Transportation Inequality for the Mean Reflected Stochastic Differential Equation

LI Yumeng (李宇勐)

(School of Statistics and Mathematics,Zhongnan University of Economics and Law,Wuhan 430073,China)

Abstract: In this paper,we study the transportation inequality for the mean reflected stochastic differential equation.By using the Girsanov transformation and the martingale representation theorem,we obtain the Talagrand’s transportation inequality on the continuous paths space with respect to the uniform metric,for the law of the equation.This extends the result obtained by WU and ZHANG(2004).

Key words: Mean reflected stochastic differential equation;Transportation inequality;Girsanov transformation

1.Introduction

Consider the Mean Reflected Stochastic Differential Equation (MR-SDE for short) described by the following system:

whereb,σandhare given Lipschitz functions from R to R,(Bt,t ≥0) stands for a standard Brownian motion defined on some complete probability space (?,F,P).The integral of the functionhwith respect to the law of the solution to the SDE is asked to be nonnegative.The solution to (1.1) is the couple of continuous processes (X,K),whereKis needed to ensure that the constraint is satisfied,in a minimal way according to the last condition namely the Skorokhod condition.

MR-SDE is a very special type of reflected Stochastic Differentials Equations (SDEs) in which the constraint is not directly on the paths of the solution to the SDE as in the usual case but on the law of the solution.This kind of processes has been introduced recently by Briand,Elie and HU[1]in backward forms under the context of risk measures.Briand,De Raynal,Guillin and Labart[2]studied the MR-SDE in forward forms,and they provided an approximation of solution to the MR-SDE with the help of interacting particles systems.

The purpose of this paper is to study the Talagrand’sT2-transportation inequality for MR-SDE(1.1).Recently,the problem of transportation inequalities and their applications to diffusion processes has been widely studied and is still a very active research area from both a theoretical and an applied point of view.

Let us first recall the transportation inequality.Let (E,d) be a metric space equipped withσ-fieldBsuch thatd(·,·) isB×Bmeasurable.Given two probability measuresμandνonE,theLp-Wassersetin distance is defined by

where the infimum is taken over all the probability measuresπonE × Ewith marginal distributionsμandν.The relative entropy ofνwith respect toμis defined as

The probability measureμsatisfies theL2-transportation inequality on (E,d) if there exists a constantC >0 such that for any probability measureνonE,

TheL2-transportation inequality was first established by Talagrand[3]for the Gaussian measure,so it is also called the Talagrand’sT2-transportation inequality.It has been brought into relation with the log-Sobolev inequality,Poincar′e inequality,Hamilton-Jacobi’s equation,the transportation-information inequality,refer to [4–7] for an overview of this theory.

With regard to the paths of SDEs,by means of Girsanov transformation and the martingale representation theorem,theT2(C) with respect (w.r.t.for to short) theL2and the Cameron-Martin distances were established by Djellout et al.[8];theT2(C)w.r.t.the uniform metric was obtained by WU and ZHANG[9].BAO,WANG and YUAN[10]established theT2(C) w.r.t.both the uniform and theL2distances on the path space for the segment process associated to a class of neutral function stochastic differential equations.Saussereau[11]studied theT2(C) for SDE driven by a fractional Brownian motion,and Riedel[12]extent this result to the law of SDE driven by general Gaussian processes by using Lyons’rough paths theory.

For the infinite dimensional stochastic partial differential equation (SPDE for short),WU and ZHANG[13]studied theT2(C) w.r.t.L2-metric by Galerkin’s approximation.By Girsanov’s transformation,Boufoussi and Hajji[14]obtained theT2(C) w.r.t.L2-metric for the stochastic heat equations driven by space-time white noise and driven by fractional noise.

In this paper,we shall study theT2(C) on the continuous paths space with respect to the uniform metric,for the law of mean reflected stochastic differential equation w.r.t.the uniform metric.

The rest of this paper is organized as follows.In Section 2,we first give the definition of the solution to Eq.(1.1),and then state the main results of this paper.In Section 3,we shall prove the main results.

