Mean-field Reflected Backward Doubly Stochastic DE With

Journal of Numerical Mathematics and Stochastics, 6 (1) : 62-72, 2014

© JNM@S Euclidean Press, LLC Online: ISSN 2151-2302

http://www.jnmas.org/jnmas6-5.pdf

Mean-field Reflected Backward Doubly Stochastic DE With Continuous Coefficients* N. CHAOUCHKHOUAN, B. LABED, and B. MANSOURI Laboratory of Applied Mathematics, University of Biskra, POB 145, Algeria, E-mail: [email protected]

Abstract. We study the existence and uniqueness of the solutions to mean-field reflected backward doubly stochastic differential equation (MF-RBDSDE), when the driver f is Lipschitzian. We also study the existence in the case where the driver is of linear growth and continuous. In this case we establish a comparison theorem. Key words : Backward Doubly SDE, Mean-field, Continuous Coefficients, Comparison Theorem. AMS Subject Classifications : 60H10, 60H05

1. Introduction After the earlier work of Pardoux & Peng (1990), the theory of backward stochastic differential equations (BSDEs in short) has a significant headway thanks to the many application areas. Several authors contributed in weakening the Lipschitzian assumption required on the drift of the equation (see Lepaltier & San Martin (1996), Kobylanski (1997), Mao (1995), Bahlali (2000)). A new kind of backward stochastic differential equations was introduced by Pardoux & Peng [5] (1994), T

T

T

t

t

t

Y t     fs, Y s , Z s ds   gs, Y s , Z s dB s −  Z s dW s ,

0≤t≤T

with two different directions of stochastic integrals, i.e., the equation involves both a standard (forward) stochastic integral dW t and a backward stochastic integral dB t . They have proved the existence and uniqueness of solutions for BDSDEs under uniformly Lipschitzian conditions. Shi et al. [6] (2005) provided a comparison theorem which is very important in studying viscosity solution of SPDEs with stochastic tools. Bahlali et al. [2] (2009) proved the existence and uniqueness of the solution to the following _________________ *This work is partially supported by the CNPRU project B01420140103

62

63

N. CHAOUCHKHOUAN, B. LABED, and B. MANSOURI

reflected backward doubly stochastic differential equations(RBDSDEs) with one continuous barrier and uniformly Lipschitzian coefficients: T

T

Y t     fs, Y s , Z s ds   gs, Y s , Z s dB s t

t

T

 K T − K t −  Z s dW s , t

0 ≤ t ≤ T.

In a recent work of Buckdahn et al. [3] (2009), a notion of mean-field backward stochastic differential equation (MF-BSDEs in short) of the form T

T

t

t

Y t     E ′ fs,  ′ , , Y s , Y s  ′ , Z s ds −  Z s dW s , with t ∈ 0, T, was introduced. The authors deepened the investigation of such mean-field BSDEs by studying them in a more general framework, with a general driver. They established the existence and uniqueness of the solution under uniformly Lipschitzian conditions. The theory of mean-field BSDE has been developed by several authors. Du et al. [4] (2001); established a comparison theorem and existence in the case linear growth and continuous condition. Shi et al. [6]; introduced and studied mean-field backward stochastic Volterra integral equations. Mean-field Backward doubly stochastic differential equations T

Y t     E ′ fs,  ′ , , Y s , Y s  ′ , Z s ds t T

T

t

t

  E ′ gs,  ′ , , Y s  ′ , Z s dB s −  Z s dW s , with t ∈ 0, T, are deduced by Ruimin Xu [7] (2012), who obtained the existence and uniqueness result of the solution with uniformly Lipschitzian coefficients and presented the connection between McKean-Vlasov SPDEs and mean-field BDSDEs. In this paper, we study the case where the solution is forced to stay above a given stochastic process, called the obstacle. We obtain the real valued mean-field reflected backward doubly stochastic differential equation : with t ∈ 0, T T

Y t     E ′ fs, ,  ′ , Y s , Y ′s , Z s , Z ′s ds t T

T

t

t

  E ′ gs, ,  ′ , Y s , Y ′s , Z s , Z ′s dB s  K T − K t −  Z s dW s .

