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HOMOGENIZATION OF BSDES WITH TWO REFLECTING BARRIERS, VARIATIONAL INEQUALITY AND STOCHASTIC GAME.

1. INTRODUCTION

Backward stochastic differential equations (BSDE's in short) is an interesting subject in stochastic calculus developed since the pioneering works of Pardoux and Peng [27], [28]. The application of such equations to finance theory and nonlinear partial differential equations has motivated many efforts to establish existence and uniqueness of the solution (see [1], [3], [9], [13], [16], [23], [26] and the references given there). In [11], El Karoui et al have introduced the notion of one barrier reflected BSDE, which is a backward equation but the solution is forced to stay above a given continuous obstacle. Moreover, the authors have established the existence and uniqueness of the solution via a penalization as well as Picard's iteration methods. The notion of double barriers reflected BSDE has been introduced by Cvitanic and Karatzas [7] where the solution is forced to remain between two described upper and lower barriers U and L. As is well known, BSDE provide probabilistic formulae for the viscosity solution of semilinear partial differential equations (PDE) (see for instance Pardoux and Peng [28], Pardoux [26] and references therein). In the present work, we wish to consider a more general problem, homogenization of two barriers reflected BSDE when the solutions are forced to stay between an upper and lower obstacles, and also provide a probabilistic formulae to limit of system of variational inequalities with two obstacles.

In this paper, we use a penalization method to show the existence and uniqueness of the weak solution for the homogenized problem associated to the reflected BSDE (3.4) when the upper barrier U and the lower barrier L are smooth Ito processes. Such equations appear when one studies the notion of zero-sum mixed problems [15] or American game options [8]. Variational inequality theory was introduced by Hartman and Stampacchia [20], as a tool for the study of partial differential equations with applications principally drawn from mechanics. Such variational inequalities were infinite-dimensional rather than finite-dimensional as we will be studying here. Equilibrium is a central concept in numerous disciplines including economics, management science, operations research, and engineering. Variational inequality theory is a powerful unifying methodology for the study of equilibrium problems. This type of inequalities also arises in zero sum stochastic differential games of mixed type where each player uses both continuous control and stopping times.

A probabilistic approach of the homogenization property has been developed since the early article of Freidlin [12] (see also [5, Chapter 3]) parallel to analytical one. The link between BSDE's and homogenization of semilinear PDE's are given since 1996 by the work of Pardoux [26].

The paper is organized as follows. The BSDE problem with reflecting barriers as well as some preliminary results are described in Section 2. In Section 3, we prove existence and uniqueness of the solution in the Markovian case. The homogenization problem is treated in Section 4 by using the result of Section 3. The last section is devoted to the homogenization problem for variational inequality and stochastic differential games of mixed type.

2. NOTATIONS AND PRELIMINARY RESULTS

Let (Q, F, P, [F.sub.t], [W.sub.t], t [member of] [0,T]) be a complete Wiener space in Rd, i.e. (Q, F, P) is a complete probability space, ([F.sub.t], t [member of] [0, T]) is a right continuous increasing family of complete sub [sigma]-algebras of F, ([W.sub.t], t [member of] [0,T]) is a standard Wiener process in Rd with respect to ([F.sub.t], t [member of] [0,T]). We assume that

[F.sub.t] = [sigma] [[W.sub.s], s [less than or equal to] t] [cross product] N

where N denotes the totality of P-null sets and [[sigma].sub.1] [cross product] [[sigma].sub.2] denotes the [sigma]-field generated by [[sigma].sub.1] [union] [[sigma].sub.2]. Let P be the [sigma]-field of predictable subsets of Q x [0,T].

Let us introduce the following spaces:

* [L.sup.2] of [F.sub.T]-measurable, real-valued random variables [member of], E [[[absolute value of ([xi])].sup.2]] < +[infinity].

* [S.sup.2] of continuous ([F.sub.t])-adapted real-valued processes [([Y.sub.t]).sub.t[less than or equal to]T], E[[sup.sub.t[less than or equal to]T] [[absolute value of ([Y.sub.t])].sup.2]] < [infinity].

* [H.sup.2,k] of ([F.sub.t])-progressively measurable processes, valued in [mathematical expression not reproducible].

* [A.sup.2] of continuous, real-valued, increasing, ([F.sub.t])-adapted process [([K.sub.t]).sub.0[less than or equal to]t[less than or equal to]T] such that K(0) = 0 and E[[absolute value of ([K.sub.T])]].sup.2] < +[infinity].

