Martingale representations for diffusion processes and backward stochastic differential equations

TL;DR

This paper proves a martingale representation theorem for continuous Hunt processes, including reflecting diffusions, and applies it to solve boundary value problems for quasi-linear parabolic equations via backward stochastic differential equations.

Contribution

It establishes a new martingale representation theorem for a class of continuous Hunt processes and applies it to boundary problems for parabolic PDEs using BSDEs.

Findings

Martingales over continuous Hunt processes are continuous.

Theorem applies to reflecting symmetric diffusions in bounded domains.

Results facilitate solving boundary value problems for PDEs using stochastic methods.

Abstract

In this paper we explain that the natural filtration of a continuous Hunt process is continuous, and show that martingales over such a filtration are continuous. We further establish a martingale representation theorem for a class of continuous Hunt processes under certain technical conditions. In particular we establish the martingale representation theorem for the martingale parts of (reflecting) symmetric diffusions in a bounded domain with a continuous boundary. Together with an approach put forward in Lyons et al(2009), our martingale representation theorem is then applied to the study of initial and boundary problems for quasi-linear parabolic equations by using solutions to backward stochastic differential equations over the filtered probability space determined by reflecting diffusions in a bounded domain with only continuous boundary.