Notes on the Cauchy Problem for Backward Stochastic Partial Differential Equations

TL;DR

This paper investigates backward stochastic parabolic PDEs with variable coefficients, providing improved existence and uniqueness results in Sobolev spaces, and includes a comparison theorem as an application.

Contribution

It offers weaker assumptions for existence and uniqueness of solutions to backward stochastic PDEs and extends prior results by Zhou.

Findings

Enhanced existence and uniqueness theorems under weaker conditions

Application of comparison theorem to backward stochastic PDEs

Solutions established in Sobolev space $H^n$

Abstract

Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in the whole Euclidean space. Improved existence and uniqueness results are given in the Sobolev space $H^n$ ($=W^n_2$) under weaker assumptions than those used by X. Zhou [Journal of Functional Analysis 103, 275--293 (1992)]. As an application, a comparison theorem is obtained.