Finally, given:

(A1): a terminal value [xi] [member of] [member of] [L.sup.2]

(A2): a coefficient "f" which is a map f: [OMEGA] x [0,T] x [R.sup.1+d] [right arrow] R such that 1. f is P x B([R.sup.1+d])-measurable and satisfies: [(f (t, 0, 0)).sub.t[less than or equal to]T] belongs to [L.sup.2]([OMEGA] x [0, T], dP [direct sum] dt) i.e.,

(2.1) E [[integral].sup.T.sub.0] [[absolute value of (f(t, 0, 0))].sup.2] dt < +[infinity]

2. f is uniformly Lipschitz with respect to (y,z), i.e., there exists a constant k [greater than or equal to] 0 such that for any y, y', z, z' [member of] R,

(2.2) P-a.s., [absolute value of (f ([omega], t, y, z) - f([omega], t, y', z'))] [less than or equal to] k([absolute value of (y - y')] + [absolute value of (z - z')]).

and

(A3): two reflecting barriers L, U [member of] [S.sup.2], i.e. real valued and P-measurable processes satisfying

(2.3) [mathematical expression not reproducible],

we shall always assume that

[for all] 0 [less than or equal to] t [less than or equal to] T, [L.sub.t] [less than or equal to] [U.sub.t] and [L.sub.T] [less than or equal to] [member of] [less than or equal to] [U.sub.T], P-a.s.

2.1. On one barrier. Let f : [OMEGA] x [0, T] x [R.sup.1+d] [right arrow] R which has the property (A2), and only the lower barrier in (A3).

Definition 2.1. A solution for one barrier reflected BSDE associated with (f, [xi],L) is a P-measurable process (Y, Z, K) := [([Y.sub.t], [Z.sub.t], [K.sub.t]).sub.t[less than or equal to]T] valued in [R.sup.1+d] x [R.sup.+] and which satisfies:

[mathematical expression not reproducible].

The following result established by El Karoui et al. [11] is concerned with the existence and uniqueness of a solution for a single barrier reflected BSDE associated with (f,[xi],L).

Theorem 2.2. Suppose that the assumptions (A1)-(A2) and (A3) hold for (f, f, L), then there exists a unique P-measurable process (Y, Z, K) solution of the one barrier reflected BSDE associated with (f, [xi], L) has a unique solution. Furthermore the following inequality holds

(2.4) [mathematical expression not reproducible]

We need also a comparison result given in the same paper, our assumptions are rather strong than required. Unfortunately we need those assumptions for homogenization purpose.

Theorem 2.3. Let (f, [xi], L) and (f', [xi]', L') be two sets of data, each one satisfying all assumptions (A1), (A2), (A3) and such that P-a.s. [xi] [less than or equal to] [xi]' [L.sub.t] [less than or equal to] [L'.sub.t]; and f(t,y,z) [less than or equal to] f'(t,y,z) dt [cross product] dP-a.e. [for all](y,z) G [R.sup.1+d]. Then [Y.sub.t] [less than or equal to] [Y'.sub.t], 0 [less than or equal to] t [less than or equal to] T P-a.s. and dK [greater than or equal to] dK'.

2.2. Double barriers reflected BSDE. Let us now introduce our double barriers reflected BSDE (DRBSDE in short)

Definition 2.4. The process [([Y.sub.t], [Z.sub.t], [K.sup.+.sub.t], [K.sup.- .sub.t]).sub.t[less than or equal to]T], with value in [R.sup.1+d] x [R.sup.+] x [R.sup.+], is called a solution for the double barriers DRBSDE associated to (f, [xi], L, U) if

[mathematical expression not reproducible].

Remark 2.5. Since our coefficient is Lipschitz in (y,z) uniformly in (u,t), the existence and uniqueness of a solution was proved (with some additional conditions) in Cvitanic and Karatzas [7] (Corollary 5.5, Theorem 6.1 or Theorem 6.5), see also Hamadene and Hassani [14], Peng and Xu [29].

3. MARKOVIAN CASE

Let ([X.sup.[epsilon].sub.t]; t [greater than or equal to] 0} a diffusion process with values in [R.sup.d] and generator L[member of], such that [X.sup.e] [??] X in C([0,T], [R.sup.d]) equipped with the topology of convergence on compact subsets of [R.sub.+], where X itself is a diffusion with generator [L.sup.0]. We suppose that the martingale problem associated to X is well posed, and there exist p,q > 0 such that

(3.1) [mathematical expression not reproducible].

