Representation theorems for backward doubly stochastic differential equations

TL;DR

This paper develops representation theorems for backward doubly stochastic differential equations (BDSDEs) with terminal conditions depending on forward diffusion history, linking them to stochastic viscosity solutions of SPDEs without derivative assumptions.

Contribution

It provides a probabilistic representation for the spatial gradient of stochastic viscosity solutions to quasilinear SPDEs, leading to explicit formulas for BDSDE martingale integrands.

Findings

Representation of the spatial gradient of stochastic viscosity solutions.

Closed-form expression for BDSDE martingale integrand.

Representation valid under standard Lipschitz conditions.

Abstract

In this paper we study the class of backward doubly stochastic differential equations (BDSDEs, for short) whose terminal value depends on the history of forward diffusion. We first establish a probabilistic representation for the spatial gradient of the stochastic viscosity solution to a quasilinear parabolic SPDE in the spirit of the Feynman-Kac formula, without using the derivatives of the coefficients of the corresponding BDSDE. Then such a representation leads to a closed-form representation of the martingale integrand of BDSDE, under only standard Lipschitz condition on the coefficients.