced by Pardoux and Peng (1994), and it was shown in various papers that stochastic differential equations (SDEs) of this type give a representation for a system of parabolic stochastic partial differential equations (SPDEs). We refer to Pardoux and Peng (1994) for the link between SPDEs and BDSDEs when the solutions of SPDEs are regular i.e the coefficients are smooth enough (at least in C3 ). The more general situation is much more delicate to treat because of the difficulties of extending the notion of viscosity solutions to SPDEs. The notion of viscosity solution for PDEs was introduced by Crandall et al. (1992) for certain first-order Hamilton-Jacobi equations. Today this theory becomes an important tool in many applied fields, especially in optimal control theory and numerous subjects related to it. The stochastic viscosity solution for semi-linear SPDEs was introduced firstly by Lions and Souganidis (1998). They use the so-called ”stochastic characteristic” to remove the stochastic integrals from a SPDEs. A few years later, two others definitions of stochastic viscosity solution of SPDEs are considered in Buckdahn and Ma (2001a,b) and Buckdahn and Ma (2002). In Buckdahn and Ma (2001a,b), they used the ”Doss-Sussman” transformation to connect the stochastic viscosity solution of SPDEs with the solution of associated BDSDEs. In Buckdahn and Ma (2002), they introduced the stochastic viscosity solution by using I The

work of the first author is supported by AUF post doctoral grant 07-08, R´ef:PC-420/2460 author Email addresses: [email protected] (Auguste Aman), [email protected] (Naoul Mrhardy)

∗ Corresponding

Preprint submitted to Elsevier

October 31, 2012

the notion of stochastic sub and super jets. Next, in order to give a representation for viscosity solution of SPDEs with nonlinear Neumann boundary condition, Boufoussi et al. (2006) introduced the so-called generalized BDSDEs. They refer the first technique (Doss-Sussman transformation) of Buckdhan and Ma. Motivated by both work of Boufoussi et al. (2006) and Boufoussi and Mrhardy (2008), we aim to establish the existence of viscosity solution for semi-linear reflected SPDEs with nonlinear Neumann boundary condition that we write formally as: for P-a.e. ω ∈ Ω, n  ← −o ∂u(t,x) ∗ (x)D u(t, x)) − g(t, x, u(t, x)). B ˙t =0  − Lu(t, x) − f (t, x, u(t, x), σ (i) min u(t, x) − h(t, x), − x  ∂t       (t, x) ∈ [0, T ] × Θ, ( f , φ, g, h, l)  (ii) ∂u   ∂n (t, x) + φ(t, x, u(t, x)) = 0, (t, x) ∈ [0, T ] × ∂Θ,      (iii) u(T, x) = l(x), x ∈ Θ. − ← − ← Here B˙ = dB dt is a ”white noise” associated to Brownian motion B, L is an infinitesimal generator of some diffusion, Θ is a connected bounded domain, and f , g, φ, l and h are some measurable functions. More precisely, we give some direct links between the stochastic viscosity solution of the above SPDE and the solution of the following reflected generalized BDSDE: Yt = ξ +

Z T t

f (s,Ys , Zs )ds +

Z T t

φ(s,Ys )dAs +

Z T t

← − g(s,Ys ) dBs −

Z T t

Zs dWs + KT − Kt , 0 ≤ t ≤ T,

where ξ is the terminal value and A is a real-valued increasing process. Our work generalize respectively the result of Ren and Xia (2006) where authors treat deterministic reflected PDEs with nonlinear Neumann boundary conditions i.e g ≡ 0; and the result appears in Boufoussi et al. (2006) where the non reflected SPDE with nonlinear Neumann boundary condition is considered. The present paper is organized as follows: An existence and uniqueness result to large class of reflected generalized BDSDEs is shown in Section 2. Section 3 is devoted to give a definition of a reflected stochastic solution to SPDEs and establishes its existence result. 2. Reflected generalized backward doubly stochastic differential equations 2.1. Notation, assumptions and definition. The scalar product of the space Rd (d ≥ 2) will be denoted by < ., . > and the associated Euclidian norm by k.k. In what follows let us fix a positive real number T > 0. First of all {Bt , 0 ≤ t ≤ T } and {Wt , 0 ≤ t ≤ T } are two mutually independent standard Brownian motions with values respectively in IR` and IRd , defined B } respectively on two complete probability spaces (Ω, F , P) and (Ω0 , F 0 , P0 ). Let FB = {Ft,T t≥0 denote a B B backward filtration generated by B, augmented by the P-null sets of F ; and let FT = F0,T . We also consider the following family of σ-fields: FtW = σ{Ws , 0 ≤ s ≤ t}. Next we consider the product space (Ω, F , P) where Ω = Ω × Ω0 , F = F ⊗ F 0 and P = P ⊗ P0 . For B ⊗ F W ∨ N and G = F B ⊗ F W ∨ N , where N denotes all the P-null each t ∈ [0, T ], we define Ft = Ft,T t t t T sets in F . The collection F = {Ft , t ∈ [0, T ]} is neither increasing nor decreasing and it does not constitute a filtration. However, (Gt ) is a filtration which contains Ft . 2

Further, we assume that random variables ξ(ω), ω ∈ Ω and ζ(ω0 ), ω0 ∈ Ω0 are considered as random variables on Ω via the following identification: ξ(ω, ω0 ) = ξ(ω); ζ(ω, ω0 ) = ζ(ω0 ). In the sequel, let {At , 0 ≤ t ≤ T } be a continuous, increasing and F-measurable real valued process such that A0 = 0. For any d ≥ 1, we consider the following spaces of processes:

1. M 2 (F, [0, T ]; Rd ) denote the Banach space of all equivalence classes (with respect to the measure dP⊗ dt) where each equivalence class contains an d-dimensional jointly measurable stochastic process ϕt ;t ∈ [0, T ], such that: for all µ > 0 (i) kϕk2M2 = E

Z T 0

eµAt |ϕt |2 dt < ∞;

(ii) ϕt is Ft -measurable , for any t ∈ [0, T ].

2. S2 (F, [0, T ]; R) is the set of one dimensional continuous stochastic processes which verify: for all µ   Z T 2 µAt 2 µAs 2 (i) kϕkS2 = E sup e |ϕt | + e |ϕs | dAs < ∞; 0

0≤t≤T

(ii) ϕt is Ft -measurable , for any t ∈ [0, T ].

3. A2 (F, [0, T ]; R) denotes a continuous, real-valued and increasing process ϕ, such that
(i) kϕk2A2 = E |ϕT |2 < ∞; (ii) ϕt is Ft -measurable, for any t ∈ [0, T ] and ϕ0 = 0.