1

We establish the existence and uniqueness of solutions for equation (1) under uniformly Lipschitz conditions on the coefficients. In the case where the coefficient f is only continuous, we establish the existence of maximal and minimal solutions. In the case where the coefficient f is continuous with linear growth, we approximate f by a sequence of Lipschitz functions f n  and use a comparison theorem established here for MF-RBDSDEs. The paper is organized as follows : In Sections 2, we give some notations, assumptions, and we define a solution of RBDSDE. In Section 3, we state our main results for existence and uniqueness in the case where the coefficients are Lipschtzian, and we present a comparison theorem. The case where the generator is continuous and linear growth is treated in section 4.

Mean-field Reflected Backward Doubly Stochastic DE with Continuous Coefficients

64

2. Notation, Assumptions and Definitions Let , F, P be a complete probability space, and T  0. Let W t , 0 ≤ t ≤ T and B t , 0 ≤ t ≤ T be two independent standard Brownian motions defined on , F, P with values in R d and R, respectively. For t ∈ 0, T, we put, B B and G t : F W F t : F W t ∨ F t,T , t ∨ FT ,

B where F W t  W s ; 0 ≤ s ≤ t and F t,T  B s − B t ; t ≤ s ≤ T, completed with P-null sets. It should be noted that F t  is not an increasing family of sub  −fields, and hence it is not a filtration. However G t  is a filtration. Let M 2T 0, T, R d  denote the set of d −dimensional, jointly measurable stochastic processes  t ; t ∈ 0, T, which satisfy : T (a) E  | t | 2 dt  . 0 (b)  t is F t −measurable, for any t ∈ 0, T. We denote by S 2T 0, T, R, the set of continuous stochastic processes  t , which satisfy : (a’) E sup 0≤t≤T | t | 2  . (b’) For every t ∈ 0, T,  t is F t −measurable. Let , F, P    , F t ⊗ F t , P ⊗ P be the (non-completed) product of , F, P with itself. We denote the filtration of this product space by F  F t  F t ⊗ F t , 0 ≤ t ≤ T. A random variable  ∈ L 0 , F, P; R n  originally defined on  is extended canonically to  :  ′  ′ ,    ′ ,  ′ ,  ∈     . For every  ∈ L 1 , F, P, the variable . ,  :  → R belongs to L 1 , F, P, Pd − a. s, . We denote its expectation by

E ′ . ,  

  ′ , Pd ′ .

Notice that E ′   E ′ . ,  ∈ L 1 , F, P, and E 

 dP   E ′ . , Pd

 EE ′ .

Then we consider the following assumptions, H1) Let f :   0, T  R  R d  R  R d → R and g :   0, T  R  R d  R  R d → R be two measurable functions and such that for every y, z, y ′ , z ′  ∈ R  R d  R  R d , f. , y, z, y ′ , z ′  and, g. , y, z, y ′ , z ′  belongs in M 2 0, T, R H2) There exist constants L  0 and 0    12 , such that for every t,  ∈   0, T and y, z, y ′ , z ′  ∈ R  R d  R  R d , |ft, y 1 , z 1 , y ′1 , z ′1  − ft, y 2 , z 2 , y ′2 , z ′2 | ≤ L|y 1 − y 2 |  |y ′1 − y ′2 |  |z 1 − z 2 |  |z ′2 − z ′2 |  |gt, y 1 , z 1 , y ′1 , z ′1  − gt, y 2 , z 2 , y ′2 , z ′2 | ≤ L |y 1 − y 2 | 2  |y ′1 − y ′2 |

2

2

  |z 1 − z 2 | 2  |z ′2 − z ′2 |

2

.

