Stochastic viscosity solutions for SPDEs with continuous coefficients

J. Math. Anal. Appl. 384 (2011) 63–69

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Stochastic viscosity solutions for SPDEs with continuous coefficients ✩ Boualem Djehiche a , Modeste N’zi b , Jean-Marc Owo b,∗,1 a b

Department of Mathematics, The Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden Université de Cocody, UFR de Mathématiques et Informatique, 22 BP 582 Abidjan, Cote d’Ivoire

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 6 April 2010 Available online 7 August 2010 Submitted by Goong Chen Keywords: Stochastic partial differential equation Backward doubly stochastic differential equation Stochastic viscosity solution

In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger conditions, we prove the existence of stochastic viscosity solutions. © 2010 Elsevier Inc. All rights reserved.

1. Introduction Nonlinear backward doubly stochastic differential equations (BDSDEs in short) have been introduced by Pardoux and Peng [15]. The original motivation for the study of this kind of equations was to provide a probabilistic interpretation for solutions of both parabolic and elliptic semilinear (stochastic) partial differential equations (see Pardoux [13], Pardoux and Peng [15,14] and Peng [17] in the particular case where solutions of SPDEs are regular). Today, this theory can be considered as one of the most useful tools in the study of SPDEs. Indeed, many recent works provide all kinds of solutions to SPDEs by means of solutions of BDSDEs. Among others, we refer to Buckdahn and Ma [4,5] for the link between stochastic viscosity solutions to SPDEs and solutions of BDSDEs. The notion of viscosity solution for PDEs was introduced by Crandall and Lions [7] for certain first order Hamilton–Jacobi equations. Today the theory has become an important tool in many applied fields, especially in optimal control theory and numerous subjects related to it (see Crandall et al. [6], Fleming and Soner [8] and Bardi et al. [1]). The stochastic viscosity solution for semilinear SPDEs was introduced for the first time in Lions and Souganidis [9]. They use the so-called ‘stochastic characteristics’ to remove the stochastic integrals from the SPDE. Another way to ‘remove’ this stochastic integral is to use a Doss–Sussman-type transformation to convert the SPDE into a PDE with random coefficients so that a stochastic viscosity solution can be defined in a pointwise sense. Buckdahn and Ma [4] were the first to use this approach in order to connect the stochastic viscosity solution of SPDEs with BDSDEs. Recently, Boufoussi et al. [2] using the technique of Buckdahn and Ma [4], established an existence result for semilinear SPDE with Neumann boundary condition via generalized BDSDEs, when the coefficients are Lipschitz continuous. In this paper, we relax the Lipschitz continuity conditions on the coefficients and we extend their results to nonlinear SPDEs with

✩

This work was initiated from a discussion while the third author was visiting the Department of Mathematics at Stockholm University, Sweden. Corresponding author. E-mail addresses: [email protected] (B. Djehiche), [email protected] (M. N’zi), [email protected] (J.-M. Owo). 1 The author wants to thank the Project (MARM) and its Director Professor Mikael Passare as well as the London Mathematical Society for initiating this research visit.

*

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.08.005

©

2010 Elsevier Inc. All rights reserved.

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B. Djehiche et al. / J. Math. Anal. Appl. 384 (2011) 63–69

continuous coefficients, but with Dirichlet boundary conditions. This result widens the range of applications of nonlinear SPDEs to control problems involving infinite dimensional systems with less regular coefficients, among others. The nonlinear SPDEs we consider in this note are of the form:

T u (t , x) = l(x) +

     Lu (s, x) + f s, x, u (s, x), σ ∗ Du (s, x) ds +

t

T



 ←−−

g s, x, u (s, x) dB s ,

t ∈ [0, T ],

(1.1)

t

where L is a second order differential operator defined by

L=

d 1 

2

σσ∗

i , j =1



 ∂ ∂2 + bi . ij ∂x ∂x ∂ xi i j d

i =1

Using a Doss–Sussman-type transformation, we provide a direct link between the SPDE (1.1) and the following BDSDE:

T Yt = ξ +

T f (s, Y s , Z s ) ds +

t

←−−

T

g (s, Y s ) dB s − t

Z s dW s ,

t ∈ [0, T ].