In addition, we give the following assumptions on the data (ξ, f , φ, g, S):
(H1) ξ is an FT -measurable, square integrable random variable; such that for all µ > 0 E eµAT |ξ|2 < ∞.

(H2) f : Ω × [0, T ] × R × Rd → R, g : Ω × [0, T ] × R × Rd → R` , and φ : Ω × [0, T ] × R → R, are three functions verifying: (a) There exists Ft -measurable processes { ft , φt , gt : 0 ≤ t ≤ T } with values in [1, +∞) such that for d each  (t, y, z) ∈ [0, T ] × R × R , and all µ > 0, the following hypotheses are satisfied for K > 0: (i) f (t, y, z), φ(t, y), and g(t, y, z) are Ft -measurable processes,     (ii) | f (t, y, z)| ≤ ft + K(|y| + kzk),    (iii) |φ(t, y)| ≤ φt + K|y|,  + kzk),  (iv) |g(t, Z y,T z)| ≤ gt + K(|y|  Z T Z T    µA 2  (v) E  e t f dt + eµAt g2 dt + eµAt φ2 dAt < ∞. 0

t

0

t

0

t

(b) There exist constants c > 0, β < 0 and 0 < α < 1 such that for any (y1 , z1 ), (y2 , z2 ) ∈ R × Rd ,   (i) | f (t, y1 , z1 ) − f (t, y2 , z2 )|2 ≤ c(|y1 − y2 |2 + kz1 − z2 k2 ), (ii) |g(t, y1 , z1 ) − g(t, y2 , z2 )|2 ≤ c|y1 − y2 |2 + αkz1 − z2 k2 ,  (iii) hy1 − y2 , φ(t, y1 ) − φ(t, y2 )i ≤ β|y1 − y2 |2 .

(H3) The obstacle {St , 0 ≤ t ≤ T }, is a continuous, Ft -measurable, real-valued process, satisfying for any µ(> 0  2  (i) E sup0≤t≤T eµAt St+ < ∞, (ii) ST ≤ ξ

a.s..

3

The first goal of this paper is to study the following reflected generalized BDSDEs: for all t ∈ [0, T ], Yt

= ξ+

Z T t

f (s,Ys , Zs )ds +

Z T t

φ(s,Ys )dAs +

Z T t

← − g(s,Ys , Zs ) dBs −

Z T t

Zs dWs + KT − Kt .

(2.1)

First of all let us give the definition of the solution to above BDSDEs. Definition 2.1. We call a solution of the reflected generalized BDSDE (ξ, f , φ, g, S) a triplet of processes (Y, Z, K) which satisfy (2.1) such that the following holds P- a.s (i) (Y, Z, K) ∈ S2 (F, [0, T ]; R) × M 2 (F, 0, T ; Rd ) × A2 (F, [0, T ]; R) (ii) the map s 7→ Ys is continuous (iii) Yt ≥ St , 0 ≤ t ≤ T , (iv)

Z T 0

(Yt − St ) dKt = 0.

In the sequel, C denotes a positive constant which may vary from one line to other. 2.2. Comparison theorem Let us give this needed comparison theorem related to the generalized respectively BSDEs and BDSDEs. Theorem 2.1. For i = 1, 2, let ξi be square integrable and GT -measurablerandom variables. Let hi , φi :  [0, T ] × Ω × R → R be such that: for every Gt -adapted process satisfying E sup |Yt |2 < ∞ the processes 0≤t≤T

hi (·,Y. ) and φi (·,Y. ) are Gt -adapted and satisfy Z T  E [|hi (s,Ys )|2 ds + |φi (s,Ys )|2 dAs ] < +∞. 0

Let (Y i , Z i ) be a solution of the following generalized BSDE:  Z T Z T Z T  i i i i i i  Y = ξ + (s,Y )ds + (s,Y )dA − h φ Zsi dWs  s s s   t t t t   Z T Z T    i 2 i 2 i 2  |Ys | dAs + kZs k ds < +∞.  E sup |Yt | + 0≤t≤T

0

0

Assume that

(i) h1 and φ1 are uniformly Lipschitz in the variable y, (ii) ξ1 ≤ ξ2 a.s., (iii) h1 (t,Yt2 ) ≤ h2 (t,Yt2 ) dP ⊗ dt a.e., φ1 (t,Yt2 ) ≤ φ2 (t,Yt2 ) dP ⊗ dAt a.e..

Then Yt1 ≤ Yt2 , 0 ≤ t ≤ T , a.s.

Theorem 2.2. Assume (H1)-(H2) and let (Y, Z) and (Y 0 , Z 0 ) be the unique solution of the non reflected generalized BDSDE associated to (ξ, f , φ, g) and (ξ0 , f 0 , φ0 , g) respectively. If ξ ≤ ξ0 , f (t,Yt0 , Zt0 ) ≤ f 0 (t,Yt0 , Zt0 ) 0 and φ(t,Yt0 ) ≤ φ (t,Yt0 ), then Yt ≤ Yt0 , ∀t ∈ [0, T ]. 4

Proof. Let us set ∆Y = Y −Y 0 , ∆Z = Z − Z 0 and (∆Y )+ = (Y −Y 0 )+ (with f + = sup{ f , 0}). Using Itˆo’s formula, we get, for all 0 ≤ t ≤ T E((∆Yt )+ )2 + E

Z T t

≤E((ξ − ξ0 )+ )2 + 2E + 2E +E

Z T t

k∆Zs k2 1{Ys >Ys0 } ds

Z T t

 (∆Ys )+ 1{Ys >Ys0 } f (s,Ys , Zs ) − f 0 (s,Ys0 , Zs0 ) ds

 (∆Ys ) 1{Ys >Ys0 } φ(s,Ys ) − φ0 (s,Ys0 ) dAs +

Z T t

g(s,Ys , Zs ) − g(s,Ys0 , Zs0 ) 2 1{Y >Y 0 } ds, s s

(2.2)

1 From (H2)(b) and the the elementary inequality 2ab ≤ a2 + εb2 , ∀ ε > 0, we have E((∆Yt )+ )2 ≤ 0 which ε leads to ∆Yt+ = 0 a.s. and so Yt0 ≥ Yt a.s. for all t ≤ T . 2.3. Existence and uniqueness result Our main goal in this section is to prove the following theorem. Theorem 2.3. Under the hypotheses (H1)-(H3), the reflected generalized BDSDE (2.1) has a unique solution (Y, Z, K). Before we start proving this theorem, let us establish the same result in case g do not depends on Y and Z. More precisely, let consider the reflected generalized BDSDE = ξ+

