Multivalued stochastic Dirichlet-Neumann problems and generalized backward doubly stochastic differential equations

TL;DR

This paper introduces a new class of generalized backward doubly stochastic differential equations involving subdifferential operators, proving existence and uniqueness, and applying these results to multivalued stochastic Dirichlet-Neumann problems.

Contribution

It develops a novel framework for solving multivalued stochastic PDEs using Yosida approximation and establishes existence and uniqueness results.

Findings

Proved existence and uniqueness of solutions for generalized backward doubly stochastic differential equations.

Derived existence of stochastic viscosity solutions for multivalued stochastic Dirichlet-Neumann problems.

Applied penalization techniques to handle subdifferential operators in stochastic differential equations.

Abstract

In this paper, a class of generalized backward doubly stochastic differential equations whose coefficient contains the subdifferential operators of two convex functions (also called generalized backward doubly stochastic variational inequalities) are considered. By means of a penalization argument based on Yosida approximation, we establish the existence and uniqueness of the solution. As an application, this result is used to derive existence result of stochastic viscosity solution for a class of multivalued stochastic Dirichlet-Neumann problems.