Backward Stochastic Differential Equations (BSDEs) Using Infinite-dimensional Martingales with Subdifferential Operator

TL;DR

This paper establishes the existence and uniqueness of solutions for a class of infinite-dimensional backward stochastic differential equations driven by martingales involving subdifferential operators, using Yosida approximations and fixed point methods.

Contribution

It introduces a novel framework for solving infinite-dimensional BSDEs with subdifferential operators driven by symmetric martingales, extending existing theories.

Findings

Existence of solutions via Yosida approximations

Uniqueness proved using Fixed Point Theorem

Application to backward stochastic PDEs with unique solutions

Abstract

In this paper, we focus on a family of backward stochastic differential equations (BSDEs) with sub-differential operators that are driven by infinite-dimensional martingales which involve symmetry, that is, the process involves a positive definite nuclear operator Q. We shall show that the solution to such infinite-dimensional BSDEs exists and is unique. The existence of the solution is established using Yosida approximations, and the uniqueness is proved using Fixed Point Theorem. Furthermore, as an application of the main result, we shall show that the backward stochastic partial differential equation driven by infinite-dimensional martingales with a continuous linear operator has a unique solution under the condition that the function F equals to zero.