Yt

Z T t

f (s,Ys , Zs )ds +

Z T t

φ(s,Ys )dAs +

Z T t

← − g(s) dBs −

Z T t

Zs dWs + KT − Kt

(2.3)

where f , φ, g and ξ satisfy the above assumptions. Proposition 2.1. There exists a unique triplet (Y, Z, K) solution of the BDSDEs (2.3). Proof. Existence For each n ∈ N∗ , and according to the work of Boufoussi et al. (2006), let (Y n , Z n ) ∈ S2 ([0, T ]; R) × M 2 (0, T ; Rd ) denote the unique pair of processes satisfying Ytn = ξ + R

Z T t

f (s,Ysn , Zsn )ds + KTn − Ktn +

Z T t

φ(s,Ysn )dAs +

Z T t

← − g(s) dBs −

Z T t

Zsn dWs ,

(2.4)

where Ktn = n 0t (Ysn −Ss )− ds. For all t ∈ [0, T ], let define fn (t, y, z) = f (t, y, z)+n(y−St )− . Then fn (t, y, z) ≤ fn+1 (t, y, z) and it follows from Theorem 2.2 that Ytn ≤ Ytn+1 , a.s.. Therefore Ytn % Yt = supn Ytn , a.s. Step 1: A priori estimate For any µ > 0, there exists C > 0 such that,   Z T Z T µAt n 2 µAs n 2 µAs n 2 n 2 sup E sup e |Yt | + e |Ys | dAs + e kZs k ds + |KT | < C n∈N∗

that

0≤t≤T

0

0

Indeed, using (H2) and the elementary inequality 2ab ≤ γa2 + 1γ b2 , ∀γ > 0, it follows from Itˆo’s formula 5

Z

Z

T T 1 E(eµAt |Ytn |2 ) + |β|E eµAs kZsn k2 ds eµAs |Ysn |2 dAs + E 2 t t   Z T Z T Z T Z T µAT 2 µAs n 2 µAs 2 µAs 2 µAs 2 ≤ CE e |ξ| + e |Ys | ds + e fs ds + e φs dAs + e kg(s)k ds t t t t   1 + E sup (eµAs Ss+ )2 + εE (KTn − Ktn )2 . ε 0≤s≤T

(2.5)

Next, we get respectively by (2.4) and H¨older inequality and the isometry equality, together with assumptions (H2)(a) that for all 0 ≤ t ≤ T  Z T Z T Z T n n 2 E(KT − Kt ) ≤ CE eµAT |ξ|2 + eµAt |Ytn |2 + eµAs fs2 ds + eµAs φ2s dAs + eµAs kg(s)k2 ds t t t   Z T  Z T Z T + eµAs |Ysn |2 ds + E sup eµAs (Ss+ )2 + eµAs |Ysn |2 dAs + eµAs kZsn k2 ds . (2.6) t

t

0≤s≤t

t

Recalling again (2.5) and taking ε small enough such that εC < min{ 12 , |β|}, Gronwall’s lemma and (2.6), we obtain   Z T Z T µAt n 2 µAs n 2 µAs n 2 n n 2 E e |Yt | + e |Ys | dAs + e kZs k ds + |KT − Kt | t t   Z T Z T Z T ≤ CE eµAT |ξ|2 + eµAs fs2 ds + eµAs φ2s dAs + eµAs kg(s)k2 ds + sup eµAt (St+ )2 . t

t

t

0≤t≤T

Finally, appliquing both forward and backward version of Burkholder-Davis-Gundy inequality use in Pardoux and Peng (1994), we have    Z T Z T Z T µAt n 2 µAs n 2 n 2 E sup e |Yt | + e kZs k ds + |KT | ≤ CE eµAT |ξ|2 + eµAs fs2 ds + eµAs φ2s dAs 0 0 0 0≤t≤T  Z T + eµAs kg(s)k2 ds + sup eµAt (St+ )2 . 0

Step 2: For each n ∈

0≤t≤T

N∗ , E



2 sup (Ytn − St )−

0≤t≤T



−→ 0, as n −→ ∞.

Indeed, since Ytn ≥ Yt0 , we can w.l.o.g. replace St by St ∨Yt0 , i.e. we may assume that E(sup0≤t≤T eµAt St2 ) < ∞. We want to compare a.s. Yt and St for all t ∈ [0, T ], while we do not know yet if Y R R R ← − ← − ← − n is a.s. continuous. Let denote ξ := ξ + 0T g(s)dBs , St := St + 0t g(s)dBs and Y t := Ytn + 0t g(s)dBs , then n

Yt = ξ+

n

Z T t

and supn Y t = Y t = Yt +

f (s,Ysn , Zsn ) ds + n

Rt

Z T t

n

Y s − Ss

← −

0 g(s)dBs .

6


−

ds +

Z T t

φ (s,Ysn ) dAs −

Z T t

Zsn dWs ,

(2.7)

On the other hand consider {(Yetn , Zetn ), 0 ≤ t ≤ T } the unique solution of the BSDE’s Yetn = ST +

Z T t

f (s,Ysn , Zsn ) ds + n

Z T t

(Ss − Yesn )ds +

Z T t

φ (s,Ysn ) dAs −

Z T t

Zesn dWs .

According to El Karoui et al. (1997), the sequence (Yen ) satisfies the equality  Z T Z T Yeνn = E e−n(T −ν) ST + e−n(s−ν) f (s,Ysn , Zsn )ds + n e−n(s−ν) Ss ds ν ν  Z T −n(s−ν) n + e φ(s,Ys )dAs | Gν , ν

(2.8)

where ν = σ ∧ T for any Gt -stopping time σ. Recall again El Karoui et al. (1997), e−n(T −ν) ST + n

Z T ν

e−n(s−ν) Ss ds → Sν

a.s. and in L2 (Ω) and it conditional expectation converges also in L2 (Ω). Moreover, in view of Ren and Xia (2006) one can derive that Z T  Z T −n(s−ν) n n −n(ν−s) n E e f (s,Ys , Zs )ds + e φ(s,Ys )dAs |Gν → 0 ν

