Probabilistic representation of parabolic stochastic variational inequality with Dirichlet-Neumann boundary and variational generalized backward doubly stochastic differential equations

TL;DR

This paper establishes the existence and uniqueness of a generalized backward doubly stochastic differential equation with convex sub-differential, providing a probabilistic representation for parabolic variational stochastic PDEs with boundary conditions.

Contribution

It introduces a novel probabilistic framework for parabolic variational stochastic PDEs with boundary conditions using generalized backward doubly stochastic differential equations.

Findings

Proved existence and uniqueness under non-Lipschitz conditions.

Provided a stochastic viscosity solution representation.

Extended the theory to include Dirichlet-Neumann boundary conditions.

Abstract

We derive the existence and uniqueness of the generalized backward doubly stochastic differential equation with sub-differential of a lower semi-continuous convex function under a non Lipschitz condition. This study allows us give a probabilistic representation (in stochastic viscosity sense) to the parabolic variational stochastic partial differential equations with Dirichlet-Neumann conditions.