We assume moreover that

1. g : [R.sup.d] [right arrow] R, f : [0, T] x [R.sup.d] x R [right arrow] R are continuous, which are such that for some C > 0, K > 0 and for all x [member of] [R.sup.d], y, y' [member of] R

(3.2) [mathematical expression not reproducible].

2. The barriers

[(L(s, [X.sup.[epsilon].sub.s])).sub.s[less than or equal to]T] and [(U(s,[X.sup.[epsilon].sub.S])).sub.s<T] [member of] [S.sup.2]

where the function U(s,x) and L(s,x) are in [C.sup.1,2] ([0,T] x [R.sup.d]; R) such that for U (resp. L),

(3.3) [mathematical expression not reproducible]

We consider the following double barriers reflected BSDE associated to

(f(s, [X.sup.[epsilon].sub.s], x), g([X.sup.[epsilon].sub.T]), L(*, [X.sup.[epsilon]]), U(*, [X.sup.[epsilon]]))

that is

(3.4) [mathematical expression not reproducible].

We need the following a priori estimates, to prove the existence and uniqueness result for (3.4)

Lemma 3.1. For each [epsilon] > 0,

[mathematical expression not reproducible].

Proof. The first inequality is a consequence of existence theorem in Lepeltier and San Martin ([22, Theorem 1]), since our assumptions are sufficient (see Remark 1 in [22]). Now using (3.2), we have [[absolute value of (f(t, [X.sup.[epsilon].sub.t], [Y.sup.[epsilon].sub.t])].sup.2] [less than or equal to] [C.sub.1](1 + [absolute value of ([[X.sup.[epsilon].sub.t])].sup.2q] + [[absolute value of ([Y.sup.[epsilon].sub.t])].sup.2]), for some [C.sub.1] > 0. By (3.1), we obtain the second.

We can now give existence and uniqueness result for the system (3.4).

Proposition 3.1. For each [epsilon] > 0, the DRBSDE (3.4) has a unique solution ([Y.sup.[epsilon]], [Z.sup.[epsilon]], [K.sup.+[epsilon]], [K.sup.-[epsilon]]), also [K.sup.+[epsilon]] and [K.sup.- [epsilon]] are absolutely continuous with respect to the Lebesgue measure.

Proof. Recall that the barriers are smooth Ito processes (in [S.sup.2]) by assumption and the process [(f (s, [X.sup.[epsilon].sub.s], [Y.sup.[epsilon],n.sub.s])).sub.0[less than or equal to]s[less than or equal to]T] [member of] [S.sup.2] from Lemma 3.1. The result follows from Theorem 6.1 in [7].

3.1. Connection with Dynkin games. As shown by Cvitanic and Karatzas ([7, Theorems 4.1 and 6.5]) see also Hamadene and Hassani [14], the existence of a solution ([Y.sup.[epsilon]], [Z.sup.[epsilon]], [K.sup.+[epsilon]], [K.sup.-[epsilon]]) to (3.4) implies that [Y.sup.[epsilon]] is the value of a certain stochastic game of stopping. Let us recall the corresponding stochastic game.

We define

[M.sub.t,[theta]] := {[tau] [member of] M/t [less than or equal to] [tau] [less than or equal to] [theta] a.s.} for 0 [less than or equal to] t [less than or equal to] [theta] [less than or equal to] T

where M is the class of F-stopping times [tau] : [OMEGA] [right arrow] [0,T], and let h(t) := f ([X.sup.[epsilon].sub.t], [Y.sup.[epsilon].sub.t]). For any [epsilon] > 0, 0 [less than or equal to] t [less than or equal to] T and any two stopping times [sigma], [tau] [member of] [M.sub.t,T], consider the payoff (3.5)

[mathematical expression not reproducible],

as well as the upper and the lower values, respectively,

(3.6) [mathematical expression not reproducible]

of the corresponding stochastic game. This game has value Vt[member of], given by the stateprocess Ye the first component of the solution to DRBSDE (3.4), that is,

(3.7) [V.sup.[epsilon].sub.t] = [[bar.V].sup.[epsilon].sub.t] = [[V.bar].sup.[epsilon].sub.t] = [Y.sup.[epsilon].sub.t] a.s. [for all]0 [less than or equal to] t [less than or equal to] T,

as well as saddle-point [mathematical expression not reproducible] given by

(3.8) [mathematical expression not reproducible],

namely

(3.9) [mathematical expression not reproducible].

for every ([sigma], [tau]) [member of] [M.sub.t,T] x [M.sub.t,T] (see [7, Theorem 4.1]).