ν

n in L2 (Ω) as n → +∞; and then Yeνn → Sν in L2 (Ω) as n → +∞. Therefore, since, using Theorem 2.1, Yetn ≤ Y t , we have Yν ≥ Sν a.s. From this and the section theorem (see Dellacherie and Meyer (1978), p.220) the last inequality is true for every t ∈ [0, T ]. Hence (Ytn − St )− ↓ 0 uniformly to t The result finally follows by dominated convergence, since (Ytn − St )− ≤ (St −Yt0 )+ ≤ |St | + Yt0 . Step 3: Convergence result  

Recalling that Ytn % Yt a.s, dominated convergence Fatou’s lemma and Step 1 ensure E Z T  |Ysn −Ys |2 ds −→ 0, as n → ∞. and E

sup eµAt |Yt |2

0≤t≤T

< +∞

0

Next, for n ≥ p ≥ 1, and from the step 1, by using again assumptions (H2), Gronwall lemma, H¨older inequality and Burkh¨older-Davis-Gundy’s inequality, we obtain   Z T Z T n p 2 n p 2 n p 2 |Ys −Ys | dAs + kZs − Zs k ds −→ 0, as n, p −→ .∞, E sup |Ys −Ys | + 0≤s≤T

0

0

Consequently, the sequence of processes (Y n , Z n ) is Cauchy in the Banach space S2 ([0, T ]; R)×M 2 (0, T ; Rd ). Hence, there exists a couple (Y, Z) ∈ S2 ([0, T ]; R) × M 2 (0, T ; Rd ) such that   Z T Z T |Ysn −Ys |2 dAs + kZsn − Zs k2 ds → 0, as n → ∞. E sup |Ysn −Ys |2 + 0≤s≤T

0

0

On the other hand, since  • supn≥0 | f (s,Ysn , Zs )| ≤ fs + K (supn≥0 |Ysn |) + kZs k ,  • supn≥0 |φ(s,Ysn )| ≤ φs + K (supn≥0 |Ysn |) , 7

• E

RT t

| f (s,Ysn , Zsn ) − f (s,Ysn , Zs )|2 ds ≤ CE

RT t

kZsn − Zs k2 ds,

there exist a process K such that for all t ∈ [0, T ], E |Ktn − Kt |2 −→ 0. and for all t ∈ [0, T ], Yt = ξ +

Z T t

f (s,Ys , Zs )ds + R

Z T t

φ(s,Ys )dAs + KT − Kt + R

Z T t

← − g(s)dBs −

Z T t

Zs dWs , a.s. R

Next, it not difficult to prove that 0T (Ysn − Ss )dKsn converges to 0T (Ys − Ss )dKs in probability and 0T (Ys − Ss )dKs = 0, which ends the proof of existence. Uniqueness Let consider (Y, Z, K) and (Y 0 , Z 0 , K 0 ) BDSDE (2.3) and define ∆Y = Yt −Yt0 , ∆Zt = Z − Z 0 , ∆Kt = K − K 0 . Use the same computation as in the proof of existence, we have   Z T Z T Z T E eµAt |∆Yt |2 + eµAs |∆Ys |dAs + eµAs k∆Zs k2 ds ≤ CE eµAs |∆Ys |2 ds, t

t

t

from which, we deduce that ∆Y = 0 and further ∆Z = 0 and ∆K = 0.

¯ ∈ S2 ([0, T ]; R) × M 2 (0, T ; Rd ), it follows from Proposition 2.1 Proof of Theorem 2.3. For any given (Y¯ , Z) that the reflected generalized BDSDE Yt = ξ +

Z T t

f (s,Ys , Zs )ds +

Z T t

φ(s,Ys )dAs +

Z T t

← − g(s, Y¯s , Z¯ s ) dBs −

Z T t

Zs dWs + KT − Kt

(2.9)

has a unique solution (Y, Z, K). Hence the mapping Φ : S2 ([0, T ]; R) × M 2 (0, T ; Rd ) −→ S2 ([0, T ]; R) × ¯ 7−→ (Y, Z) = Φ(Y¯ , Z) ¯ is well defined. Let (Y, Z) = Φ(Y¯ , Z) ¯ and (Y 0 , Z 0 ) = Φ(Y¯0 , Z¯0 ) and M 2 (0, T ; Rd ), (Y¯ , Z) 0 ¯ ¯ set ∆η = η − η for η = Y, Y , Z, Z. Use the same calculation as previous, there exist α < α0 < 1 such that Z T

Z T

Z T

¯ eµs+βAs |∆Ys |2 ds + βE eµs+βAs |∆Ys |2 dAs + E eµs+βAs k∆Zs k2 ds t t t  Z T  Z T Z T α 2 2 2 ¯ ¯ eµs+βAs |∆Y¯s | ds + βE eµs+βAs |∆Y¯s | dAs + E eµs+βAs k∆Z¯ s )k ds ≤ 0 cE α t t t cE ¯

which implies that Φ is a strict contraction on the space L2 (F, [0, T ]; R) × M 2 (F, (0, T ); Rd ) equipped with the equivalent norm k(Y, Z)k2 = cE ¯

Z T 0

¯ eµs+βAs |Ys |2 ds + βE

Z T 0

eµs+βAs |Ys |2 dAs + E

Therefore Φ admit a unique fixed point (Y, Z) satisfying Yt = ξ +

Z T t

f (s,Ys , Zs )ds +

Z T t

φ(s,Ys )dAs +

Z T t

Z T 0

← − g(s,Ys , Zs ) dBs −

eµs+βAs kZs k2 ds. Z T t

Zs dWs + KT − Kt

and; hence (Y, Z, K) is the unique solution of BDSDE (2.1). 3. Connection to stochastic viscosity solution for reflected SPDEs with nonlinear Neumann boundary condition In this section we will investigate the reflected generalized BDSDEs studied in the previous section in order to give a interpretation for the stochastic viscosity solution of a class of nonlinear reflected SPDEs with nonlinear Neumann boundary condition. 8

3.1. Notion of stochastic viscosity solution for reflected SPDEs with nonlinear Neumann boundary condition B } With the same notations as in Section 2, let FB = {Ft,T 0≤t≤T be the filtration generated by B. The set B B M0,T denote all the F -stopping times τ such 0 ≤ τ ≤ T , a.s. For generic Euclidean spaces E and E1 we introduce the following: 1. The symbol C k,n ([0, T ] × E; E1 ) stands for the space of all E1 -valued functions defined on [0, T ] × E which are k-times continuously differentiable in t and n-times continuously differentiable in x, and Cbk,n ([0, T ] × E; E1 ) denotes the subspace of C k,n ([0, T ] × E; E1 ) in which all functions have uniformly bounded partial derivatives. B , C k,n (G , [0, T ] × E; E ) (resp. C k,n (G , [0, T ] × E; E )) denotes 2. For any sub-σ-field G ⊆ FTB = F0,T 1 1 b

the space of all C k,n ([0, T ] × E; E1 ) (resp. Cbk,n ([0, T ] × E; E1 )-valued random variable that are G ⊗ B ([0, T ] × E)-measurable; 3. C k,n (FB , [0, T ]×E; E1 ) (resp.Cbk,n (FB , [0, T ]×E; E1 )) is the space of all random fields α ∈ C k,n (FTB , [0, T ]× E; E1 ) (resp. C k,n (FTB , [0, T ] × E; E1 ), such that for fixed x ∈ E, the mapping (ω,t) 7→ α(ω,t, x) is FB progressively measurable. B and a real number p ≥ 0, L p (G ; E) to be all E-valued G 4. For any sub-σ-field G ⊆ FTB = F0,T measurable random variable ξ such that E|ξ| p < ∞.