H3) Let  be a square integrable random variable which is F T −mesurable.

65

N. CHAOUCHKHOUAN, B. LABED, and B. MANSOURI

H4) The obstacle S t , 0 ≤ t ≤ T, is a continuous F t −progressively measurable real-valued process satisfying E sup 0≤t≤T S t  2  . We assume also that S T ≤  a. s. Definition 2.1. A solution of equation (1) is a R  R d  R   -valued F t −progressively measurable process Y t , Z t , K t  0≤t≤T which satisfies equation (1) and i) Y, Z, K T  ∈ S 2  M 2  L 2 . ii) Y t ≥ S t . T iii) K t  is continuous and nondecreasing, K 0  0 and  Y t − S t dK t  0. 0

3. Existence of a Solution to the RBDSDE With a Lipschitz Condition Theorem 3.1. Under conditions, H1), H2), H3) and H4), the MF-RBDSDE (1) has a unique solution. Proof. For any y, z we consider the following MF-RBDSDE, with t ∈ 0, T T

T

Y t     E ′ fs, ,  ′ , Y s , y ′s , Z s , z ′s ds   E ′ gs, ,  ′ , Y s , y ′s , Z s , z ′s  dB s t

t

T

 K T − K t −  Z s dW s . t

According to Theorem 1 in Bahlali et al. [2], there exists a unique solution Y, Z ∈ S 2  M 2 i.e., if we define the process t

K t  Y 0 − Y t −  E ′ fs, ,  ′ , Y s , y ′s , Z s , z ′s ds 0

t

−  E ′ gs, ,  ′ , Y s , y ′s , Z s , z ′s  dB s  0

t

 0 Z s dW s ,

then Y, Z, K satisfies Definition 2.1. Hence, if we define Θy, z  Y, Z), then Θ maps S 2  M 2 itself. We show now that Θ is contractive. To this end, take any y i , z i  ∈ S 2  M 2 (i  1, 2, and let Θy i , z i   Y i , Z i . We denote Y, Z, K  Y 1 − Y 2 , Z 1 − Z 2 , K 1 − K 2  and 2 y, z,   y 1 − y 2 , z 1 − z 2 . Therefore, Itô’s formula applied to |Y| e t where   0, and the inequality 2ab ≤ 1 a 2  b 2 , lead to 2

E|Y t | e t   − 3L −

8L

2

T

E  |Y s | e s ds 

1−2

2

t

1 2

T

E  e s |Z s | ds 2

t

T

≤  E  e s Y s dK 1s − dK 2s  t T

 E  e s t

Choosing   3L  yield

L 8L

2

1−2

1−2 2L



1 2

2 |y s |   124  |z s | 2 ds 4 12

L

1−2 2L

and setting M 

4 12

L

1−2 2L

Mean-field Reflected Backward Doubly Stochastic DE with Continuous Coefficients

E|Y t | e t  2

1 2

T

ME  |Y s | e s ds  2

t

T

≤ E  e s Y s dK 1s − dK 2s   t

1 2

T

66

E  e s |Z s | ds 2

t

12 4

T

E  e s M|y s | 2  |z s | 2 ds. t

As T

E  e s Y s dK 1s − dK 2s   0, t

and T

E  e s M|Y s |  |Z s | 2

2

t

12 2

ds ≤

T

E  e s M|y s | 2  |z s | 2 ds, t

consequently the mapping Θ is a strict contraction on S 2  M 2 equipped with the norm ‖Y, Z‖  

T

E e t

s

2

M|Y s |  |Z s |

2

ds

1 2

.

Moreover, it has a unique fixed point, which is the unique solution of the MF-RBDSDE with data , f, g, S. 

