Strong Solution of Backward Stochastic Partial Differential Equations in $C^2$ Domains

TL;DR

This paper establishes existence and uniqueness of strong solutions for backward stochastic PDEs in $C^2$ domains, using continuation methods and weighted Sobolev spaces, with applications to comparison theorems and semi-linear equations.

Contribution

It provides new existence and uniqueness results for backward stochastic PDEs in $C^2$ domains under weak conditions, including in weighted Sobolev spaces.

Findings

Proved existence and uniqueness of strong solutions.

Developed comparison theorem for solutions.

Analyzed semi-linear equations in $C^2$ domains.

Abstract

This paper is concerned with the strong solution to the Cauchy-Dirichlet problem for backward stochastic partial differential equations of parabolic type. Existence and uniqueness theorems are obtained, due to an application of the continuation method under fairly weak conditions on variable coefficients and $C^2$ domains. The problem is also considered in weighted Sobolev spaces which allow the derivatives of the solutions to blow up near the boundary. As applications, a comparison theorem is obtained and the semi-linear equation is discussed in the $C^2$ domain.