Furthermore, regardless their dimensions we denote by < ., . > and |.| the inner product and norm in E and E1 , respectively. For (t, x, y) ∈ [0, T ] × Rd × R, we denote Dx = ( ∂x∂1 , ...., ∂x∂d ),

∂ Dxx = (∂2xi x j )di, j=1 , Dy = ∂y , Dt = ∂t∂ . The meaning of Dxy and Dyy is then self-explanatory. Let Θ be an open connected and smooth bounded domain of Rd (d ≥ 1) such that for a function ψ ∈ Cb2 (Rd ), Θ and its boundary ∂Θ are characterized by Θ = {ψ > 0}, ∂Θ = {ψ = 0} and, for any x ∈ ∂Θ, ∇ψ(x) is the unit normal vector pointing towards the interior of Θ. In this section, let consider the following standing assumptions:

(A1) The functions σ : Rd → Rd×d and b : Rd → Rd are uniformly Lipschitz continuous, with a common Lipschitz constant K > 0. (A2) The functions f : Ω × [0, T ] × Θ × R × Rd → R and φ : Ω × [0, T ] × Θ × R → R are continuous random field such that for fixed (x, y, z) and f (·, x, y, z), φ(·, x, y) and h(·, x) are FB -progressively measurable; and there exist β < 0 and K > 0, such that for P-a.e ω, (i) | f (ω,t, x, y, z)| ≤ K(1 + |x| + |y| + kzk),

(ii) | f (ω,t, x, y, z) − f (ω,t, x0 , y0 , z0 )|2 ≤ K(|x − x0 |2 + |y − y0 |2 + kz − z0 k2 ), (iii) |φ(ω,t, x, y)| ≤ K(1 + |x| + |y|),

(iv) hy − y0 , φ(ω,t, x, y) − φ(ω,t, x, y0 )i ≤ β|y − y0 |2

(v) |φ(ω,t, x, y) − φ(ω,t, x0 , y0 )|2 ≤ K(|x − x0 |2 + |y − y0 |2 ) (A3) The function l : Rn → R is Lipschitz continuous with the common Lispchitz constant. (A4) The function h : Ω × [0, T ] × Θ → R is continuous random field such that for fixed x, h(·, x) is FB progressively measurable; and there exists K > 0, such that for P-a.e ω, (i) |h(ω,t, x)| ≤ K(1 + |x|), (ii) h(ω, T, x) ≤ l(x).

9

(A5) The function g ∈ Cb0,2,3 ([0, T ] × Θ × R; R).

Consider the related obstacle problem for SPDE with nonlinear Neumann boundary condition: n  ← −o ∂u(t,x) ∗   (i) min u(t, x) − h(t, x), − ∂t − Lu(t, x) − f (t, x, u(t, x), σ (x)Dx u(t, x)) − g(t, x, u(t, x)). B˙ t = 0     (t, x) ∈ [0, T ] × Θ,   ( f , φ, g, h, l)

 (ii) ∂u   ∂n (t, x) + φ(t, x, u(t, x)) = 0, (t, x) ∈ [0, T ] × ∂Θ,      (iii) u(T, x) = l(x), x ∈ Θ.

∂ψ ∂ where L = 21 ∑di, j=1 (σ(x)σ∗ (x))i, j ∂x∂i ∂x j + ∑di=1 bi (x) ∂x∂ i , ∀ x ∈ Θ, ∂n = ∑di=1 ∂x (x) ∂x∂ i , ∀ x ∈ ∂Θ. i We now define the notion of stochastic viscosity solution for the SPDE( f , φ, g, h, l). We are inspired by the work of Buckdahn and Ma (2001a,b) and we refer to their paper for a lucid discussion on this topic. Let η, ε ∈ C (FB , [0, T ] × Rd × R) be stochastic flows respectively defined as unique solution of the SDE which, in Stratonowich form, reads as follows: 2

η(t, x, y) = y +

Z T t

← − hg(s, x, η(s, x, y)), ◦dBs i, ε(t, x, y) = y −

Z T t

← − hDy ε(s, x, y)g(s, x, y), ◦dBs i.

To simplify the notation we write: A f ,g (ϕ(t, x)) = −Lϕ(t, x)− f (t, x, ϕ(t, x), σ∗ Dx ϕ(t, x))+ 12 (g, Dy g)(t, x, ϕ(t, x)) and Ψ(t, x) = η(t, x, ϕ(t, x)); and define the notion of a stochastic viscosity solution of the SPDE( f , φ, g, h, l) as follows.
Definition 3.1. A random field u ∈ C FB , [0, T ] × Θ is called a stochastic viscosity subsolution of the B , stochastic obstacle problem OP ( f ,φ,g,h,l) if u (T, x) ≤ l (x), for all x ∈ Θ, and if for any stopping time τ ∈ M0,T

0 B 1,2 B d any state variable ξ ∈ L Fτ , Θ , and any random field ϕ ∈ C Fτ , [0, T ] × R such that for P-almost all ω ∈ {0 < τ < T }, u (ω,t, x) − Ψ (ω,t, x) ≤ 0 = u (τ(ω), ξ(ω)) − Ψ (τ(ω), ξ(ω))

for all (t, x) in some neighborhood V (ω, τ (ω) , ξ (ω)) of (τ (ω) , ξ (ω)), it holds:

(a) on the event {0 < τ < T } ∩ {ξ ∈ Θ}  min u(τ, ξ) − h(τ, ξ), A f ,g (Ψ (τ, ξ)) − Dy Ψ (τ, ξ) Dt ϕ (τ, ξ) ≤ 0

