Backward doubly stochastic differential equations with random coefficients and quasilinear stochastic PDEs

TL;DR

This paper extends the nonlinear stochastic Feynman-Kac formula to non-Markovian cases by linking backward doubly stochastic differential equations with random coefficients to quasilinear stochastic PDEs using Malliavin calculus.

Contribution

It introduces a novel connection between backward doubly stochastic differential equations with random coefficients and quasilinear stochastic PDEs, expanding existing stochastic analysis frameworks.

Findings

Established a relationship between backward doubly stochastic differential equations and stochastic PDEs.

Extended the nonlinear stochastic Feynman-Kac formula to non-Markovian cases.

Utilized Malliavin calculus to achieve these theoretical advancements.

Abstract

In this paper, by virtue of Malliavin calculus, we establish a relationship between backward doubly stochastic differential equations with random coefficients and quasilinear stochastic PDEs, and thus extend the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [14] to non-Markovian case.