(1.2)

t

More precisely, we prove that the BDSDE (1.2) provides in a Markovian framework, a sub- and super-stochastic viscosity solution for the above SPDE when the coefficient f is continuous and linear growth. Our result, on the one hand, generalizes the work of Buckdahn and Ma [4,5] and of Boufoussi et al. ([2] without the Neumann term) and on the other hand, provides some applications of Theorem 4.1 in Shi et al. [18] and Theorem 3.1 in N’zi and Owo [10]. The rest of the paper is organized as follows. In Section 2, we introduce some notations, set up our framework assumptions and recall the notion of stochastic viscosity solution. In Section 3, we establish our main result, Theorem 3.2 and give a special case where one can reach the existence of stochastic viscosity solutions, Corollary 3.4. In the sequel, we will denote SPDE( f , g ) for the SPDE (1.1) with coefficients f and g, and BDSDE(ξ, f , g ) for the BDSDE in (1.2), with terminal condition ξ and coefficients f and g. 2. Preliminaries 2.1. Notations and assumptions Throughout this paper, |x| will denote the Euclidean norm of a vector x ∈ Rk , ·,· the inner product on Rk , T is a fixed final time and { W t ; 0  t  T } and { B t ; 0  t  T } are two mutually independent standard Brownian motions, with values, respectively in Rd and Rl defined on the complete probability spaces (Ω 1 , F 1 , P1 ) and (Ω 2 , F 2 , P2 ), respectively. η η η For any process {ηt ; t ∈ [0, T ]} defined on (Ω i , F i , Pi ), i = 1, 2, we set Fs,t = σ {ηr − ηs ; s  r  t } ∨ N , Ft = F0,t , where

N denotes the class of Pi -null sets of F i . Note that {F0W,t , t ∈ [0, T ]} is an increasing filtration and {FtB, T , t ∈ [0, T ]} is a decreasing filtration, but, the collection {FtW ⊗ FtB, T , t ∈ [0, T ]} is neither increasing nor decreasing so it does not constitute

a filtration. We will denote by (Ω, F , P, {Ft }0t  T ) the global complete probability space, where Ω = Ω 1 × Ω 2 , F = F 1 ⊗ F 2 , 

P = P1 ⊗ P2 , Ft = FtW ⊗ FtB, T . Any random variable ζ (ω1 ), ω1 ∈ Ω 1 (resp. η(ω2 ), ω2 ∈ Ω 2 ) is considered as a random variable on Ω via the following identification: ζ (ω) = ζ (ω1 ) (resp. 

 {Ft }0t  T , F W =



η(ω) = η(ω2 )), ω = (ω1 , ω2 ).

 {FtW }0t  T and F B =

Set F = {FtB, T }0t  T . For any n ∈ N, let M2 ( F , [0, T ]; Rn ) denote the set of (dP ⊗ dt a.e. equal) n-dimensional jointly measurable random processes {ϕt ; 0  t  T } which satisfy:

T

(i) ϕ 2M2 = E( 0 |ϕt |2 dt ) < ∞; (ii) ϕt is Ft -measurable, for a.e. t ∈ [0, T ]. Endowed with the norm · M2 , M2 ( F , [0, T ]; Rn ) is a Banach space. Similarly by S 2 ( F , [0, T ]; Rn ) we denote the set of continuous n-dimensional random processes which satisfy: (i) ϕ 2S 2 = E(sup0t  T |ϕt |2 ) < ∞; (ii) ϕt is Ft -measurable, for any t ∈ [0, T ]. The space S 2 ( F , [0, T ]; Rn ) is complete with the norm · S 2 , and therefore a Banach space. Let M0B, T denote the set of all F B -stopping times τ such that 0  τ  T , P2 -almost surely. For generic Euclidean spaces E and E 1 , we introduce the following vector spaces of functions:

