Backward Doubly Stochastic Differential Equations Research Articles

Doubly-stochastic interpretation for nonlocal semi-linear backward stochastic partial differential equations

Motivated by P.L. Lions' seminal course on Mean Field Games (MFG) and the associated Master equation at Collège de France [19], we investigate in the case of absence of control mean-field backward doubly stochastic differential equations (BDSDEs), i.e., BDSDEs whose driving coefficients also depend on the joint law of the solution process. This BDSDE can be interpreted as randomly perturbed recursive cost functional which we associate with a forward mean-field SDE. Defining the value function V=V(t,x,μ) through this BDSDE, we prove that V=V(t,x,μ) is the unique classical solution of a backward stochastic PDE over [0,T]×Rd×P2(Rd). This backward SPDE extends in the non-control case the Master equation studied in earlier works by different authors. The associated MFG can be interpreted as a forward dynamics endowed with a randomly perturbed recursive cost functional, the solution of the MFG system (u(t,x),m(t)) remains also in this extended case related with V=V(t,x,μ) by the relation u(t,x)=V(t,x,m(t)).Our main objective is to characterise the random field V as classical solution of the backward SPDE. For this we study the regularity of V over [0,T]×Rd×P2(Rd). However, unlike the pioneering paper on BDSDEs by Pardoux-Peng [25] where V does not depend on the measure, here we have only an L2-regularity of V with respect to its variables. Other difficulties to overcome and subtle technical proofs are related with the study of the L2-regularity and, hence, that of the solution of the underlying mean-field BDSDE. It adds that our BDSDE needs weaker assumptions on the coefficients than in [25]. Finally, let us point out that for our approach we extend the classical mean-field Itô formula to smooth functions of solutions of mean-field BDSDEs. This formula contains earlier (mean-field) Itô formulas as special cases.

Backward doubly-stochastic differential equations with mean reflection

Backward doubly-stochastic differential equations with mean reflection

Backward doubly stochastic differential equations (BDSDEs): Existence and Uniqueness

In this paper, we present a class of stochastic differential equations with terminal condition, called backward doubly stochastic differential equations (BDSDEs). Precisely, we will prove the existence and uniqueness of the solutions of FBDSDEs but under weaker conditions

Anticipated Backward Doubly Stochastic Differential Equations with Non-Lipschitz Coefficients

The work presented in this paper focuses on a type of differential equations called anticipated backward doubly stochastic differential equations (ABDSDEs) whose generators not only depend on the anticipated terms of the solution (Y·,Z·) but also satisfy one kind of non-Lipschitz assumption. Firstly, we give the existence and uniqueness theorem. Further, two comparison theorems for the solutions of these equations are obtained after finding a new comparison theorem for backward doubly stochastic differential equations (BDSDEs) with non-Lipschitz coefficients.

General mean-field BDSDEs with continuous coefficients

In this paper we consider general mean-field backward doubly stochastic differential equations (BDSDEs), i.e., the generator of our mean-field BDSDEs depends not only on the solution but also on the law of the solution. The first part of the paper is devoted to the existence and the uniqueness of solutions for general mean-field BDSDEs under the Lipschitz conditions. We emphasize that in the mean-field BDSDEs we study, from Pardoux-Peng's pioneering work [17] the expected condition that the Lipschitz constant of the coefficient for the backward integral with respect to z is strictly less than 1 is considerably weakened for the Z-component of the law variable of the coefficient. We also establish higher order moment estimates and obtain a comparison theorem of one dimensional general mean-field BDSDEs. Furthermore, we study an existence theorem and a comparison theorem for general mean-field BDSDEs with a continuous coefficient f. Finally, we investigate uniqueness theorems for general mean-field BDSDEs with a uniformly continuous coefficient f.

Comparison Theorems for Multi-Dimensional General Mean-Field BDSDES

In this paper we study multi-dimensional mean-field backward doubly stochastic differential equations (BDSDEs), that is, BDSDEs whose coefficients depend not only on the solution processes but also on their law. The first part of the paper is devoted to the comparison theorem for multi-dimensional mean-field BDSDEs with Lipschitz conditions. With the help of the comparison result for the Lipschitz case we prove the existence of a solution for multi-dimensional mean-field BDSDEs with an only continuous drift coefficient of linear growth, and we also extend the comparison theorem to such BDSDEs with a continuous coefficient.

Backward Doubly Stochastic Differential Equations with Markov Chains and a Comparison Theorem

In this paper we study the existence and uniqueness of solutions for one kind of backward doubly stochastic differential equations (BDSDEs) with Markov chains. By generalizing the Itô’s formula, we study such problem under the Lipschitz condition. Moreover, thanks to the Yosida approximation, we solve such problem under monotone condition. Finally, we give the comparison theorems for such equations under the above two conditions respectively.

Backward Doubly SDEs with weak Monotonicity and General Growth Generators

We deal with backward doubly stochastic differential equations (BDSDEs) with a weak monotonicity and general growth generators and a square integrable terminal datum. We show the existence and uniqueness of solutions. As application, we establish the existenceand uniqueness of Sobolev solutions to some semilinear stochastic partial differential equations (SPDEs) with a general growth and a weak monotonicity generators. By probabilistic solution, we mean a solution which is representable throughout a BDSDEs.