(3.1)

holds, P-almost surely;

(b) on the event {0 < τ < T } ∩ {ξ ∈ ∂Θ} the inequality   min min u(τ, ξ) − h(τ, ξ), A f ,g (Ψ (τ, ξ)) − Dy Ψ (τ, ξ) Dt ϕ (τ, ξ) , i − ∂Ψ (τ, ξ) − φ (τ, ξ, Ψ (τ, ξ)) ≤ 0 (3.2) ∂n holds, P-almost surely.
A random field u ∈ C FB , [0, T ] × Θ is called a stochastic viscosity supersolution of the stochastic obstacle B , any state variable problem OP ( f ,φ,g,h,l) if u (T, x) ≥ l (x), for all x ∈ Θ, and if for any stopping time τ ∈ M0,T

0 B 1,2 B d ξ ∈ L Fτ , Θ , and any random field ϕ ∈ C Fτ , [0, T ] × R such that for P-almost all ω ∈ {0 < τ < T } u (ω,t, x) − Ψ (ω,t, x) ≥ 0 = u (τ(ω), ξ(ω)) − Ψ (τ(ω), ξ(ω))

for all (t, x) in some neighborhood V (ω, τ (ω) , ξ (ω)) of (τ (ω) , ξ (ω)), it holds: 10

(a) on the event {0 < τ < T } ∩ {ξ ∈ Θ}  min u(τ, ξ) − h(τ, ξ), A f ,g (Ψ (τ, ξ)) − Dy Ψ (τ, ξ) Dt ϕ (τ, ξ) ≥ 0

(3.3)

holds, P-almost surely;

(b) on the event {0 < τ < T } ∩ {ξ ∈ ∂Θ}   max min u(τ, ξ) − h(τ, ξ), A f ,g (Ψ (τ, ξ)) − Dy Ψ (τ, ξ) Dt ϕ (τ, ξ) ,  ∂Ψ − (τ, ξ) − φ (τ, ξ, Ψ (τ, ξ)) ≥ 0 ∂n

(3.4)

holds, P-almost surely.


Finally, a random field u ∈ C FB , [0, T ] × Θ is called a stochastic viscosity solution of the stochastic obstacle problem OP ( f ,φ,g,h,l) if it is both a stochastic viscosity subsolution and a stochastic viscosity supersolution. Remark 3.1. Observe that if f , φ, l and h are deterministic and g ≡ 0, the flow η becomes η(t, x, y) = y and Ψ(t, x) = ϕ(t, x), ∀ (t, x, y) ∈ [0, T ] × Rd × R. Thus, definition 3.1 coincides with the definition of (deterministic) viscosity solution of PDE( f , φ, 0, h, l) given in Ren and Xia (2006). We now recall a notion of random viscosity solution which will be a bridge linking the stochastic viscosity solution and its deterministic counterpart. Definition 3.2. A random field u ∈ C(FB , [0, T ] × Rn ) is called an ω-wise viscosity solution if for P-almost all ω ∈ Ω, u(ω, ·, ·) is a (deterministic) viscosity solution of PDE( f , φ, 0, h, l).

Next we introduce the Doss-Sussman transformation which enables us to convert an reflected SPDE( f , φ, g, h, l) to an classical PDE( fe, e φ, 0, e h, l) where fe, e φ and e h are certain well-defined random fields, which are defined in terms of f , φ and h.

Proposition 3.1. Assume (A1)-(A5). A random field u is a stochastic viscosity solution to the OP ( f ,φ,g,h,l) ee e if and only if v(·, ·) = ε(·, ·, u(·, ·)) is a stochastic viscosity solution to the SPDE OP ( f ,φ,0,h,l) , where ( fe, e φ, e h) are three coefficients that will be made precise later. Proof. We shall only prove that if u is a stochastic viscosity sub-(resp. super-) solution to SPDE OP ( f ,φ,g,h,l) , ee e v(·, ·) = ε(·, ·, u(·, ·)) is a stochastic viscosity sub-(resp. super-) solution to SPDE OP ( f ,φ,0,h,l) . The converse part of the proposition can be proved very similarly. We shall only argue for the stochastic subsolution case, as the supersolution part is similar. Let us assume that u ∈ C (FB , [0, T ] × Θ) is a stochastic viscosity subsolution of the SPDE OP ( f ,φ,g,h,l) . By it definition, v belongs to C (FB , [0, T ] × Θ) and set e h(t, x) = B , ξ ∈ L0 (F B ; Θ) and ϕ ∈ ε(t, x, h(t, x)), we have v(t, x) ≥ e h(t, x), ∀ (t, x) ∈ [0, T ] × Θ a.s.. Let τ ∈ M0,T τ
C 1,2 FτB , [0, T ] × Θ be such that for P-almost all ω ∈ {0 < τ < T } ∩ {v(τ, ξ) > eh(τ, ξ)} the inequality v(ω,t, x) − ϕ(ω,t, x) ≤ 0 = v(ω, τ(ω), ξ(ω)) − ϕ(ω, τ(ω), ξ(ω))

(3.5)

holds for all (t, x) in some neighborhood V (ω, τ(ω), ξ(ω)) of (ω, τ(ω)). For all (t, x) ∈ V (τ, ξ), if we set Ψ(t, x) = η(t, x, ϕ(t, x)), hence u(t, x) − Ψ(t, x) = η(t, x, v(t, x)) − η(t, x, ϕ(t, x)) ≤ 0

= η(τ, v(τ, ξ)) − η(τ, ϕ(τ, ξ))

= u(τ, ξ) − Ψ(τ, ξ) 11

holds for all P-almost all ω ∈ {0 < τ < T } ∩ {v(τ, ξ) > e h(τ, ξ)}. Moreover, since u is a stochastic viscosity subsolution of the SPDE OP ( f ,φ,g,h,l) and {v(τ, ξ) > e h(τ, ξ)} = {u(τ, ξ) > h(τ, ξ)} we obtain: (a) on the event {0 < τ < T } ∩ {u(τ, ξ) > h(τ, ξ)} ∩ {ξ ∈ Θ}

A f ,g (Ψ (τ, ξ)) − Dy Ψ (τ, ξ) Dt ϕ (τ, ξ) ≤ 0 holds, P-almost surely; (b) on the event {0 < τ < T } ∩ {u(τ, ξ) > h(τ, ξ)} ∩ {ξ ∈ ∂Θ} the inequality   ∂Ψ (τ, ξ) − φ (τ, ξ, Ψ (τ, ξ)) ≤ 0 min A f ,g (Ψ (τ, ξ)) − Dy Ψ (τ, ξ) Dt ϕ (τ, ξ) , − ∂n holds, P-almost surely. By the similarly calculation used in Boufoussi et al. (2006), we have: (a) on the event {0 < τ < T } ∩ {v(τ, ξ) > e h(τ, ξ)} ∩ {ξ ∈ Θ} the inequality holds, P-almost surely;