B. Djehiche et al. / J. Math. Anal. Appl. 384 (2011) 63–69

65

• The symbol C k,n ([0, T ] × E ; E 1 ) denotes the space of all functions defined on [0, T ] × E with values in E 1 , which are k,n k-times continuously differentiable in t and n-times continuously differentiable in x and Cb ([0, T ] × E ; E 1 ) denotes the k,n subspace of C ([0, T ] × E ; E 1 ) of all functions with uniformly bounded partial derivatives. • For any sub-σ -field G ⊂ F TB , C k,n (G , [0, T ] × E ; E 1 ) (resp. Cbk,n (G , [0, T ] × E ; E 1 )) stands for the space of all random k,n variables with values in C k,n ([0, T ] × E ; E 1 ) (resp. Cb ([0, T ] × E ; E 1 )) which are G ⊗ B ([0, T ] × E )-measurable. k , n B B • The symbol C k,n ( F , [0, T ] × E ; E 1 ) (resp. Cb ( F , [0, T ] × E ; E 1 )) is the space of random fields ϕ ∈ C k,n (F TB , [0, T ] × k,n E ; E 1 ) (resp. Cb (F TB , [0, T ] × E ; E 1 )) such that for any x ∈ E, the mapping (ω2 , t ) → ϕ (ω2 , t , x) is F B -progressively measurable.

• The symbol LSC ([0, T ] × E ; E 1 ) (resp. U SC ([0, T ] × E ; E 1 )) denotes the space of all lower (resp. upper) semi-continuous functions defined on [0, T ] × E with values in E 1 . • For any sub-σ -field G ⊂ F TB , LSC (G , [0, T ] × E ; E 1 ) (resp. U SC (G , [0, T ] × E ; E 1 )) stands for all random variables with values in LSC ([0, T ] × E ; E 1 ) (resp. U SC ([0, T ] × E ; E 1 )) which are G ⊗ B ([0, T ] × E )-measurable. • The symbol LSC ( F B , [0, T ] × E ; E 1 ) (resp. U SC ( F B , [0, T ] × E ; E 1 )) denotes the space of random fields ϕ ∈ LSC (F TB , [0, T ] × E ; E 1 ) (resp. U SC (F TB , [0, T ] × E ; E 1 )) such that for any x ∈ E, the mapping (ω2 , t ) → ϕ (ω2 , t , x) is F B -progressively measurable.

• For any sub-σ -field G ⊂ F TB , and for any p  0, L p (G , E ) denotes the space of G -measurable random variables ξ with values in E such that E|ξ | p < ∞. We note that

      C 0,0 F B , [0, T ] × E ; E 1 = LSC F B , [0, T ] × E ; E 1 ∩ U SC F B , [0, T ] × E ; E 1 . For any (t , x, y ) ∈ [0, T ] × Rq × R, we denote D = D x = ( ∂∂x , . . . , ∂∂x ), D y = ∂∂y , D t = ∂∂t , and D xx = (∂x2i x j )i , j =1,...,q . 1

q

Let f : Ω 2 × [0, T ] × Rq × R × Rd → Rk , g : Ω 2 × [0, T ] × Rq × R → Rl be two measurable functions and b : [0, T ] × q R → Rd , σ : [0, T ] × Rq → Rq×d and l : Rq → R satisfy the following assumptions: 0,2,3

(A1) g ∈ C b

([0, T ] × Rq × R; Rl ),

(A2) there exists a process ϕ. ∈ M2 ( F B , [0, T ]; R) with positive values and positive constant K such that, for any (ω2 , t , x, y , z) ∈ Ω 2 × [0, T ] × Rq × R × Rd ,

 2      f ω , t , x, y , z  ϕt ω2 + K |x| + | y | + | z| ,

(A3) for any (ω2 , t , x) ∈ Ω 2 × [0, T ] × Rq , f (ω2 , t , x, ·, ·) is continuous and for any (ω2 , y , z) ∈ Ω 2 × R × Rd and (t 1 , x1 ), (t 2 , x2 ) ∈ [0, T ] × Rq ,

 2      f ω , t 1 , x1 , y , z − f ω2 , t 2 , x2 , y , z  K |t 1 − t 2 | + |x1 − x2 | ,

(A4) the functions σ and b are uniformly Lipschitz continuous with Lipschitz constant K > 0, (A5) the function l is continuous such that for p > 0, we have

  l(x)  K 1 + |x| p ,

x ∈ Rq .