Lp (1 < p < 2) solutions of backward doubly stochastic differential equations with locally monotone coefficients

The article considers the existence and uniqueness of solutions for backward doubly stochastic differential equations (BDSDEs) under globally (locally) monotone conditions. Some useful and important prior estimates are established. By using truncation method, Lp solutions for BDSDEs under globally monotone assumptions are obtained. With the help of suitable approximation techniques, we derive the existence and uniqueness of Lp solutions for BDSDEs with locally monotone coefficients.

Lp solutions of infinite time interval backward doubly stochastic differential equations

In this paper, we study the existence and uniqueness theorem for Lp (1 &lt; p &lt; 2) solutions to a class of infinite time interval backward doubly stochastic differential equations (BDSDEs). Furthermore, we obtain the comparison theorem for 1-dimensional infinite time interval BDSDEs in Lp.

Backward doubly SDEs and SPDEs with superlinear growth generators

We deal with multidimensional backward doubly stochastic differential equations (BDSDEs) with a superlinear growth generator and a square integrable terminal datum. We introduce new local conditions on the generator and then show that they ensure the existence and uniqueness as well as the stability of solutions. Our work goes beyond the previous results on the subject. Although we are focused on multidimensional case, the uniqueness result we establish is new in one-dimensional too. As an application, we establish the existence and uniqueness of probabilistic solutions to some semilinear stochastic partial differential equations (SPDEs) with superlinear growth gernerator. By probabilistic solution, we mean a solution which is representable throughout a BDSDEs.

Numerical computation for backward doubly SDEs with random terminal time

Abstract In this article, we are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time τ. The main motivations are giving a probabilistic representation of the Sobolev’s solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when τ is the first exit time of a forward SDE from a cylindrical domain. Euler schemes and bounds for the discrete-time approximation error are provided.

Wong–Zakai approximations of backward doubly stochastic differential equations

In this paper we obtain a Wong–Zakai approximation to solutions of backward doubly stochastic differential equations (BDSDEs). The situation is quite different from the case of stochastic differential equations because the integrands in the stochastic integral are also part of the solutions.

A first order semi-discrete algorithm for backward doubly stochastic differential equations

Numerical solutions of backward doubly stochastic differential equations (BDSDES) and the related stochastic partial differential equations (Zakai equations) are considered. First order algorithms are constructed using a generalized Itô-Taylor formula for two-sided stochastic differentials. The convergence order is proved through rigorous error analysis. Numerical experiments are carried out to verify the theoretical results and to demonstrate the efficiency of the proposed numerical algorithms.

Backward doubly stochastic differential equations with a superlinear growth generator

We deal with backward doubly stochastic differential equations (BDSDEs) with a superlinear growth generator and a square integrable terminal datum. We introduce a new local condition on the generator, then we show that it ensures the existence and uniqueness as well as the stability of solutions. Our work goes beyond the previous results on the multidimensional BDSDEs. Although we are focused on the multidimensional case, our uniqueness result is new for one-dimensional BDSDEs, too. As an application, we establish the existence of a Sobolev solution to SPDEs with superlinear growth generator. Some illustrative examples are also presented.

SPDEs with polynomial growth coefficients and the Malliavin calculus method

In this paper we study the existence and uniqueness of the Lρ2p(Rd;R1)×Lρ2(Rd;Rd) valued solutions of backward doubly stochastic differential equations (BDSDEs) with polynomial growth coefficients using weak convergence, equivalence of norm principle and Wiener–Sobolev compactness arguments. Then we establish a new probabilistic representation of the weak solutions of SPDEs with polynomial growth coefficients through the solutions of the corresponding BDSDEs. This probabilistic representation is then used to prove the existence of stationary solutions of SPDEs on Rd via infinite horizon BDSDEs. The convergence of the solution of a finite horizon BDSDE, when its terminal time tends to infinity, to the solution of the infinite horizon BDSDE is shown to be equivalent to the convergence of the pull-back of the solution of corresponding SPDE to its stationary solution. This way we obtain the stability of the stationary solution naturally.

A numerical scheme for backward doubly stochastic differential equations

In this paper we propose a numerical scheme for the class of backward doubly stochastic differential equations (BDSDEs) with possible path-dependent terminal values. We prove that our scheme converges in the strong $L^{2}$-sense and derives its rate of convergence. As an intermediate step we derive an $L^{2}$-type regularity of the solution to such BDSDEs. Such a notion of regularity, which can be thought of as the modulus of continuity of the paths in an $L^{2}$-sense, is new.

Backward doubly stochastic differential equations with infinite time horizon

We give a sufficient condition on the coefficients of a class of infinite horizon backward doubly stochastic differential equations (BDSDES), under which the infinite horizon BDSDES have a unique solution for any given square integrable terminal values. We also show continuous dependence theorem and convergence theorem for this kind of equations.

Comparison Theorems for the Multidimensional BDSDEs and Applications

A class of backward doubly stochastic differential equations (BDSDEs) are studied. We obtain a comparison theorem of these multidimensional BDSDEs. As its applications, we derive the existence of solutions for this multidimensional BDSDEs with continuous coefficients. We can also prove that this solution is the minimal solution of the BDSDE.

A class of backward doubly stochastic differential equations with discontinuous coefficients

In this work the existence of solutions of one-dimensional backward doubly stochastic differential equations (BDSDEs) with coefficients left-Lipschitz in y (may be discontinuous) and Lipschitz in z is studied. Also, the associated comparison theorem is obtained.