In this paper we prove that the solution of a backward stochastic differential equation, which involves a subdifferential operator and associated to a family of reflecting diffusion processes, converges to the solution of a deterministic backward equation and satisfes a large deviation principle

Essaky, E. H.

Universitat Autònoma de Barcelona. Centre de Recerca Matemàtica

Centre de Recerca Matemàtica

Diposit Digital de Documents de la UAB

LARGE DEVIATION FOR BSDE WITHSUBDIFFERENTIAL OPERATORE. H. ESSAKY 1Abstract. In this paper we prove that the solution of a backwardstochastic differential equation, which involves a subdifferential op-erator and associated to a family of reflecting diffusion processes,converges to the solution of a deterministic backward equation andsatisfies a large deviation principle.1. IntroductionLet Xs,x,ε be the diffusion process that is the unique solution of thestochastic differential equation(1) Xs,x,εt = x+∫ tsb(Xs,x,εr )dr +√ε∫ tsσ(Xs,x,εr )dBr, 0 ≤ s ≤ t ≤ T,where b : IRd −→ IRd is a uniformly Lipschitz continuous function, allthe elements of the diffusion matrix σ are bounded, uniformly lipschitzcontinuous functions, and B is a standard Brownian motion in IRd. Theexistence and uniqueness of the strong solution Xs,x,ε of (1) is standard(see, for example, see Dembo and Zeitouni [4]). It is known, thanks to theworks of Freidlin and Wentzell [7], that Xs,x,ε converges in probability, asε goes to 0, to the solution χs,x of the following deterministic equation(2) χs,xt = x+∫ tsb(χs,xr )dr, 0 ≤ s ≤ t ≤ T,and satisfies a large deviation principle. This result has been generalizedrecently by Rainero [15] to the case of backward stochastic differential equa-tion (BSDE for short), related to the family of diffusion processes {Xs,x,ε},1Supported by Ministerio de Educacion y Ciencia, grant number SB2003-0117.Current adress : Centre de Recerca Matema`tica Apartat 50 E-08193 Bellaterra,Barcalona.1991 Mathematics Subject Classification. 60H10, 60H20.Key words and phrases. Backward stochastic differential equation; Reflecting diffu-sion process; Large deviation principle; Contraction principle.12 E. H. ESSAKYof the form(3) Y s,x,εt = h(Xs,x,εT ) +∫ Ttf(r,Xs,x,εr , Ys,x,εr , Zs,x,εr )dr −∫ TtZs,x,εr dBr,0 ≤ s ≤ t ≤ T,where h and f are a given functions and satisfy some appropriate assump-tions. The author has proved that the solution (Y s,ε, Zs,ε) of equation (3)converges, as ε goes to 0, to (Y s,x, 0) solution of the following backwarddeterministic equationY s,xt = h(χs,xT ) +∫ Ttf(r, χs,xr , Ys,xr , 0)dr, 0 ≤ s ≤ t ≤ T,and satisfies a large deviation principle.Backward stochastic differential equations of type (3) have been first in-troduced by Pardoux and Peng [12]. A solution for such equation is a coupleof adapted processes (Y,Z) with values in IRk × IRk×d which mainly satis-fies equation (3). The aim of Pardoux and Peng was to give a probabilisticinterpretation of a solution of second order quasi-linear partial differentialequation. Since then, those