2.2. Stochastic viscosity solutions In order to give the definition of stochastic viscosity solutions for SPDE( f , g ), we consider the process R × R; R) as the solution to the following Stratonovich SDE q

T



η(t , x, y ) = y +



←−−

g s, x, η(s, x, y ) , ◦ dB s ,

0 t  T.

η ∈ C 0,0,0 ([0, T ] ×

(2.1)

t

As an Itô SDE, (2.1) reads

η(t , x, y ) = y +

1

T

2 t

   g , D y g  s, x, η(s, x, y ) ds +

T





←−−

g s, x, η(s, x, y ) , dB s ,

0 t  T.

t

Under condition (A1), the mapping y → η(t , x, y ) is a diffeomorphism for all (t , x), P2 -a.s. such that η ∈ C 0,2,2 ( F B , [0, T ] × Rq × R; R) (cf. Boufoussi et al. [2]). Let ε (t , x, y ) denote the y-inverse of η(t , x, y ). Thus, ε ∈ C 0,2,2 ( F B , [0, T ] × Rq × R; R). Moreover, for any (t , x, y ) ∈ [0, T ] × Rq × R, P2 -a.s., it holds that

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B. Djehiche et al. / J. Math. Anal. Appl. 384 (2011) 63–69





ε t , x, η(t , x, y ) = y and

T

ε(t , x, y ) = y −



←−−

D y ε (s, x, y ) g (s, x, y ), ◦ dB s ,

0 t  T.

t

Furthermore, if ϕ (t , x) = ε (t , x, ψ(t , x)), for (t , x) ∈ [0, T ] × Rq , then C 0, p ( F B , [0, T ] × Rq ; R), for p = 0, 1, 2. For ease of notation, we set

ϕ ∈ C 0, p ( F B , [0, T ] × Rq ; R) if and only if ψ ∈

  1     A f , g ψ(t , x) = −Lψ(t , x) − f t , x, ψ(t , x), σ ∗ (t , x) D x ψ(t , x) +  g , D y g  t , x, ψ(t , x) . 2

We now introduce the notion of stochastic viscosity solution for SPDE( f , g ). Definition 2.1.

• A random field u ∈ LSC ( F B , [0, T ] × Rq ; R) is said to be a stochastic viscosity sub-solution of SPDE( f , g ) if u ( T , x)  l(x), for all x ∈ Rq ; and if for any stopping time τ ∈ M0B, T , any state variable ξ ∈ L 0 (FτB , Rq ), and any random field ϕ ∈ C 1,2 (FτB , [0, T ] × Rq ; R) such that, for P2 -almost all ω2 ∈ {0 < τ < T }, it holds u







 









 







 





ω2 , t , x − η ω2 , t , x, ϕ (t , x)  0 = u τ ω2 , ξ ω2 − η τ ω2 , ξ ω2 , ϕ τ ω2 , ξ ω2



,

for all (t , x) in a neighborhood of (τ (ω ), ξ(ω )), then we have, P -a.s. on {0 < τ < T }, 2

2

2

    A f , g ψ(τ , ξ ) − D y η τ , ξ, ψ(τ , ξ ) D t ϕ (τ , ξ )  0,

(2.2)



where ψ(t , x) = η(t , x, ϕ (t , x)). • A random field u ∈ U SC ( F B , [0, T ] × Rq ; R) is said to be a stochastic viscosity super-solution of SPDE( f , g ) if u ( T , x)  l(x), for all x ∈ Rq ; and if for any stopping time τ ∈ M0B, T , any state variable ξ ∈ L 0 (FτB , Rq ), and any random field

ϕ ∈ C 1,2 (FτB , [0, T ] × Rq ; R) such that, for P2 -almost all ω2 ∈ {0 < τ < T }, it holds                    u ω2 , t , x − η ω2 , t , x, ϕ (t , x)  0 = u τ ω2 , ξ ω2 − η τ ω2 , ξ ω2 , ϕ τ ω2 , ξ ω2 , for all (t , x) in a neighborhood of (τ (ω2 ), ξ(ω2 )), then we have, P2 -a.s. on {0 < τ < T },

    A f , g ψ(τ , ξ ) − D y η τ , ξ, ψ(τ , ξ ) D t ϕ (τ , ξ )  0.