3.2. Link with variational inequality. Let us recall some known results about this link.

Let x [member of] [R.sup.d] and {[X.sup.t,x,[epsilon].sub.s]; 0 [less than or equal to] t [less than or equal to] s [less than or equal to] T} the diffusion process defined as above, starting at x at time t. We denote by ({[Y.sup.t,x,[epsilon].sub.s], [Z.sup.t,x,+[epsilon].sub.s], [K.sup.t,x,+[epsilon].sub.s], [K.sup.t,x,-[epsilon].sub.s]}; 0 [less than or equal to] t [less than or equal to] s [less than or equal to] T) be the unique solution associated to DRBSDE (f (s, [X.sup.t.sub.t,x,[epsilon].sub.s], *), g([X.sup.t,x,[epsilon].sub.T]), L(*, [X.sup.t,x,[epsilon]]), U(*, [X.sup.t,x,[epsilon]])). We assume moreover the polynomial growth condition on the barriers

(3.10) [absolute value of (U(t,x))] + [absolute value of (L(t,x))] [less than or equal to] C(1 + [[absolute value of (x)].sup.p])

for any t [member of] [0, T] and the constants C and p are already used in (3.2). Let us consider the following double obstacles variational inequality:

(3.11) [mathematical expression not reproducible].

Definition 3.2. Let ue be a function which belongs to C ([0,T] x [R.sup.d]; R); ue is said to be a viscosity

(i) subsolution of (3.11) if u [member of] (T, *) [less than or equal to] g(*) and for any [phi] [member of] [C.sup.1,2] ([0,T] x [R.sup.d]; R) and any local maximum point (t, x) [member of] [0, T] x [R.sup.d] of [u.sup.[epsilon]] - 0, we have

min (([u.sup.[epsilon]] - L); max [[u.sup.[epsilon]] - U); ([partial derivative][phi]/[partial derivative]t - [L.sup.[epsilon].sub.t][phi]) - (*, *, u[epsilon])]) t,x) [less than or equal to] 0

(ii) supersolution of (3.11) if [u.sup.[epsilon]](T, *) [greater than or equal to] g(*) and for any [phi] [member of] [C.sup.1,2] ([0,T] x [R.sup.d]; R) and any local minimum point (t, x) [member of] [0, T] x [R.sup.d] of [u.sup.[epsilon]] - [phi], we have

min (([u.sup.[epsilon]] - L); max [([u.sup.[epsilon]] - U); (-[partial derivative][phi]/[partial derivative]t - [L.sup.[epsilon].sub.t][phi]) - f(*, *, [u.sup.[epsilon]])]) (t, x) [greater than or equal to] 0

(iii) solution of (3.11) if it is both a viscosity subsolution and supersolution.

Since in our setting, the generator f do not depend on z, by virtue of Hamadene and Hassani ([14, Theorem 7.2]), the function [u.sup.[epsilon]]: [0,T] x [R.sup.d] [right arrow] R defined by [u.sup.[epsilon]](t, x) = [Y.sup.t,x,[epsilon].sub.t], is a viscosity solution of (3.11).

4. THE DRBSDE HOMOGENIZATION RESULT

Let {([Y.sup.[epsilon].sub.s], [Z.sup.[epsilon].sub.s], [K.sup.+[epsilon].sub.s], [K.sup.-[epsilon].sub.s]); 0 [less than or equal to] s [less than or equal to] t} the unique solution of double barriers reflected BSDE (3.4). We want to prove that ([X.sup.[epsilon]], [Y.sup.[epsilon]], [Z.sup.[epsilon]], [K.sup.+[epsilon]], [K.sup.-[epsilon]]) converge in law to (X, Y, Z, [K.sup.+], [K.sup.+]) where (Y, Z, [K.sup.+], [K.sup.+]) is the unique solution of double barriers reflected BSDE.

(4.1) [mathematical expression not reproducible].

We give now a useful Lemmas, the first give us some tightness criteria for sequence of quasi-martingales.

Lemma 4.1 (See Meyer-Zheng [25] or Kurtz [24]). The sequence of quasi-martingale {[V.sup.n.sub.s]; 0 [less than or equal to] s [less than or equal to] T} defined on the filtered probability space {[OMEGA]; [F.sub.s], 0 [less than or equal to] s [less than or equal to] T; P} is tight whenever

[mathematical expression not reproducible]

where C[V.sub.T]([V.sup.n]), denotes the "conditional variation of [V.sup.n] on [0,T]" defined by

[mathematical expression not reproducible],

with "sup" meaning that the supremum is taken over all partitions of the interval [0,T].