equations have been intensively investigateddue to their connections with financial mathematics, optimal control andstochastic game, non-linear PDEs and homogenization (see, for example,[5, 6, 8, 9, 14, 12, 2, 3, 10, 1] and the references therein).In this paper, we are interested to the system of forward-backward sto-chastic differential equations(4)Xs,x,εt == x+∫ tsb(Xs,x,εr )dr+√ε∫ tsσ(Xs,x,εr )dBr+ρs,x,εt −ρs,x,εs , 0≤s≤1≤T,ρs,x,εt =∫ t0∇ψ(Xs,x,εr )d|ρs,x,ε|r, |ρs,x,ε|t =∫ t01{Xs,x,εr ∈∂Θ}d|ρs,x,ε|r,(5)Y s,x,εt == h(Xs,x,εT )+∫ Ttf(r,Xs,x,εr , Ys,x,εr , Zs,x,εr )dr−∫ TtZs,x,εr dBr−∫ TtUs,x,εr dr(Y s,x,εt , Us,x,εt ) ∈ ∂Π, and IE∫ T0Π(Y s,x,εr )dr < +∞,where ψ is C2b (IRd) function and ρ is a bounded variation process such thatρ0 = 0, Π is a proper lower semicontinuous convex function and ∂Π isa subdifferntial operator. Equations of type (5) have been introduced byPardoux and Rascanu [13]. A solution of such equations is a triple of process(Y, Z, U) with values in IRk×IRk×d×IRk and satisfies equation (5). Our aimLARGE DEVIATION FOR BSDE WITH SUBDIFFERENTIAL OPERATOR 3is to prove that the solution (Xs,x,ε, ρs,x,ε, Y s,x,ε, Zs,x,ε, Us,x,ε) of system(4)-(5) converges, as ε goes to 0, to the solution (χs,x, ρs,x, Y s,x, Zs,x, Us,x)of the following system of forward-backward deterministic equationχs,xt = x+∫ tsb(χs,xr )dr + ρs,xt − ρs,xsρs,xt =∫ t0∇ψ(χs,xr )d|ρs,x|r, |ρs,x|t =∫ t01{χs,xr ∈∂Θ}d|ρs,x|rY s,xt = h(χs,xT ) +∫ Ttf(r, χs,xr , Ys,xr , 0)dr −∫ TtUs,xr dr(Y s,xt , Us,xt ) ∈ ∂Π, and IE∫ T0Π(Y s,xr )dr < +∞,and that Y s,x,ε satisfies a large deviation principle. Our paper is, in fact, ageneralization of the two works cited before.2. Assumptions and problem formulationLet (Ω,F , (Ft)t≤1)) be a stochastic basis such that F0 contains all P -nullsets of F , Ft+ =⋂²>0Ft+² = Ft, ∀t ≤ 1, and suppose that the filtration isgenerated by a d-dimensional Brownian motion (Bt)t≤1.On the other hand, let• Θ be an open connected bounded subset of IRd, which is such that for afunction ψ ∈ C2b (IRd), Θ = {ψ > 0}, ∂Θ = {ψ = 0}, and | ∇ψ(x) |= 1,x ∈ ∂Θ. Note that at any boundary point x ∈ ∂Θ, ∇ψ(x) is a unit normalvector to the boundary, pointing towards the interior of Θ. The aboveassumptions imply that there exists a constant δ > 0 such that for allx ∈ ∂Θ, x′ ∈ Θ(6) 2〈x′ − x,∇ψ(x)〉+ δ|x− x′|2 ≥ 0.• b : Θ −→ IRd, σ : Θ −→ IRd×d be functions such that :(A1) There exists a constant C > 0 such that|b(x)|+ |σ(x)| ≤ C, ∀x ∈ Θ|b(x)− b(x′)|+ |σ(x)− σ(x′)| ≤ C|x− x′ |,∀x, x′ ∈ Θ.(A2) The matrix a = σσ∗ is uniformly elliptic, that is, there exists a con-stant γ > 0 such thata(x) ≥ γ|x|2, ∀x ∈ Θ.• h ∈ C(Θ; IRk), f ∈ C([0, 1]×Θ× IRk × IRk×d; IRk) be functions satisfyingthe following assumptions :4 E. H. ESSAKY(A3) There exist constants α ∈ IR, K > 0, c > 0, µ > 0 such that(i)∀t, ∀x, ∀y, ∀(z, z′),| f(t, x, y, z)− f(t, x′, y, z′) |≤ µ(| z − z′ | + | x− x′ |)(ii)∀t, ∀x, ∀z, ∀(y, y′),〈y − y′, f(t, x, y, z)− f(t, x, y′, z)〉 ≤ α | y − y′ |2(iii)∀x,∀x′, |h(x)− h(x′)| ≤ c|x− x′|,(vi)∀t,∀x, ∀y, ∀z, | f(t, x, y, z) |≤ K(1+ | y | + | z |)(v)∀x, |h(x)| ≤ K(1 + |x|).