(2.3)

• A random field u is said to be a stochastic viscosity solution of SPDE( f , g ) if u ∈ C 0,0 ( F B , [0, T ] × Rq ; R) and is both a stochastic viscosity sub-solution and super-solution. Remark 2.2. If, in SPDE( f , g ), g ≡ 0, then for all (t , x, y ), η(t , x, y ) = y and ψ(t , x) = ϕ (t , x). Hence, if f is deterministic, the above definition coincides with the deterministic case (cf. E. Pardoux and S. Zhang [16] and Boufoussi and Mrhardy [3]). Thus, any stochastic viscosity (sub- or super-) solution is viewed as a (deterministic) viscosity (sub- or super-) solution for each fixed ω2 ∈ {0 < τ < T }, modulo the FτB -measurability requirement of the test function ϕ . 3. Stochastic viscosity sub- and super-solution We need to put BDSDE(ξ, f , g ) in the Markovian framework, i.e. ξ , f and g are explicit functions of the solution to a progressive SDE driven by ( W t )0t  T . To this end, we consider the following SDE:

s X st ,x = x +





s

b r , X rt ,x dr + t





σ r , X rt ,x dW r , s ∈ [t , T ],

(3.1)

t

with (t , x) ∈ [0, T ] × Rq , where b and σ verify condition (A4). We know that under this condition, SDE (3.1) has a unique t ,x solution { X s ; t  s  T , x ∈ Rq } (see e.g. Øksendal [12]). t ,x t ,x t ,x Then, taking ξ = l( X T ), f (s, y , z) = f (s, X s , y , z) and g (s, y ) = g (s, X s , y ), the BDSDE (1.2) becomes



t ,x 

Y st ,x = l X T

T





T

f r , X rt ,x , Y rt ,x , Z rt ,x dr +

+ s



 ←−−

T

g r , X rt ,x , Y rt ,x dB r − s

Z rt ,x dW r , s

s ∈ [t , T ].

(3.2)

B. Djehiche et al. / J. Math. Anal. Appl. 384 (2011) 63–69

67

It is easy to check that, under assumptions (A1)–(A5), BDSDE (3.2), satisfies the conditions of Theorem 3.1 in N’zi t ,x t ,x and Owo [10]. Then BDSDE (3.2) has at least one solution and a minimal solution {(Y ∗s , Z ∗s ); t  s  T , x ∈ Rq } in t ,x t ,x 2 k 2 d×k q S ( F , [0, T ]; R ) × M ( F , [0, T ]; R ), in the sense that if {(Y s , Z s ); t  s  T , x ∈ R } is any other solution, then t ,x t ,x Y ∗s  Y s , t  s  T a.s. for (t , x) ∈ [0, T ] × Rq (see the proof of Lemma 3.2 in [10]). In the same way, it is easy to see that (3.2) has a maximal solution. Lemma 3.1. Let u ∗ : Ω 2 × [0, T ] × Rq → R be a random field defined by 

t ,x

for all (t , x) ∈ [0, T ] × Rq .

u ∗ (t , x) = Y ∗t ,

Then, u ∗ ∈ LSC ( F B , [0, T ] × Rq ; R). Proof. Let ( f n )n0 be a sequence of functions defined by

f n (t , x, y , z) =



inf

(u , v )∈Q×Qd





f (t , x, u , v ) + ( K + n) | y − u | + | z − v | ,

n ∈ N.

