Obstacle problem for SPDE with nonlinear Neumann boundary condition via reflected generalized backward doubly SDEs

TL;DR

This paper establishes a probabilistic framework linking reflected generalized backward doubly stochastic differential equations to the stochastic viscosity solutions of semi-linear SPDEs with nonlinear Neumann boundary conditions.

Contribution

It introduces a novel probabilistic representation for solutions of complex SPDEs with boundary conditions using reflected backward doubly stochastic differential equations.

Findings

Probabilistic representation for stochastic viscosity solutions.

Connection between reflected backward doubly SDEs and SPDEs with boundary conditions.

Framework applicable to semi-linear reflected SPDEs.

Abstract

This paper is intended to give a probabilistic representation for stochastic viscosity solution of semi-linear reflected stochastic partial differential equations with nonlinear Neumann boundary condition. We use it connection with reflected generalized backward doubly stochastic differential equation.