• Π : IRk →]−∞,+∞], be a proper lower semicontinuous convex functionsuch that(A4) There exists a constant C > 0 such thatΠ(h(x)) ≤ C(1 + |x|), ∀x ∈ Θ,Π(y) ≥ Π(0) = 0, ∀y ∈ IRk.We need also the following notations :• C[0, T ] denotes the space of continuous functions Φ : [0, T ] −→ IRd suchthat f(0) ∈ Θ.• C[0, T ] denotes the space of continuous functions Ψ : [0, T ] −→ Θ.• V[0, T ] denotes the space of functions ρ : [0, T ] −→ IRd with boundedvariation and ρ(0) = 0.For ρ ∈ V[0, T ], |ρ|t denotes the total variation of ρ in the interval [0, t].Consider the system of forward-backward stochastic differential equations(7)Xs,x,εt == x+∫ tsb(Xs,x,εr )dr+√ε∫ tsσ(Xs,x,εr )dBr+ρs,x,εt −ρs,x,εs , 0 ≤ s ≤ t ≤ T,ρs,x,εt =∫ t0∇ψ(Xs,x,εr )d|ρs,x,ε|r, |ρs,x,ε|t =∫ t01{Xs,x,εr ∈∂Θ}d|ρs,x,ε|r(8)Y s,x,εt == h(Xs,x,εT )+∫ Ttf(r,Xs,x,εr , Ys,x,εr , Zs,x,εr )dr−∫ TtZs,x,εr dBr−∫ TtUs,x,εr dr(Y s,x,εt , Us,x,εt ) ∈ ∂Π, and IE∫ T0Π(Y s,x,εr )dr < +∞,where∂Π(u) = {u∗ ∈ IRk : 〈u∗, v − u〉+Π(u) ≤ Π(v),∀v ∈ IRk}LARGE DEVIATION FOR BSDE WITH SUBDIFFERENTIAL OPERATOR 5Note that the subdifferential operator ∂Π : IRk −→ 2IRk is a maximalmonotone operator, that is〈u′ − v′, u− v〉 ≥ 0. ∀(u, u′), (v, v′) ∈ ∂Π.The existence and uniqueness of the strong solution Xs,x,ε, under assump-tion (A1), for equation (7) is standard (see, for example, Lions and Sznit-man [11] or Saisho [16]). It follows also from the result of Pardoux andRascanu [13] that, under assumptions (A3) and (A4), there exists a uniquetriple (Y s,x,ε, Zs,x,ε, Us,x,ε) for equation (8).The objective of this work is to prove that the solution of forward-backwardstochastic differential equation (7)-(8) converges and satisfies a large devia-tion principle.For the sake of simplicity, we put, in general, s = 0. Of course, theresults hold true for all s ∈ [0, T ]. We denote then by Xx,ε := X0,x,ε,Y 0,x,ε := Y x,ε,...3. Large deviation principle and convergence of the solutionof the forward equationBefore giving a large deviation principle for the reflecting diffusion processXs,x,ε, we recall the followingDefinition 3.1. The family of processes (Xt, 0 ≤ t ≤ T ) which dependson a parameter ε is said to satisfy a large deviation principle with a ratefunction S(Ψ) if the following condition hold for every Borel set A ⊆ C[0, T ]1. lim supε→0ε ln(P (Xε ∈ A)) ≤ infΨ∈AS(Ψ)2. lim infε→0ε ln(P (Xε ∈ A)) ≥ − infΨ∈A˚S(Ψ),where A is the closure of A and A˚ the interior of A.Let Φ ∈ C[0, T ], Ψ ∈ C[0, T ], ρ ∈ V[0, T ] such that(9)Ψ(t) = Φ(t) + ρ(t),ρt =∫ t0∇ψ(Ψr)d|ρ|r, |ρ|t =∫ t01{Ψ(r)∈∂Θ}d|ρ|rFor Φ and Ψ defined as above, let Ψ = F (Φ). It is known from Lions andSznitman [11] or Saisho [16] that F is continuous. We have the followingtheorem6 E. H. ESSAKYTheorem 3.1. The process Xx,ε given by equation (7) satisfies a largedeviation principle with rate function S(Ψ) defined byS(Ψ) =12infΦ∈F−1(Ψ)∫ T0(Φ˙(s)− b(Ψ(s)))∗a−1(Ψ(s))(Φ˙(s)− b(Ψ(s)))ds,with the fact that S(Ψ) =∞ if F−1(ψ) = ∅ or Φ is not absolutely continu-ous.Proof. The