Anticipated Backward Doubly Stochastic Differential Equations with Non-Lipschitz Coefficients

Tie Wang,Siyu Cui

doi:10.3390/math10030396

Abstract

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.

Highlights

In 1990, the pioneer research of Pardoux and Peng [1] proposed the theory of nonlinear backward stochastic differential Equations (BSDEs)
Lipschitz assumptions, obtained a comparison theorem for their solutions under some specific condition, and investigated the duality between anticipated BSDEs and delayed stochastic differential equations
We mainly focus on one dimensional anticipated backward doubly stochastic differential equations (ABDSDEs), that is, k = 1

Summary

Introduction

In 1990, the pioneer research of Pardoux and Peng [1] proposed the theory of nonlinear backward stochastic differential Equations (BSDEs). Lipschitz assumptions, obtained a comparison theorem for their solutions under some specific condition, and investigated the duality between anticipated BSDEs and delayed stochastic differential equations. BDSDEs with one kind of non-Lipschitz coefficients, in which generator g does not depend on the anticipated term of y, z They obtained the existence and uniqueness result and a comparison theorem in the one dimensional case. We will prove that under proper assumptions, the solution of the above ABDSDE with non-Lipschitz coefficients exists uniquely, and two comparison theorems are given for the one dimensional ABDSDEs with non-Lipschitz coefficients These results are the cornerstones of ABDSDEs witn non-Lipschitz coefficients applied to some stochastic optimal control problems with delay effect.

Preliminaries

Existence and Uniqueness Theorem

A Li δ0i L

Comparison Theorems

Conclusions