2.Framework and Main Results

We consider the following assumptions:

Assumption (A.1)The functionsb:R→R andσ:R→R are Lipschitz continuous.

Assumption (A.2)(i) The functionh: R→R is an increasing function and there exist two constantsm,M,0

(ii) The initial conditionX0=xsatisfies:h(x)≥0.

Definition 2.1A couple of continuous processes(X,K)is said to be a flat deterministic solution to Eq.(1.1) if (X,K) satisfies (1.1) withKbeing a non-decreasing deterministic function withK0=0.

The proof of the existence and uniqueness result for the case of backward SDE is given in [1],and the proof for the the forward case is given in [2].

Theorem 2.1[6?7]Under Assumptions (A.1) and (A.2),the mean reflected SDE (1.1)has a unique deterministic flat solution (X,K).Moreover

where (Ut)0≤t≤Tis the process defined by:

LetC:=C([0,T];R) be the space of all continuous functions on [0,T],which is endowed with

Let Pxbe the law of{X(t);t ∈[0,T]}onC.The main result of this paper is the following transportation inequality for Px.

Theorem 2.2Assume Assumptions (A.1),(A.2) and|σ| ≤‖σ‖∞hold,there exists some constantC=C(T)>0 such that the probability measure PxsatisfiesT2(C) on the spaceCendowed with the metricd∞.

As indicated in [7],many interesting consequences can be derived from Theorem 2.2,see also Corollary 5.11 in [8].

Corollary 2.1Assume Assumptions (A.1),(A.2) and|σ| ≤‖σ‖∞hold,then we have for anyT >0

(a) for any smooth cylindrical functionFonC,that is,

the following Poincar′e inequality

holds,where VarPx(F) is the variance ofFunder law Px,and?F(γ)∈Gis the gradient ofFatγ.

(b) For anyg ∈C∞b(R),

(c) (Inequality of Hoeffding type) For anyV:R→R such that‖V ‖Lip≤α,

3.The Proof

To use the representation formula (2.1) of the processK,we recall a result from [2].Define the function

and

With these notation,denoting by(μt)t∈[0,T]the family of marginal laws of(Ut)t∈[0,T],we have

Lemma 3.1[6]Under Assumption (A.2),for anyν,ν′ ∈P(R),

Proof of Theorem 2.2Clearly,it is enough to prove the result for any probability measure Q onC([0,T];R) such that Q?PxandH(Q|Px)<∞.We divide the proof into two steps.

Step 1 We shall closely follow the arguments in Djellout et al.[8].Letbe a complete probability space on which (Bt,t ≥0) is a standard one-dimensional Brownian motion and let

Let (X·(x),K) be the unique solution of (1.1) starting fromx.Then the law ofX·(x)under

Consider

(Mt) can and will be chosen as a continuous martingale.Let

with the convention that inf ?=T+,whereT+is an artificially added element larger thanT,but smaller than anyt >T.Thenτis an (Ft)-stopping time and(τ=T+) = 1.Thus,we can write

where

Letτn=inf{t ∈[0,τ);[L]t=n}with the same convention that inf ?:=T+.It is elementary thatτn ↑τ,-a.s..Hence by the martingale convergence theorem,

By Girsanov’s theorem,is a-square integrable martingale with respect to the filtration (Ft).Consequently,

Substituting it into the preceding equality and noting thatwe get by the monotone convergence theorem

Step 2 By Girsanov’s theorem,

Moreover

where

Now,we consider the solutionYt(under) of

Moreover

where

The law of (Yt)t∈[0,T]underis exactly Px.Thus,(X,Y) underis a coupling of (Q,Px).

By Cauchy-Schwartz and Doob’s inequalities,we have

By (3.6),(3.9) and Lemma 3.1,we have

Therefore,for anyt ≤T,there exists a constantC1>0 satisfying that

By Gronwall’s inequality,we have

The proof is complete.