A fe,0 (ϕ (τ, ξ)) − Dt ϕ (τ, ξ) ≤ 0

(b) on the event {0 < τ < T } ∩ {v(τ, ξ) > e h(τ, ξ)} ∩ {ξ ∈ ∂Θ} the inequality   ∂ϕ min A fe,0 (ϕ (τ, ξ)) − Dt ϕ (τ, ξ) , − (τ, ξ) − e φ (τ, ξ, ϕ (τ, ξ)) ≤ 0 ∂n

(3.6)

(3.7)

holds, P-almost surely.

where fe(t, x, y, z) =

e φ(t, x, y) =

1 [ f (t, x, η(t, x, y), σ(x)∗ Dx η(t, x, y) + Dy η(t, x, y)z) Dy η(t, x, y) 1 − gDy g(t, x, η(t, x, y)) + Lx η(t, x, y) + hσ(x)∗ Dxy η(t, x, y), zi 2  1 + Dyy η(t, x, y)|z|2 2 1 [φ(t, x, η(t, x, y)) + Dx η(t, x, y)∇ψ(x)] . Dy η(t, x, y)

Combining inequality (3.6) and (3.7), we obtain that the random field v is a stochastic viscosity subsolution ee e of the SPDE OP ( f ,φ,0,h,l) .

12

3.2. Existence of stochastic viscosity solutions for SPDE with nonlinear Neumann boundary condition Now, in order to give a link between the stochastic obstacle problem OP ( f ,φ,g,h,l) and BDSDE (2.1), let consider this SDE: s 7→At,x s is increasing Xst,x = x + At,x s =

Z s∨t

Z s∨t t

t

Z
b Xrt,x dr +

t

I{Xrt,x ∈∂Θ} dAt,x r .

s∨t

Z
σ Xrt,x dWr +

t

s∨t


∇ψ Xrt,x dAt,x r ,

∀ s ∈ [0, T ] , (3.8)

It is clear (see Lions and Sznitman (1984)) that under conditions (A1), SDE (3.8) has a unique strong FW adapted solution and with similar arguments as in Pardoux and Zhang (1998) ( Propositions 3.1 and 3.2), we can provide the following regularity results. Proposition 3.2. There exists a constant C > 0 such that for all for all 0 ≤ t < t 0 ≤ T and x, x0 ∈ Θ, the following inequalities hold: for any p > 4  p  n  o 0 0 p t 0 ,x0 E sup Xst,x − Xst ,x + At,x − A ≤ C |t 0 − t| p/2 + |x − x0 | p . s s 0≤s≤T

Moreover, for all p ≥ 1, there exists constant Cp , C(µ,t) such that for all (t, x) ∈ [0, T ] × Θ,  t,x  p
≤ Cp (1 + t p ), E eµAs ≤ C(µ,t). E At,x s

The main topic of this section will be the generalization of the Kolmogorov continuity criterion to the reflected generalized BDSDE:  Z T Z T

− Z T
t,x
 t,x t,x t,x t,x t,x t,x ←  Y = l X + f r, X ,Y , Z dr + g r, X ,Y dBr + φ r, Xrt,x ,Yrt,x dAt,x  s r r r r r r T   s∨t s∨t s∨t  Z T  (3.9) + KTt,x − Kst,x − Zrt,x dWr , 0 ≤ s ≤ T  s   Z  T 
  Yst,x ≥ h(s, Xst,x ) such that Yrt,x − h(r, Xrt,x ) dKrt,x = 0. t

where the coefficients l, f , g, φ and h satisfy the hypotheses (A2)-(A5).

Proposition 3.3. Let the ordered triplet (Yst,x , Zst,x , Kst,x ) be a solution of the BDSDE (3.9). Then the random field (t, x) 7→ Yst,x is almost surely continuous on [0, T ] × Θ for all s ∈ [0, T ]. 0

0

0

0

Proof. Let (t, x) and (t 0 , x0 ) be two elements of [0, T ]×Θ and denote Y s = Yst,x −Yst ,x , Z s = Zst,x −Zst ,x , K s = 0 0 t,0 x0 t,x Kst,x −Kst ,x , As = At,x and kt,x o’s formula and assumptions (A2), (A3) s −As s = |A|s +As . It follows from Itˆ and (A5), and using Young’s that   Z t,x p(p − 1) T λr+µkt,x r e |Y r | p−2 kZ r k2 dr E eλs+µks |Y s | p + 2 s  Z T Z t0 Z T t,x t,x t,x t,x p ≤ Cp E eλT +µkT X T + eλr+µkr |X r | p dr + eλr+µkr |Y r | p−1 (1 + |x0 | + |Ytt,x eλr+µkr |X r | p dAt,x 0 |)dr + r 0

t

13

0

 Z T t,x t,x 0 0 0 0 (3.10) eλr+µkr |Y r | p−1 (1 + |Xrt,x | + |Yrt,x |) d|A|r + eλr+µkr |Y r | p−1 (1 + |Xrt ,x | + |Yrt ,x |) d|A|r 0 0  Z T  Z T Z T Z T t,x t,x t,x 1 λr+µkt,x p r +c p E eλr+µkr |Y r | p−2 kZ r k2 dr + eλr+µkr |Y r | p dr + eλr+µkr |Y r | p dAt,x + e | d|A| |Y r r r γ s s s 0   Z T Z T Z t0 t,x t,x t,x p(p − 1) eλr+µkr |Y r | p dkt,x −E λ eλr+µkr |Y r | p dr + µ + eλr+µkr |Y r | p−2 |g(r, Xrt,x ,Yrt,x )|2 dr. r 2 s s t +

Z T

Next, from both forward and backward version of Burkholder-Davis-Gundy inequality use in [14], using again young inequality and choosing γ, λ and µ convenably, together with Proposition 3.2, it suffice to choose p = γ convenably to get   0 0 γ E sup Yst,x −Yst ,x ≤ Cp (|t − t 0 |1+β + |x − x0 |d+δ ). 0≤s≤T