result follows by using the contraction principle (see Demboand Zeitouni [4]) and a large deviation principle for diffusion processes (seeStroock [18] or [4], see also Sheu [17] for other assumptions on Θ).Remark 3.1. The function S(Ψ) has the following properties1. S(Ψ) is lower semi-continuous in Ψ.2. If S(Ψ) <∞, then there exists Φ ∈ C[0, T ] such that F (Φ) = Ψ andS(Ψ) =12∫ T0(Φ˙(s)− b(Ψ(s)))∗a−1(Ψ(s))(Φ˙(s)− b(Ψ(s)))ds.Let (χs,x, ρs,x) be the solution of the following deterministic equationχs,xt = x+∫ tsb(χs,xr )dr + ρs,xt − ρs,xsρs,xt =∫ t0∇ψ(χs,xr )d|ρs,x|r, |ρs,x|t =∫ t01{χs,xr ∈∂Θ}d|ρs,x|r.We get the followingLemma 3.1. For all ε ∈]0, 1], there exists a constant C > 0, independentof x and ε, such that(10) IE sup0≤t≤T|Xx,εt − χxt |2 ≤ Cε.Proof. Applying Itoˆ’s formula toe−δ(ψ(Xx,εt )+ψ(χxt ))|Xx,εt − χxt |2,LARGE DEVIATION FOR BSDE WITH SUBDIFFERENTIAL OPERATOR 7where δ is given by the inequality (6), we get a.s for all t ∈ [0, T ](11)e−δ(ψ(Xx,εt )+ψ(χxt ))|Xx,εt − χxt |2= 2∫ t0e−δ(ψ(Xx,εr )+ψ(χxr ))[〈Xx,εr − χxr ,√εσ(Xx,εr )dBr〉++〈Xx,εr − χxr , b(Xx,εr )− b(χxr )〉dr]+2∫ t0e−δ(ψ(Xx,εr )+ψ(χxr ))[〈Xx,εr − χxr ,∇ψ(Xx,εr )〉d|ρx,ε|r−−〈Xx,εr − χxr ,∇ψ(χxr )〉d|ρx|r]+ε∫ t0e−δ(ψ(Xx,εr )+ψ(χxr ))|σ(Xx,εr )|2dr−δ∫ t0e−δ(ψ(Xx,εr )+ψ(χxr ))|Xx,εr − χxr |2((∇ψ(Xx,εr ))∗√εσ(Xx,εr )++ ε2 tr(D2ψ(Xx,εr )σσ∗(Xx,εr )+〈∇ψ(Xx,εr ), b(Xx,εr )〉++〈∇ψ(χxr ), b(χxr )〉)dr − δ∫ t0e−δ(ψ(Xx,εr )+ψ(χxr ))|Xx,εr −−χxr |2(|∇ψ(Xx,εr )|2d|ρx,ε|r + |∇ψ(Xxr )|2d|ρx|r)+ εδ22∫ t0e−δ(ψ(Xx,εr )+ψ(χxr ))|Xx,εr − χxr |2|(σ(Xx,εr ))∗∇ψ(Xx,εr )|2dr−2εδ∫ t0e−δ(ψ(Xx,εr )+ψ(χxr ))〈Xx,εr − χxr , σ(Xx,εr )〉(σ(Xx,εr )∗∇ψ(Xx,εr )dr.Since |∇φ| = 1 for all x ∈ ∂Θ, by inequality (6) we have(12)2∫ t0e−δ(ψ(Xx,εr )+ψ(χxr ))〈Xx,εr − χxr ,∇ψ(Xx,εr )〉d|ρx,ε|r−δ∫ t0e−δ(ψ(Xx,εr )+ψ(χxr ))|Xx,εr − χxr |2|∇ψ(Xx,εr )|2d|ρx,ε|r ≤ 0,and(13)−2∫ t0e−δ(ψ(Xx,εr )+ψ(χxr ))〈Xx,εr − χxr ,∇ψ(Xxr )〉d|ρx|r−δ∫ t0e−δ(ψ(Xx,εr )+ψ(χxr ))|Xx,εr − χxr |2|∇ψ(Xxr )|2d|ρx|r ≤ 0.The result is then a consequence of the boundeness of b, σ, ψ, ∇ψ, D2ψ,inequalities (12)–(13) and Burkholder-Davis-Gundy inequality.Remark 3.2. As a consequence of Lemma 3.1, the solution of the reflectingdiffusion process Xx,ε converges to the deterministic path χx in L2.8 E. H. ESSAKY4. Convergence and large deviation principle for the solutionof the backward equationLet (χ(s,x), ρs,x, Y (s,x), 0, U (s,x)) be the solution of the following deter-ministic equationsχs,xt = x+∫ tsb(χs,xr )dr + ρs,xt − ρs,xsρs,xt =∫ t0∇ψ(χs,xr )d|ρs,x|r, |ρs,x|t =∫ t01{χs,xr ∈∂Θ}d|ρs,x|r(14)Y s,xt = h(χs,xT ) +∫ Ttf(r, χs,xr , Ys,xr , 0)dr −∫ TtUs,xr dr(Y s,xt , Us,xt ) ∈ ∂Π, and IE∫ T0Π(Y s,xr )dr < +∞.We have the following theoremTheorem 4.1. ∀ε ∈]0, 1], there exists a constant C > 0, independent of s,x and ε, such that(15)IE[ sups≤t≤T|Y s,x,εt − Y s,xt |2 +∫ Ts|Zs,x,εr |2dr]≤ C[IE(Xs,x,εT − χs,xT |2) + IE∫ Ts|Xs,x,εr − χs,xr |2dr].Proof. Applying Itoˆ’s formula to |Y s,x,εt − Y s,xt |2, we getIE|Y s,x,εt −Y s,xt |2+IE∫ Ts|Zs,x,εr |2dr+2IE∫ Ts〈Y s,x,εr −Y s,xr , Us,x,εr −Us,xr 〉dr≤ IE(h(Xs,x,εT )− h(χs,xT |2)+2IE∫ Ts〈Y s,x,εr − Y s,xr , f(r,Xs,x,εr , Y s,x,εr , Zs,x,εr )− f(r, χs,xr , Y s,xr , 0)〉dr.But 〈Y s,x,εr −Y s,xr , Us,x,εr −Us,xr 〉 ≥ 0, dP ×dr a.e and f satisfies conditionsA3(i)− (ii), thenIE|Y s,x,εt − Y s,xt |2 + IE∫ Ts|Zs,x,εr |2dr≤ IE(h(Xs,x,εT )− h(χs,xT |2) + 2αIE∫ Ts|Y s,x,εr − Y s,xr |2dr+2µIE∫ Ts|Y s,x,εr −Y s,xr ||Xs,x,εr −χs,xr |dr+2µIE∫ Ts|Y s,x,εr −Y s,xr ||Zs,x,εr |drLARGE DEVIATION FOR BSDE WITH SUBDIFFERENTIAL OPERATOR 9HenceforthIE[|Y s,x,εt − Y s,xt |2 +∫ Ts|Zs,x,εr |2dr]≤ C[IE(Xs,x,εT − χs,xT |2) + IE∫ Ts|Xs,x,εr − χs,xr |2dr],where C is a positive constant. The result then follows from Burkholder-Davis-Gundy inequality.Remark 4.1. As a consequence of Theorem 4.1 and Lemma 3.1, we get(16) IE[ sups≤t≤T|Y s,x,εt − Y s,xt |2 +∫ Ts|Zs,x,εr |2dr] ≤ Cε,where C is a positive constant and then the solution of the BSDE (8) con-verges to the deterministic solution of the equation (14).We now consider the BSDE in the case k = 1. We want to prove thatthe process Y s,x,ε satisfies a large deviation principle. For that reason, werecall the link between Variational Inequality (VI, for short) and BSDE. Forall ε ≥ 0, we consider the following VI(17)∂uε∂t(t, x) + Lx,εuε (t, x)++f(t, x, uε (t, x) ,((∇uε)∗√εσ) (t, x)) ∈ ∂Π(uε (t, x)) ,t ∈ ]0, T [ , x ∈ Θ∂uε∂n(t, x) ∈ ∂Π(uε (t, x)) , x ∈ ∂Θuε (T, x) = h (x) , x ∈ Θ,where Lx,ε is the second order partial differential operatorLx,ε := ε2d∑i,j=1(σσ∗)ij∂2∂xi∂xj+d∑i=1bi∂∂xi,and at point x ∈ ∂Θ∂∂n:=d∑i=1∂ψ∂xi(x)∂∂xi,then we have, for each (t, x) ∈ [0, T ]×Θ,(18) uε(t, x) = Y t,x,εt ,both in the sense that any classical solution of the VI (17) is equal to Y t,x,εt ,and Y t,x,εt is, in the case where all coefficients are continuous, a viscositysolution of the VI (17) (see Pardoux and Rascanu [13]). Moreover, we havealso thatY s,x,εt = uε(t,Xs,x,εt ).10 E. H. ESSAKYLet s ∈ [0, T ] and ε ≥ 0, we define the following applications :F ε(Ψ) := [t→ uε(t,Ψt)], t ∈ [s, T ], Ψ ∈ C[s, T ] satisfying equation (9).Hence Y s,x,εt = F ε(Xs,x,ε)(t), for all t ∈ [0, T ], and Y s,x,ε = F ε(Xs,x,ε).For ε = 0, u and F stand for u0 and F 0. We have the following theoremTheorem 4.2. Y x,ε satisfies a large deviation principle with a rate functionS′(Ψ′) = inf{S(Ψ)|Ψ′t = F (Ψ)(t) = u(t,Ψt), ∀t ∈ [0, T ]}.Proof. In order to apply the contraction principle, we need to prove thatF ε, ε ≥ 0 are continuous and {F ε} converges uniformly to F on every com-pact of C[0, T ]. Since uε is continuous, it is not hard to prove that F ε is alsocontinuous. The uniform convergence of {F ε} is a consequence of Remark4.1.Acknowledgments. The author would like to thank the ”Centre de Re-cerca Matema`tica” for their extraordinary hospitality and facilities for doingthis work. The author would like also to thank Prof. Youssef Ouknine forvarious discussions on BSDEs.References[1] K. Bahlali, E.H. Essaky, M. Hassani, E. Pardoux, Lp-solutions to BSDEs with super-linear growth coefficient. Application to degenerate semilinear PDEs, CRM Preprintnumber 682, (2006).