We use the penalization method to prove the convergence result. This second Lemma is the core, which explain each step.

Lemma 4.2. Let [U.sup.[epsilon]] be a family of random variables defined on the same probability spaces. For each [epsilon] [greater than or equal to] 0, we assume the existence of a family of random variables [([U.sup.[epsilon],n]).sup.n], such that

* [U.sup.[epsilon],n] [??] [U.sup.0,n] as [member of] goes to zero.

* [U.sup.[epsilon],n] [??] [U.sup.[epsilon]] as n [right arrow] +[infinity], uniformly in [epsilon].

* [U.sup.0,n] [??] [U.sup.0] as n [right arrow] +[infinity]

then, [U.sup.[epsilon]] converge in distribution to [U.sup.0].

Proof. This lemma is a simplified version of Theorem 3.2 in Billingsley [2, p. 28].

We put

[M.sup.[epsilon].sub.t] = -[[integral].sup.t.sub.0] [Z.sup.[epsilon].sub.s] d[B.sub.r].

We denote by

* C([0, T], [R.sup.d]) the space of functions of [0, T] with values in Rd equipped with the topology of uniform convergence.

* D([0, T], [R.sup.k]) is the space of cadlag functions of [0, T] with values in Rk equipped with the topology of Meyer-Zheng.

Theorem 4.3. Under the above conditions, as [epsilon] tends to zero, the family of processes ([X.sup.[epsilon]],[Y.sup.[epsilon]],[M.sup.[epsilon]],[K.sup.+[epsilon]],[K.sup .-[epsilon]]) converge in law to the processes (X, Y, M, [K.sup.+],[K.sup.-]) in C([0,t],Rd) x D([0,t],[R.sup.2]) x (C[0,t],R) x (C[0,t],R).

The proof of this theorem follow the above Lemma. We first need some extra lemmas.

Let define

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

Consider the backward stochastic differential equation

(4.2) [mathematical expression not reproducible]

and let ([Y.sup.n], [Z.sup.n]), be the unique solution of the backward stochastic differential equation

(4.3) [Y.sup.n.sub.s] = g([X.sub.T]) + [[integral].sup.T.sub.s] f(r, [X.sub.r], [Y.sup.n.sub.r])dr - [[integral].sup.T.sub.s] [Z.sup.n.sub.r]d[B.sub.r] + [[integral].sup.T.sub.s] [k.sup.0,n.sub.r]([Y.sup.n])dr.

We set

[mathematical expression not reproducible]

Lemma 4.4. Under the above conditions, we have for each n > 0

[mathematical expression not reproducible]

Proof. Let ([[Y.bar].sup.[epsilon],n], [[Z.bar].sup.[epsilon],n.sub.r]) and ([[bar.Y].sup.[epsilon],n] [[bar.Z].sup.[epsilon],n.sub.r]) be the solution of the following BSDE

[mathematical expression not reproducible]

then by the comparison theorem for BSDE we have for all n > 0, [epsilon] > 0,

[[Y.bar].sup.[epsilon],n] < [Y.sup.[epsilon],n] [less than or equal to] [[bar.Y].sup.[epsilon],n]

Since [f.sub.n](t,x, y) [member of] {f (t,x, y) - n[(y - U(t, y)).sup.+], f (t, x, y) + n[(L(t, x)- y).sup.+]} is Lipschitz in y uniformly in ([omega], t, x), so using (3.1), we have [sup.sub.[epsilon]]E [[sup.sub.t[less than or equal to]T] < [infinity] and [sup.sub.[epsilon]]E [[sup.sub.t[less than or equal to]T] [[absolute value of ([[bar.Y].sup.[epsilon],n.sub.t])].sup.2]] < [infinity]. Hence

[mathematical expression not reproducible]

From (3.2) and (3.1) we have [sup.sub.[epsilon]]E[[sup.sub.t[less than or equal to]T][[absolute value of (f(t, [X.sup.[epsilon].sub.t], [Y.sup.[epsilon],n.sub.t]))].sup.2]] < [infinity].

Lemma 4.5. There exists a constant C such that for each n > 0,

[mathematical expression not reproducible]

Proof. For proof, see the proof of the following lemma.

Lemma 4.6. Under the above conditions, we have for each n > 0,

[mathematical expression not reproducible].

