Stochastic viscosity solution for stochastic PDIEs with nonlinear Neumann boundary condition

TL;DR

This paper extends the concept of viscosity solutions to nonlinear stochastic PDEs with Neumann boundary conditions, utilizing backward doubly stochastic differential equations driven by Lévy processes to establish existence and extend the Feynman-Kac formula.

Contribution

It introduces a new framework for stochastic viscosity solutions for PDEs with boundary conditions, leveraging generalized backward doubly stochastic differential equations.

Findings

Proves existence of stochastic viscosity solutions for the class of equations.

Extends the nonlinear Feynman-Kac formula to these equations.

Provides a new approach for boundary value problems in stochastic PDEs.

Abstract

This paper is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential integral equations with nonlinear Neumann boundary condition. Using the recently developed theory on generalized backward doubly stochastic differential equations driven by a L\'evy process, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman-Kac formula.