[2] R. Buckdahn, Y. Hu, Probabilistic approach to homogenizations of systems of quasi-linear parabolic PDEs with periodic structures, Nonlinear Anal., 32, no. 5, pp.609–619, (1998).[3] R. Buckdahn, Y. Hu, S. Peng, Probabilistic approach to homogenization of viscositysolutions of parabolic PDEs, Nonlinear Differential Equations Appl., 6, no. 4, pp.395–411, (1999).[4] A. Dembo, O. Zeitouni: Large Deviations Techniques And Applications, SpringerVerlag, New York, second edition, (1998).[5] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equationsin finance, Mathematical Finance, 7, pp. 1-71, (1997).[6] N. El-Karoui, C. Kapoudjian, E. Pardoux, S. Peng, M.C. Quenez, Reflected solutionsof backward SDE’s and related obstacle problems for PDE’s, Annals of Probability,25, 2, pp. 702-737, (1997).[7] M.I. Freidlin, A.D. Wentzell, Random Perturbations of dynamical systems, SpringerVerlag, (1984).[8] S, Hamade`ne, Reflectd BSDE’s with discountinous barrier and application, Stochas-tics and Stochastics reports, 74, 3-4, pp. 571-596, (2002).[9] S, Hamade`ne, J.P, Lepeltier, Zero-sum stochastic differential games and BSDEs,Systems and control letteres, 24, 259-263, (1995).[10] S. Hamade`ne, Y. Ouknine, Reflected backward stochastic differential equation withjumps and random obstacle, EJP, 8 pp. 1-20, (2003).LARGE DEVIATION FOR BSDE WITH SUBDIFFERENTIAL OPERATOR 11[11] P.L. Lions, A.S. Sznitman, Stochastic differential equations with reflecting boundaryconditions, Comm. Pure Appl. Math., 37, pp. 511-537, (1984).[12] E. Pardoux, S. Peng, Adapted solutions of backward stochastic differential Equa-tions, Systems and Control Letters 14, pp. 51-61, (1990).[13] E. Pardoux and A. Rascanu, Backward SDE’s with maximal monotone operator,Stoch. Proc. Appl. 76, (2), pp. 191-215, (1998).[14] E. Pardoux, S. Peng, Backward stochastic differential equations and quasilinearparabolic partial differential equations. Stochastic partial differential equations andtheir applications (Charlotte, NC, 1991), Lecture Notes in Control and Inform. Sci.,176, 200–217, Springer, Berlin, (1992).[15] S. Rainero, Un principe de grandes de´viations pour une e´quation diffe´rentiellestochastique progressive re´trograde, Comptes Rendus Mathematique, 343, Issue 2,pp. 141-144, (2006).[16] Y. Saisho, Stochastic differential equations for multidimensional domains with re-flecting boundary, Prob. Theory and Rel. Fields, 74, pp. 455-477, (1987).[17] S. S. Sheu, Large deviation principle of reflecting diffusions, Taiwanese Journal ofMathematics, 2, Issue 2, pp. 251-256, (1998).[18] D. Stroock, An introduction to the theory of large deviations, Springer, New York,(1984).Universite´ Cadi Ayyad,, Faculte´ Poly-Disciplinaire,, De´partement deMathe´matiques et d’Informatique,, B.P 4162, Safi, Morocco.E-mail address: essaky@ucam.ac.ma

Large deviation for BSDE with subdifferential operator

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Large deviation for BSDE with subdifferential operator

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