Forward-backward doubly stochastic systems and classical solutions of path-dependent stochastic PDEs

TL;DR

This paper explores non-Markovian forward-backward doubly stochastic systems using path-dependent calculus, establishing their connection to path-dependent SPDEs and extending the nonlinear stochastic Feynman-Kac formula to non-Markovian cases.

Contribution

It introduces a novel approach linking non-Markovian stochastic systems with path-dependent SPDEs, extending classical results to non-Markovian frameworks.

Findings

Established the relationship between non-Markovian systems and path-dependent SPDEs.

Proved differentiability of solutions to the stochastic systems.

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

Abstract

In this paper, a class of non-Markovian forward-backward doubly stochastic systems is studied. By using the technique of functional It\^o (or path-dependent) calculus, the relationship between the systems and related path-dependent quasi-linear stochastic partial differential equations (SPDEs in short) is established, and the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98 (1994), pp. 209--227] is developed to the non-Markovian situation. Moreover, we obtain the differentiability of the solution to the forward-backward doubly stochastic systems and some properties of solutions to the path-dependent SPDEs.