4. RBDSDEs With a Continuous Coefficient In this section we prove the existence of a solution to the MF-RBDSDE where the coefficient is only continuous. Towards this end, we consider the following assumption. H5) i) for a. e t, w, the mapping y, y ′ , z, z ′   ft, y, y ′ , z, z ′  is continuous. ii) There exist constants L  0 and  ∈ 0, 12 , such that for every t,  ∈   0, T and y, z, y ′ , z ′  ∈ R  R d  R  R d , |ft, y, y ′ , z, z ′ | ≤ L1  |y|  |y ′ |  |z|  |z ′ | |gt, y 1 , y ′1 , z 1 , z ′1  − gt, y 2 , y ′2 , z 2 , z ′2 | ≤ L |y ′1 − y ′2 |  |y 1 − y 2 | 2 2

2

 |z ′1 − z ′2 |  |z 1 − z 2 | 2 2

Theorem 4.1. Under assumption H1), H3), H4) and H5), the MF-RBDSDE (1) has an adapted solution Y, Z, K which is a minimal one, in the sense that, if Y ∗ , Z ∗  is any other solution we have Y ≤ Y ∗ , P − a. s. Before giving a proof to this theorem, we invoke first the following classical lemma, which can be proved by adapting the proof given in Alibert and Bahlali [1]. Lemma 4.1. Let f : 0, T    R  R  R d  R be a measurable function such that: (a) For almost every t,  ∈ 0, T  , x  ft, , x is continuous, (b) There exists a constant K  0 such that for every t, y ′ , y, z ∈ 0, T  R  R  R d

67

N. CHAOUCHKHOUAN, B. LABED, and B. MANSOURI

|ft, y ′ , y, z|≤ K1  |y ′ ||y|  |z| a. s. (c) For almost every y, z, ft, y ′ , y, z is increasing in y ′ . Then, the sequence of functions f n t, y ′ , y, z  inf ft, u, v, w  ny ′ − u   n|y − v|n|z − w| u,v,w∈Q 2d

is well defined for each n ≥ K and satisfies: (1) for every t, y ′ , y, z ∈ 0, T  R 2d , |f n t, y ′ , y, z|≤ K1  |y ′ ||y|  |z|, (2) for every t, y ′ , y, z ∈ 0, T  R 2d , n → f n t, x is increasing, (3) for every t, y ′ , y, z ∈ 0, T  R 2d , y ′ → f n t, y ′ , y, z is increasing, (4) for every n ≥ K, t, y ′1 , y 1 , z 1  ∈ 0, T  R 2d ,t, y ′2 , y 2 , z 2  ∈ 0, T  R 2d |f n t, y ′1 , y 1 , z 1  − f n t, y ′2 , y 2 , z 2 |≤ n|y ′1 − y ′2 ||y 1 − y 2 ||z ′1 − z ′2 |, (5) If y ′n , y n , z n  → y ′ , y, z, as n →  then for every t ∈ 0, T f n t, y ′n , y n , z n  → ft, y ′ , y, z as n → . Since  satisfies H3), we get from theorem 3.1, that for every n ∈ N ∗ , there exists a unique solution Y nt , Z nt , K nt , 0 ≤ t ≤ T for the following MF-RBDSDE T

T

t T

t

Y nt     f n s, Y ns  ′ , Y ns , Z ns ds  K nT − K nt   gs, Y ns  ′ , Y ns , Z ns dB s −  Z ns dW s , t

0 ≤ t ≤ T,

T

Y nt ≥ S t ,  Y ns − S s dK ns  0. 0

2

Since, |f 1 t, u, v, w − f 1 t, u ′ , v ′ , w ′ |≤ L|u − u ′ ||v − v ′ |  |w − w ′ |, we consider the function defined by f 1 t, u, v, w : L1  |u||v|  |w|, then a similar argument as before shows that there exists a unique solution U s , V s , K s , 0 ≤ s ≤ T to the following MF-RBDSDE: T

T

T

t

t

t

U t     f 1 s, U ′s , U s , V s ds  K T − K t   gs, U ′s , U s , V s dB s −  V s dW s Ut ≥ St, T