Then, for all n  0, f n verifies the following properties: (i) (ii) (iii) (iv) (v)

| f n (t , x, y , z)|  ϕt + K (|x| + | y | + | z|); | f n (t , x, y 1 , z1 ) − f n (t , x, y 2 , z2 )|  ( K + n)(| y 1 − y 2 | + | z1 − z2 |); ( f n (t , x, y , z))n0 is non-decreasing; if ( yn , zn ) → ( y , z), then f n (t , x, yn , zn ) → f (t , x, y , z) as n → +∞; | f n (t 1 , x1 , y , z) − f n (t 2 , x2 , y , z)|  K (|t 1 − t 2 | + |x1 − x2 |). t ,x,n

, Z st ,x,n , t  s  T ) be the solution of BDSDE (l( X tT,x ), f n , g ). As in the proof of Theorem 3.1 in N’zi and Owo [10], t ,x,n t ,x,n t ,x,n one can show that (Y s )n0 is a non-decreasing sequence such that (Y s , Z s , t  s  T ) converges to the minimal t ,x t ,x solution (Y ∗s , Z ∗s , t  s  T ) of BDSDE (3.2). Let us define, for any n ∈ N,

Let (Y s

t ,x,n

un (t , x) = Y t

,

(t , x) ∈ [0, T ] × Rq .

Then, (un )n0 is a non-decreasing sequence of continuous random fields which converges to u ∗ in [0, T ] × Rq . Therefore, u ∗ is lower semi-continuous, i.e. u ∗ ∈ LSC ( F B , [0, T ] × Rq ; R). 2 Our main result is the following 

t ,x

Theorem 3.2. Under conditions (A1)–(A5), the random field u ∗ ∈ LSC ( F B , [0, T ] × Rq ; R) defined by u ∗ (t , x) = Y ∗t , for all (t , x) ∈ [0, T ] × Rq , is a stochastic viscosity sub-solution for SPDE (1.1). Moreover, if v ∈ C 0,2 ( F B , [0, T ] × Rq ; R) is any classical solution for SPDE (1.1), then v (t , x)  u ∗ (t , x). Proof. First, it is clear that u ( T , x)  l(x), for all x ∈ Rq . Now, let us consider C

1, 2

(Fτ , [0, T ] × R ; R) such that B

τ ∈ M0B,T , ξ ∈ L 0 (FτB , Rq ) and ϕ ∈

q





u ∗ (t , x) − η t , x, ϕ (t , x)  0 = u ∗ (τ , ξ ) − η





τ , ξ, ϕ (τ , ξ ) ,

for all (t , x) in a neighborhood V (τ , ξ ) of (τ , ξ ), P2 -a.s., on {0 < τ < T }. We know from the result of Buckdahn and Ma [4,5] or from Boufoussi et al. [2] (without the Neumann term) that un is a stochastic viscosity solution for SPDE( f n , g ). Therefore, if we let τn ∈ M0B, T , ξn ∈ L 0 (FτB , Rq ) and ϕn ∈ C 1,2 (FτB , [0, T ] × Rq ; R) be sequences satisfying





un (t , x) − η t , x, ϕn (t , x)  0 = un (τn , ξn ) − η





τn , ξn , ϕn (τn , ξn ) ,

for all (t , x) in a neighborhood V (τn , ξn ) of (τn , ξn ), P2 -a.s., on {0 < τn < T } such that (τn , ξn , ϕn ) converges to (τ , ξ, ϕ ), we get P2 -a.s., on {0 < τn < T },

    A f n , g ψn (τn , ξn )  D y η τn , ξn , ψn (τn , ξn ) D t ϕn (τn , ξn ),

(3.3)

where ψn (t , x) = η(t , x, ϕn (t , x)) converges to ψ(t , x) = η(t , x, ϕ (t , x)) in [0, T ] × R . Moreover, we have A f n , g (ψn (τn , ξn )) → A f , g (ψ(τ , ξ )), as n → ∞. Then, passing to the limit in (3.3), we obtain q

    A f , g ψ(τ , ξ )  D y η τ , ξ, ψ(τ , ξ ) D t ϕ (τ , ξ ).

