Probabilistic interpretation for solutions of fully Nonlinear Stochastic PDEs

TL;DR

This paper develops a well-posedness theory for second order backward doubly stochastic differential equations (2BDSDEs) and links their solutions to classical and stochastic viscosity solutions of fully nonlinear stochastic PDEs, providing a probabilistic interpretation.

Contribution

It introduces a new well-posedness framework for 2BDSDEs and establishes their connection to solutions of fully nonlinear stochastic PDEs, enhancing understanding of their probabilistic structure.

Findings

Existence and uniqueness of solutions for 2BDSDEs under Lipschitz conditions

Probabilistic interpretation of fully nonlinear SPDE solutions via 2BDSDEs

Link between Markovian solutions of 2BDSDEs and viscosity solutions of SPDEs

Abstract

In this article, we propose a wellposedness theory for a class of second order backward doubly stochastic differential equation (2BDSDE). We prove existence and uniqueness of the solution under a Lipschitz type assumption on the generator, and we investigate the links between our 2BDSDEs and a class of parabolic fully nonLinear Stochastic PDes. Precisely, we show that the Markovian solution of 2BDSDEs provide a probabilistic interpretation of the classical and stochastic viscosity solution of fully nonlinear SPDEs.