 0 U s − S s dK s  0.

3

We would also need the following comparison theorem. Theorem 4.2. (Comparison theorem) Let  1 , f 1 , g, S 1  and  2 , f 2 , g, S 2  be two MF-RBDSDEs. Each one satisfying all the previous assumptions H1), H2), H3) and H4). Assume moreover that : i)  1 ≤  2 a.s. ii) f 1 t, y ′ , y, z ′ , z ≤ f 2 t, y ′ , y, z ′ , z dP  dt a. e. ∀y ′ , y, z ′ , z ∈ R  R d . iii) S 1t ≤ S 2t , 0 ≤ t ≤ T a. s. Let Y 1 , Z 1 , K 1  be a solution of MF-RBDSDE  1 , f 1 , g, S 1  and Y 2 , Z 2 , K 2  be a solution of MF-RBDSDE  2 , f 2 , g, S 2 . We suppose also : a) One of the two generators is independent of z ′ .

Mean-field Reflected Backward Doubly Stochastic DE with Continuous Coefficients

68

b) One of the two generators is nondecreasing in y ′ . then Y 1t ≤ Y 2t , 0 ≤ t ≤ T a. s. Proof. Suppose that (a) is satisfied by f 1 and (b) by f 2 . Applying Itô’s formula to  2 |Y 1t − Y 2t  | , and passing to expectation, we have T

E|Y 1t − Y 2t   |  E  1 Y 1s Y 2s  |Z 1s − Z 2s | 2 ds 2

t

T

 2 E  Y 1s − Y 2s   E ′ f 1 s, Y 1s  ′ , Y 1s , Z 1s − f 2 s, Y 2s  ′ , Y 2s , Z 2s  ′ , Z 2s

ds

t T

 2 E  Y 1s − Y 2s   dK 1s − dK 2s t

T

 E  E ′ g s, Y 1s  ′ , Y 1s , Z 1s  ′ , Z 1s − g s, Y 2s  ′ , Y 2s , Z 2s  ′ , Z 2s t

2

1 Y 1s Y 2s  ds.

Since on the set Y 1s  Y 2s , we have Y 1t  S 2t ≥ S 1t , then T

T

 t Y 1s − Y 2s   dK 1s − dK 2s   −  t Y 1s − Y 2s   dK 2s ≤ 0. Since f 1 and f 2 are Lipschitzian, we have on the set Y s  Y ′s , T

E|Y 1t − Y 2t   |  E  1 Y 1s Y 2s  |Z 1s − Z 2s | 2 ds 2

≤ E

t T

2

 2 |Y 1t − Y 2t  |  |Z 1s − Z 2s | ds ,

2

 |Y 1t − Y 2t  | ds.

L 6L  1−2 

t

2

then E|Y 1t − Y 2t   | ≤ E  2

T t

L 6L  1−2 

2



The required result follows by using Gronwall’s lemma. ≤ U t . ii) There exists Z ∈ M 2 , such that Z n Lemma 4.2. i) a. s. for all t , Y 0t ≤ Y nt ≤ Y n1 t converges to Z.

Proof. Assertion i) follows from the comparison theorem. We therefore need to prove ii) only. Itô’s formula yields T

T

T

E|Y n0 | 2  E  |Z ns | 2 ds  E|| 2  2E  Y ns E ′ f n s, Y ns  ′ , Y ns , Z ns  ds  2E  S s dK ns 0

0

T

 E  E′ 0

gs, Y ns  ′ , Y ns , Z ns 

0

2

ds.

2 From assumption H5) , and the inequality 2ab ≤ a  b 2 for   0, we get :

69

N. CHAOUCHKHOUAN, B. LABED, and B. MANSOURI T T 2 L  4LE  T|Y n | 2 ds E  |Z ns | 2 ds ≤ E|| 2  LT  E 0, 0, 0 | ds   3 L  |gs,  s   0 0 0

L   E  T 1 |Z n | 2 ds  2E  T S s dK n . s   s 0 0



On the other hand, we have from (2) T

T

0

0

K nT  Y n0 −  −  E ′ f n s, Y ns  ′ , Y ns , Z ns ds −  E ′ gs, Y ns  ′ , Y ns , Z ns dB s  

T

Z ns

0

4

dW s .