Proof. Recall that the barriers are smooth Ito processes (in [S.sup.2]) and from Lemma 4.4 [(f(s,[X.sup.[epsilon].sub.s], [Y.sup.[epsilon],n.sub.s])).sub.0[less than or equal to]s[less than or equal to]T] [member of] [S.sup.2]. For each n [greater than or equal to] 0, let [[bar.Y].sup.[epsilon],n] := [Y.sup.[epsilon],n] - U, [f.sup.*](s) := f(s, [X.sup.[epsilon].sub.s], [Y.sup.[epsilon],n.sub.s]) and [U.sub.t]:= [U.sub.0] + [[integral].sup.t.sub.0] [v.sub.s]ds + [[integral].sup.t.sub.0][v.sub.s]d[W.sub.s], recall that

[mathematical expression not reproducible],

then

[mathematical expression not reproducible].

For each n [member of] N, let [D.sup.n] the class of P-measurable processes z : [OMEGA] x [0,T] [right arrow] [0,n]. For v [member of] [D.sup.n] and [mu] [member of] [D.sup.n], by applying Ito's formula to the product of [[bar.Y].sup.[epsilon],n] and exp (-[[integral].sub.0]([mu](r) + v(r))dr) and using the same arguments as in Cvitanic and Karatzas [7] (see also Matoussi et al [16]), one can show that

[mathematical expression not reproducible].

Therefore

[mathematical expression not reproducible],

since [sup.sub.0[less than or equal to]s[less than or equal to]T] [absolute value of ([f.sup.*](s) - [u.sub.s])] [member of] [L.sup.2], from Doob's maximal inequality, we have

[mathematical expression not reproducible].

where C is a constant which is independent in [epsilon] and can change line by line, so

[mathematical expression not reproducible].

A similar analysis yields

[mathematical expression not reproducible].

Thus

[mathematical expression not reproducible].

The above proof, give us also the following

Lemma 4.7. There exists a constant C > 0 such that for all n > 0,

[mathematical expression not reproducible]

This (uniform in [epsilon]) a priori estimate lead to

Proposition 4.1. Under the above conditions, the family of processes ([Y.sup.[epsilon],n], [M.sup.[epsilon],n]) which converge in law to the the family of processes ([Y.sup.n], [M.sup.n]) on D[([0,T], R).sup.2].

Proof. Step 1: Tightness.

Clearly

[mathematical expression not reproducible], and it follows from Lemma 4.4 and assumptions that

[mathematical expression not reproducible]

hence the sequence {(Yse,n, Mfn); 0 < s < T} satisfy Meyer-Zheng's tightness criterion for quasi-martingales under P.

Step 2: Convergence in law.

By tighness, there exists a subsequence (which we still denote ([Y.sup.[epsilon],n], [M.sup.[epsilon],n])) such that

([X.sup.[epsilon]], [Y.sup.[epsilon],n], [M.sup.[epsilon],n]) (X, [Y.sup.n], [M.sup.n]),

on C([0,T],[R.sup.d]) x [(D([0,T],R)).sup.2], where the first factor is equipped with the topology of uniform convergence, and the second with the topology of convergence in ds measure. Clearly, for each 0 [less than or equal to] s [less than or equal to] T, (x, y) [right arrow] [[integral].sup.T.sub.s] f(x(r), y(r))dr is continuous for C([0,T],[R.sup.d]) x D[([0,T],R).sup.2] equipped with the same topology as above, and y [right arrow] [[integral].sup.T.sub.s] ([k.sup.+[epsilon],n.sub.r] - [k.sup.- [epsilon],n.sub.r]) (y)dr is continuous in D([0,T],R) as [epsilon] goes to 0. We can now take the limit in (4.2), yielding

[Y.sup.n.sub.s] = g([X.sub.s]) + [[integral].sup.T.sub.s] f(r, [X.sub.r], [Y.sup.n.sub.r]) dr + [M.sup.n.sub.t] - [M.sup.n.sub.s] + [[integral].sup.T.sub.s] ([k.sup.+0,n.sub.r] - [k.sup.- 0,n.sub.r]) ([Y.sup.n]])dr.

Moreover, for any 0 [less than or equal to] [s.sub.1] [less than or equal to] [s.sub.2] [less than or equal to] T, [phi] [member of] [C.sup.[infinity].sub.b] and [[psi].sub.s] a functional of [X.sup.[epsilon].sub.r], [Y.sup.[epsilon],n.sub.r], [K.sup.[epsilon],n.sub.r] [less than or equal to] r [less than or equal to] T, bounded and continuous in C([0,T],Rd) x D([0,T],Rk) x C([0,T],Rd), we have

[mathematical expression not reproducible],

and for each [epsilon] > 0,

[mathematical expression not reproducible].

