Semi-linear Degenerate Backward Stochastic Partial Differential Equations and Associated Forward Backward Stochastic Differential Equations

TL;DR

This paper establishes existence, uniqueness, and regularity results for semi-linear degenerate backward stochastic PDEs and links them to forward-backward stochastic differential equations, extending the Feynman-Kac formula to non-Markovian settings.

Contribution

It provides the first comprehensive analysis of semi-linear degenerate BSPDEs without restrictive coefficient assumptions and connects them to FBSDEs in a non-Markovian framework.

Findings

Proved existence and uniqueness of solutions.

Established regularity properties of solutions.

Extended Feynman-Kac formula to non-Markovian cases.

Abstract

In this paper, we consider the Cauchy problem of semi-linear degenerate backward stochastic partial differential equations (BSPDEs in short) under general settings without technical assumptions on the coefficients. For the solution of semi-linear degenerate BSPDE, we first give a proof for its existence and uniqueness, as well as regularity. Then the connection between semi-linear degenerate BSPDEs and forward backward stochastic differential equations (FBSDEs in short) is established, which can be regarded as an extension of Feynman-Kac formula to non-Markov frame.