Then T

EK nT  2 ≤ C 1  E  ‖Z ns ‖ 2 ds . 0

We also have T

2E

 0 S s dK ns ≤

1 

≤

1 

 EK nT  2

E sup |S t | 2 t

T

 C 1  E  ‖Z ns ‖ 2 ds ,

E sup |S t | 2

0

t

which leads to T T 2 L  4LE  T|Y n | 2 ds E  |Z ns | 2 ds ≤ E|| 2  LT   C  E 0, 0, 0 | ds   3 L  |gs,  s   0 0 0



L    C E  T 1 |Z n | 2 ds  1 E sup|S t | 2 .   s  t 0

Choosing ,  such that  L    C  1, we obtain T

E  ‖Z ns ‖ 2 ds ≤ C. 0

For n, p ≥ K, Itô’s formula gives, T

EY n0 − Y 0  2  E  ‖Z ns − Z s ‖ 2 ds p

p

0

T

 2 E  Y ns − Y s E ′ f n s, Y ns , Y ns  ′ , Z ns  − f p s, Y s , Y s  , Z s ds p

0 T

p

p ′

p

 2 E  Y ns − Y s dK ns  2 E  Y s − Y ns dK s T

p

 E

T 0

p

p

0

0

p ′

E ′ gs, Y ns , , Y ns  ′ , Z ns  − gs, Y s , Y s  , Z s  p

p

But T

T

E  Y ns − Y s dK ns  E  S s − Y s dK ns ≤ 0. p

0

Similarly, we have E 

0 T p Y s 0

p

p

− Y ns dK s ≤ 0. Therefore,

2

ds.

Mean-field Reflected Backward Doubly Stochastic DE with Continuous Coefficients

T

T

70

E  ‖Z ns − Z s ‖ 2 ds ≤ 2E  Y ns − Y s E ′ f n s, Y ns , Y ns  ′ , Z ns  − f p s, Y s , Y s  , Z s ds p

0

p

p ′

p

p

0

 E

T 0

p ′

2

E ′ gs, Y ns , Y ns  ′ , Z ns  − gs, Y s , Y s  , Z s  p

p

ds.

By Hôlder’s inequality and the fact that g is Lipschitzian, we get T

E  ‖Z ns − Z s ‖ 2 ds p

0

T

1 2

E  Y ns − Y s  2 ds

≤

 LE 

0 T

0 T

p

T

E  E ′ f n s, Y ns , Y ns  ′ , Z ns  − f p s, Y s , Y s  , Z s  2 ds p

p ′

p

1 2

0

T

|Y ns − Y s | 2  |Y ns  ′ − Y s  | 2 ds  E  |Z ns − Z s | 2 ds p ′

p

p

0

Since sup E  |f n s, Y ns , Y ns  ′ , Z ns | 2 ≤ C, we obtain, 0

n

E

T 0

‖Z ns

−

p Z s ‖ 2 ds

≤ C E

T 0

Y ns

−

p Y s  2 ds

1 2

.

Hence T

E  ‖Z ns − Z s ‖ 2 ds  0, as n, p → . p

0

Thus Z n  n≥1 is a Cauchy sequence in M 2 R d .