From weak convergence, the fact that E([sup.sub.0<s<T] [[absolute value of ([M.sup.[epsilon],n.sub.s])].sup.2]) < +[infinity], dividing the second identity by [alpha] and for [alpha] [right arrow] 0, we have

[mathematical expression not reproducible].

Therefore, both [M.sup.n] and [M.sup.X] (the martingale part of X) are [mathematical expression not reproducible] martingales. It follows from the first statement that the process X satisfies the martingale problem with respect to the filtration [mathematical expression not reproducible], hence [M.sup.X] is [mathematical expression not reproducible]-martingale.

Step 3: Identification of the limit.

Since the martingale problem in Step 2 is well-posed, let ([[bar.Y].sup.n], [[bar.U].sup.n]) denote the unique solution of the BSDE

[mathematical expression not reproducible],

satisfying ETr [[integral].sup.t.sub.s] [[bar.U].sup.n.sub.r][<[M.sup.X]>.sub.r][[bar.U].sup.n.sub.r] < +[infinity], and let [[??].sup.n.sub.s] = [[integral].sup.s.sub.0][[bar.U].sup.n.sub.r]d[M.sup.X.sub.r]. Since T7(tm) and IT(tm) are [F.sup.X.sub.t] adapted, and [M.sup.X] is [mathematical expression not reproducible] martingale, [[??].sup.n] is [mathematical expression not reproducible] martingale. It follows from Ito formula that

[mathematical expression not reproducible]

(we use the fact that the operator is n-Lipschitz). Hence from Gronwall's lemma [[bar.Y].sup.n.sub.r] = [Y.sup.n.sub.r], 0 [less than or equal to] s [less than or equal to] t, and [M.sup.n] = [[??].sup.n], and so all sequence converge.

Now we deal with the uniform convergence of the process [([Y.sup.[epsilon],n], [M.sup.[epsilon],n], [K.sup.[epsilon],n]).sub.n] to ([Y.sup.[epsilon]], [M.sup.[epsilon]], [K.sup.[epsilon]]) as n goes to +[infinity].

Proposition 4.2. The family of processes [([Y.sup.[epsilon],n], [M.sup.[epsilon],n], [K.sup.[epsilon],n]).sub.n] converges uniformly of [epsilon] [member of] [0,1] in probability to the family of processes ([Y.sup.[epsilon]], [M.sup.[epsilon]], [K.sup.[epsilon]]) as n goes to +[infinity].

Proof. By the same proof as in Lemma 4.4, we have (since one can choose all constants independently on [epsilon] and n),

[mathematical expression not reproducible]

Now, let us prove the convergence of [([Y.sup.[epsilon],n], [Z.sup.[epsilon],n]).sub.n] for every (n,m) [member of] [N.sup.*] x [N.sup.*]. By Ito's formula, one has

[mathematical expression not reproducible],

using (3.2), we deduce that

[mathematical expression not reproducible].

Using Schwartz inequality, we have

[mathematical expression not reproducible].

We let C to be a constant changing line by line and we have used the facts that [square root of (a + b)] [less than or equal to] [square root of (a)] + [square root of (b)], [[bar.Y].sup.[epsilon],n.sub.r] - [[bar.Y].sup.[epsilon],m.sub.r]) = [[bar.Y].sup.[epsilon],n.sub.r] - [[bar.Y].sup.[epsilon],n.sub.r], 2ab [less than or equal to] [a.sup.2] + [b.sup.2] and Lemma 4.5.

Hence, from Gronwall's lemma and above Lemmas, we deduce that

[mathematical expression not reproducible].

Using Bulkholder-Davis-Gundy inequality, we obtain

[mathematical expression not reproducible]

We set

[mathematical expression not reproducible].

If we return to the equation satisfied by the ([Y.sup.[epsilon],n], [Z.sup.[epsilon],n]), we find also that [([K.sup.[epsilon],n]).sub.n] converges uniformly in [L.sup.2]([OMEGA]) to the limit [[bar.K].sup.[epsilon]] where

[mathematical expression not reproducible].

