Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions

TL;DR

This paper introduces a new class of generalized backward doubly stochastic differential equations involving an integral with respect to an increasing process, providing a probabilistic representation for viscosity solutions of semi-linear SPDEs with Neumann boundary conditions.

Contribution

It develops a novel class of backward doubly stochastic differential equations and links them to viscosity solutions of semi-linear SPDEs with nonlinear boundary conditions.

Findings

Established a probabilistic representation for viscosity solutions of SPDEs.

Extended the theory of backward doubly stochastic differential equations.

Provided insights into SPDEs with Neumann boundary conditions.

Abstract

In this paper a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for viscosity solutions of semi-linear stochastic partial differential equations with a Neumann boundary condition is given.