4.1. Proof of Theorem 4.1. Let Y t sup Y nt , and we have Y n , Z n  → Y, Z in S 2 R d   M 2 R d . Then, along a n

subsequence which we will still denote as Y n , Z n , we have Y n , Z n  → Y, Z, dt ⊗ dP a. e. Then, by using Lemma 4.1, we get f n t, Y nt , Y ns  ′ , Z nt  → ft, Y t , Y t  ′ , Z t  dPdt a. e. On the other hand, since Z n  Z in M 2 R d , then there exists an  ∈ M 2 R and a subsequence, which we continue to denotes as denote Z n , such that ∀n, |Z n |≤ , Z n  Z, dt ⊗ dP a. e. Moreover, from H5), and Lemma 4.2, we have |f n t, Y nt , Y nt  ′ , Z nt | ≤ 1 sup |Y nt | sup |Y nt  ′ | t  ∈ L 2 0, T, dt, P − a. s. n

n

It follows from the dominated convergence theorem that, T

E  E ′ f n s, Y ns , Y nt  ′ , Z ns  − fs, Y s , Y s  ′ , Z s  0

Subsequently,

2

ds  0,

n → .

71

N. CHAOUCHKHOUAN, B. LABED, and B. MANSOURI T

E  ‖E ′ gs, Y ns , Z ns  − gs, Y s , Z s ‖ 2 ds 0

T

≤ CE  E ′ |Y ns − Y s | 2  |Y ns  ′ − Y s  ′ | 2 ds 0 T

  E  ‖Z ns − Z s ‖ 2 ds  0, 0

as n → .

It is not difficult to show that Y, Z is a solution to our MF-RBDSDE. Indeed, let T

Y t     E ′ f s, Y s , Y s  ′ , Z s ds  K T − K t t

T

T

t

t

  E ′ g s, Y s , Y s  ′ , Z s dB s −  Z s dW s , where Z ∈ M , Y ∈ S , K T ∈ L , Y t ≥ S t , K t  is continuous and nondecreasing, K 0  0 and T  0 Y t − S t dK t  0, and Y ∗ , Z ∗ , K ∗  be a solution of (1). Then, by theorem 4.2, we have for every n ∈ N ∗ , Y n ≤ Y ∗ . Therefore, Y is a minimal solution of (1). 2

2

2

Remark 4.1. Using the same arguments and the following approximating sequence Eh n t, x, y, z  sup hu, v, w − n|x − u|−n|y − v|−n|z − w| , u,v,w∈Q p

one can prove that the MF-RBDSDE (1) has a maximal solution. Acknowledgements The authors would like to thank the referee for some comments which have led to an improvement of the paper.

References [1] J. J. Alibert, and K. Bahlali, Genericity in deterministic and stochastic differential equations, Séminaire de Probabilités XXXV, Lecture Notes in Mathematics 1755, (2001), 220-240. [2] K. Bahlali, M. Hassani, B. Mansouri, and N. Mrhardy, One barrier reflected backward doubly stochastic differential equations with continuous generator, Comptes Rendus Mathematique 347(19), (2009), 1201-1206. [3] R. Buckdan, B. Djehiche, J. Li, and S. Peng, Mean-field backward stochastic differential equations : a limit approach, Annals of Probability 37, (2009), 1524-1565. [4] H. Du, J. Li, and Q. M. Wei, Mean-field backward stochastic differential equations with continuous coefficients, In Proceedings of the 30th Chinese Control Conference, July 22-24, Yantai, China, 2012. [5] E. Pardoux, and S. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probability Theory and Related Fields 98, (1994), 209-227.

Mean-field Reflected Backward Doubly Stochastic DE with Continuous Coefficients

72

[6] Y. Shi, Y. Gu, and K. Liu, Comparison theorems of backward doubly stochastic differential equations and applications, Stochastic Analysis and Applications 23, (2005), 97-110. [7] R. Xu, Mean-field backward doubly stochastic differential equation and related SPDEs, Boundary Value Problems 2012:114, (2012). [8] L. Zhi, and J. Luo, Mean-field reflected backward stochastic differential equations, Statistics and Probability Letters 82, (2012), 1961-1968. ______ Article history: Submitted October, 11, 2014; Accepted February, 05, 2015.