We have shown

[mathematical expression not reproducible]

where [H.sup.1]([0,T], [R.sup.d]) is the Sobolev space. Hence the sequence [([K.sup.[epsilon],n]).sub.n] is bounded independently of [epsilon] in [L.sup.2]([OMEGA]; [H.sup.1] ([0,T], [R.sup.d])) and there exist a subsequence of (KT,n)n which converges weakly. The limiting process Ke belong to [L.sup.2]([OMEGA]; [H.sub.1] ([0, T], [R.sup.d])), hence [[bar.K].sup.[epsilon]] is absolutely continuous. By uniqueness of solution of the reflected BSDE, we can find that [[bar.Y].sup.[epsilon]] = [Y.sup.[epsilon]], [[bar.Z].sup.[epsilon]] = [Z.sup.[epsilon]], [[bar.K].sup.[epsilon]] = [K.sup.[epsilon]].

Proposition 4.3. Under the assumption of the above lemma, the family of processes ([Y.sup.n], [M.sup.n], [K.sup.n]) converge in probability to (Y,M,K) as n goes to +[infinity].

Proof. Similar to the above one.

Proof of Theorem 4.3. Combining the above lemmas, we find that (XT, YT, MT, KT) converge in law to (X, Y, M, K) in the sense defined as above, where

[Y.sub.s] = g([X.sub.T]) + [[integral].sup.T.sub.s] f(r, [X.sub.r], [Y.sub.r])dr - [[integral].sup.T.sub.s] [Z.sub.r]d[B.sub.r] + [K.sub.T] - [K.sub.s].

Corollary 4.1. Under the assumptions of theorem, {[Y.sup.[epsilon].sub.0]} converge to [Y.sub.0] as [epsilon] goes to 0.

Proof. Since [Y.sup.[epsilon].sub.0] is deterministic, we have

[Y.sup.[epsilon].sub.0] = E[g([X.sup.[epsilon].sub.T]) + [[integral].sup.T.sub.0] f(s, [X.sup.[epsilon].sub.s], [Y.sup.[epsilon].sub.s])ds + [K.sup.[epsilon].sub.T]].

Put

[A.sub.[epsilon]] = g([X.sup.[epsilon].sub.T]) + [[integral].sup.T.sub.0] f(s, [X.sup.[epsilon].sub.s], [Y.sup.[epsilon].sub.s])ds + [K.sup.[epsilon].sub.T],

we have

[mathematical expression not reproducible],

by above assumptions and estimates, we have

[mathematical expression not reproducible],

and [A.sub.[epsilon]] converge in law, as [epsilon] goes to 0, toward

g([X.sub.T]) + [[integral].sup.T.sub.0] f(r, [X.sub.r], [Y.sub.r])dr + [K.sub.T],

the uniform integrability of Ae implies that

[mathematical expression not reproducible],

this means that Yf converge to

[Y.sub.0] = g([X.sub.T]) + [[integral].sup.T.sub.0] f(r, [X.sub.r], [Y.sub.r])dr + [K.sub.T].

We applied the previous results to the homogenization of a class of variational inequality as well as to stochastic game.

5. APPLICATIONS

Let [u.sup.[epsilon]]: [0,T] x R [right arrow] R be a solution of the variational inequality (3.11) The homogenization problem consists in computing the limit as [epsilon] [down arrow] 0 of [u.sub.[epsilon]](t,x).

Theorem 5.1. If the barriers satisfies also the polynomial growth conditions (3.10), then

u[member of](t, x) [right arrow] u(t, x), as [epsilon] goes to 0,

where u is the solution of the solution (in the viscosity sense) of the variational inequality

(5.1) [mathematical expression not reproducible].

For any [epsilon] > 0, 0 [less than or equal to] t [less than or equal to] T and any two stopping times [sigma], [tau] [member of] [M.sub.t,T], consider the pay off of the stochastic game [R.sup.[epsilon].sub.t] ([sigma], [tau]), its value function [V.sup.[epsilon].sub.t] (see (3.5), (3.7)). The homogenization problem consists also in computing the limit as [epsilon] [down arrow] 0,

Theorem 5.2. For any 0 [less than or equal to] t [less than or equal to] T and any two stopping times a,r [member of] Mt,T

[R.sup.[epsilon].sub.t] ([sigma], [tau]) [right arrow] [R.sub.t] ([sigma], [tau]) as [epsilon] goes to 0,

[V.sup.[epsilon].sub.t] [right arrow] [V.sub.t] as [epsilon] goes to 0,

where

[mathematical expression not reproducible]

is the payoff of a stochastic game, which value function Vt is given by the state-process Y given in the Theorem 4.3.

Received April 1, 2017

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ABOUBAKARY DIAKHABY AND YOUSSEF OUKNINE

Universite Gaston Berger, UFR SAT, Saint-Louis, Senegal

Universite Cadi Ayyad, Faculte des Sciences Semlalia, Maroc
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