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Hence, we conclude that u ∗ ∈ LSC ( F B , [0, T ] × Rq ; R) is a stochastic viscosity sub-solution for the SPDE (1.1). Finally, since t ,x t ,x v ∈ C 0,2 ( F B , [0, T ] × Rq ; R), then, by Pardoux and Peng [15], { v (s, X s ), (σ ∗ D v )(s, X s ), 0  s  t  T } is a solution to the BDSDE (3.2). t ,x t ,x Therefore, v (s, X s )  Y ∗s , for all s ∈ [t , T ], in particular, v (t , x)  u ∗ (t , x). 2 

∗t ,x

∗t ,x

∗t ,x

Remark 3.3. Let u ∗ (t , x) = Y t , for all (t , x) ∈ [0, T ] × Rq , where {(Y s , Z s ); t  s  T , x ∈ Rq } is the maximal solution of the BDSDE (3.2). Then u ∗ ∈ U SC ( F B , [0, T ] × Rq ; R) and following a similar argument as above it follows that u ∗ is a stochastic viscosity super-solution for the SPDE (1.1). Moreover, if v ∈ C 0,2 ( F B , [0, T ] × Rq ; R) is any classical solution for the SPDE (1.1), then u ∗ (t , x)  v (t , x). In particular, we easily see that the Lipschitz condition is a trivial case of our proposed conditions. Now, we assume that the coefficients f and g in the SPDE (1.1) satisfy the following assumptions: (A6) f (·, 0, 0, 0) ∈ M2 (0, T , R), g (·, 0, 0) ∈ M2 (0, T , Rl ) such that

 T E

   f (s, 0, 0, 0) 2 + g (s, 0, 0) 2 ds > 0.

0

(A7) For all ( y 1 , z1 ), ( y 2 , z2 ) ∈ R × Rd , t ∈ [0, T ] and x ∈ Rq ,

   f (t , x, y 1 , z1 ) − f (t , x, y 2 , z2 ) 2  ρ t , | y 1 − y 2 |2 + C | z1 − z2 |2 ,   g (t , x, y 1 ) − g (t , x, y 2 ) 2  ρ t , | y 1 − y 2 |2 ,

where C > 0 and 0 < α < 1 are two constants and ρ : [0, T ] × R+ → R+ satisfies: • For fixed t ∈ [0, T ], ρ (t , .) is a concave non-decreasing function such that ρ (t , 0) = 0. T • For fixed u, 0 ρ (t , u ) dt < +∞. • For any M > 0, the following ODE



u  = − M ρ (t , u ), u(T ) = 0

has a unique solution u (t ) ≡ 0, t ∈ [0, T ]. t ,x

t ,x

Under these conditions, N’zi and Owo [11] have proved that the BDSDE (3.2) has a unique solution {(Y s , Z s ); t  s  T , x ∈ Rq } in S 2 ( F , [0, T ]; R) × M2 ( F , [0, T ]; Rd ) when the terminal condition is square integrable (see Theorem 3.4 in [11]). Then, as a consequence, we have the following existence result for the SPDE (1.1). t ,x

Corollary 3.4. Under conditions (A1)–(A7), the random field (t , x) → Y t

is a stochastic viscosity solution for the SPDE (1.1). t ,x

t ,x

Proof. The coefficients f and g being continuous, the BDSDE (3.2) has a minimal solution (Y ∗ , Z ∗ ) and a maximal solution (Y ∗t ,x , Z ∗t ,x ) both in S 2 ( F , [0, T ]; R) × M2 ( F , [0, T ]; Rd ). t ,x Therefore, by Theorem 3.2, the random field (t , x) → Y ∗t is a stochastic viscosity sub-solution for the SPDE (1.1). Also, ∗t ,x by Remark 3.3, the random field (t , x) → Y t is a stochastic viscosity super-solution for the SPDE (1.1). Finally, by the t ,x ∗t ,x uniqueness of the solution of the BDSDE (3.2) (Theorem 3.4 in [11]), we get that Y ∗t = Y t , for all (t , x) ∈ [0, T ] × Rq . 2 Acknowledgment We would like to thank an anonymous referee for his insightful suggestions that helped to improve the content of the paper.

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