Anticipated Generalized Backward Doubly Stochastic Differential Equations

Tie Wang,Jiaxin Yu

doi:10.3390/sym14010114

Tie Wang, Jiaxin Yu

Open Access

https://doi.org/10.3390/sym14010114

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Journal: Symmetry	Publication Date: Jan 9, 2022
Citations: 3	License type: CC BY 4.0

Affiliation: Shanghai University, Liaoning University

Abstract

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process but also include generators depending on the anticipated terms of the solution (Y, Z). Firstly, we prove the existence and uniqueness theorem for AGBDSDEs. Further, two comparison theorems are obtained after finding a new comparison theorem for GBDSDEs.

Highlights

We will prove that the solution of the above AGBDSDE exists uniquely under proper assumptions, and two versions of one dimensional comparison theorems are given
These results are the cornerstones of AGBDSDEs applied to the obstacle problem for some semilinear partial differential equations (SPDEs) with the nonlinear Neumann boundary condition and some stochastic control problems with delay
With the help of Propositions 1 and 2, we can establish the following existence and uniqueness theorem in this part

Summary

Introduction

Xu [14] and zhang [15] introduced the so-called anticipated BDSDES (ABDSDEs) They proved the existence and uniqueness of the solution to these equations, obtained some comparison theorems in the one dimensional case, and studied the duality between ABDSDEs and delayed SDDEs. Reference [16] investigated a coupled system which is composed by a delayed forward doubly stochastic differential equation and an anticipated backward doubly SDE. We will prove that the solution of the above AGBDSDE exists uniquely under proper assumptions, and two versions of one dimensional comparison theorems are given These results are the cornerstones of AGBDSDEs applied to the obstacle problem for some SPDEs with the nonlinear Neumann boundary condition and some stochastic control problems with delay.

Preliminaries p

Existence and Uniqueness Theorem

Comparison Theorems

Conclusions