Stochastic viscosity solutions of reflected stochastic partial differential equations with non-Lipschitz coefficients

TL;DR

This paper extends the concept of stochastic viscosity solutions to reflected semi-linear SPDEs with non-Lipschitz coefficients using probabilistic methods and RBDSDEs, broadening the applicability of the nonlinear Feynman-Kac formula.

Contribution

It introduces a probabilistic approach to establish solutions for reflected SPDEs with non-Lipschitz coefficients, generalizing previous results that required Lipschitz conditions.

Findings

Proved existence of stochastic viscosity solutions for non-Lipschitz coefficients.

Extended nonlinear Feynman-Kac formula to a broader class of reflected SPDEs.

Connected solutions with reflected backward doubly stochastic differential equations.

Abstract

This paper, is an attempt to extend the notion of stochastic viscosity solution to reflected semi-linear stochastic partial differential equations (RSPDEs, in short) with non-Lipschitz condition on the coefficients. Our method is fully probabilistic and use the recently developed theory on reflected backward doubly stochastic differential equations (RBDSDEs, in short). Among other, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman-Kac formula to reflected SPDEs, like one appear in \cite{2}. Indeed, in their recent work, Aman and Mrhardy \cite{2} established a stochastic viscosity solution for semi-linear reflected SPDEs with nonlinear Neumann boundary condition by using its connection with RBDSDEs. However, even Aman and Mrhardy consider a general class of reflected SPDEs, all their coefficients are at least Lipschitz. Therefore, our current work can be thought of as a new generalization of a now well-know Feymann-Kac formula to SPDEs with large class of coefficients, which does not seem to exist in the literature. In other words, our work extends (in non boundary case) Aman and Mrhardy's paper.