Therefore, (t, x) ∈ [0, T ] × Θ 7→ Yst,x has an a.s. continuous version, for all s ∈ [0, T ]. Let {nYst,x , n Zst,x , t ≤ s ≤ T } denotes the solution of the following BDSDE: n t,x Ys

= l(XTt,x ) + +

Z T s

Z T s

f (r, Xrt,x , nYrt,x , n Zrt,x )dr + n

φ(r, Xrt,x ,n Yrt,x )dAt,x r +

Z T s

Z T s

(nYrt,x − h(t, Xrt,x ))− dr

← − g(r, Xrt,x ,n Yrt,x )dBr −

Z T s

n t,x Zr dWr ,

(3.11)

and define un (t, x) = nYtt,x , it is shown in Boufoussi et al. (2006) that the function vn (t, x) = ε(t, x, un (t, x)) is an ω-wise viscosity solution to the SPDE  n  (i) ∂v ∂t(t,x) + Lvn (t, x) + fen (t, x, vn (t, x), σ∗ (x)Dx vn (t, x)) = 0, (t, x) ∈ [0, T ] × Θ      n n e (ii) ∂v ∂n (t, x) + φ(t, x, v (t, x)) = 0, (t, x) ∈ [0, T ] × ∂Θ,       (iii) vn (T, x) = l(x), x ∈ Θ, where fen (t, x, y, z) = fe(t, x, y, z) + n (η(t,x,y)−h(t,x)) . Dy η(t,x,y) −

Let us define for all (t, x) ∈ [0, T ] × Θ, u(ω,t, x) = Ytt,x and v(t, x) = ε(t, x, u(t, x)), It follows from penalization argument, that (along a subsequence), we have |un (τ, ξ) − u(τ, ξ)| → 0, a.s. for all (τ, ξ) ∈ B × L0 (F B ; Θ). Hence we have |vn (τ, ξ) − v(τ, ξ)| → 0 as n goes to infinity. M0,T τ Our main result in this section is the following. Theorem 3.1. Let assumptions (A1)-(A5) be satisfied. Then the function u(t, x) defined above is a stochastic viscosity solution of obstacle problem OP ( f ,φ,g,h,l) . B -measurable, u(t, x) is F B -measurable. Then, combining Proof. We have u(T, x) = l(x) and since Ytt,x is Ft,T t,T B B × L0 (F B ; Θ), u(τ(ω), ξ(ω)) = this result with Proposition 3.3, u ∈ C(F ; [0, T ] × Θ) and for all (τ, ξ) ∈ M0,T τ τ(ω),ξ(ω)

Yτ(ω) ≥ h(τ(ω), ξ(ω)) a.s. Using Proposition 3.1, to show that u satisfies (3.1)-(3.2) and (3.3)-(3.4), it suffices to prove that v satisfies (3.6) and (3.7). Let ω ∈ Ω be fixed satisfies |vn (ω,t, x) − v(ω,t, x)| → 0 as n → ∞. 14

(3.12)



B × L0 F B , Θ × C 1,2 F B , [0, T ] × Θ such that 0 < τ(ω) < T , For this fixed ω, let consider (τ, ξ, ϕ) ∈ M0,T τ τ v(ω, τ(ω), ξ(ω)) > e h(ω, τ(ω), ξ(ω)) and the the following inequality: v(ω,t, x) − ϕ(ω,t, x) ≤ 0 = v(ω, τ(ω), ξ(ω)) − ϕ(ω, τ(ω), ξ(ω))

(3.13)

for all (t, x) in some neighborhood V (ω, τ (ω) , ξ (ω)) of (τ (ω) , ξ (ω)). From Example 8.2 in El Karoui et al. (1997) and Lemma 6.1
in Crandall et al. (1992), there exists sequence (τ j (ω), ξ j (ω), ϕ j (ω)) j≥1 ∈ [0, T ] × Θ × C 1,2 [0, T ] × Θ such that n j → +∞, τ j (ω) → τ(ω), ξ j (ω) → ξ(ω), ϕ j (ω) → ϕ(ω) and vn j (ω,t, x) − ϕ j (ω,t, x) ≤ vn j (ω, τ j (ω), ξ j (ω)) − ϕ j (ω, τ j (ω), ξ j (ω))

for all (t, x) in some neighborhood V (τ j (ω) , ξ j (ω)) ⊂ V (τ (ω) , ξ (ω)) and a suitable subsequence (vn j ) j≥1 . Using the fact that vn j (ω, ·, ·) is a (deterministic) viscosity solution of the PDE ( fen j (ω, ·), e φ(ω, ·), 0, e h, l), we obtain: (a) if ξ j (ω) ∈ Θ the inequality holds;

A fen

j ,0

(ϕ j (ω, τ j (ω), ξ j (ω))) − Dt ϕ j (ω, τ j (ω), ξ j (ω)) ≤ 0

(b) if ξ j (ω) ∈ ∂Θ, the inequality h min A fen holds.

(ϕ j (ω, τ j (ω), ξ j (ω))) − Dt ϕ j (ω, τ j (ω), ξ j (ω)) ,  ∂ϕ j (ω, τ j (ω), ξ j (ω)) − e φ(ω, τ j (ω), ξ j (ω), Ψn (ω, τ j (ω), ξ j (ω))) ≤ 0 − ∂n j ,0

Since v(ω, τ(ω), ξ(ω)) > e h(ω, τ(ω), ξ(ω)), it follows from (3.12) that vn j (ω, τ j (ω), ξ j (ω)) > e h(ω, τ j (ω), ξ j (ω)) for j large enough; hence taking the limit as j → +∞ in the above inequalities yields: (a) if ξ(ω) ∈ Θ, the inequality holds;

A fe,0 (ϕ (ω, τ(ω), ξ(ω))) − Dt ϕ (ω, τ(ω), ξ(ω)) ≤ 0

(b) if ξ(ω) ∈ ∂Θ, the inequality

holds.

h min A fe,0 (ϕ (ω, τ(ω), ξ(ω))) − Dt ϕ (ω, τ(ω), ξ(ω)) ,  ∂ϕ e − (ω, τ(ω), ξ(ω)) − φ(ω, τ(ω), ξ(ω), Ψ(ω, τ(ω), ξ(ω))) ≤ 0 ∂n

Acknowledgments This work is partially done when the first author was post doctoral internship at Cadi Ayyad University of Marrakech. He would like to express his deep gratitude to B. Boufoussi, Y. Ouknine and UCAM Mathematics Department for their friendly hospitality. The authors would like to thank also the Editor for carefully handling their paper and the anonymous referees for their valuable comments and suggestions for improving the quality of this work. 15

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