Martingale representations for diffusion processes and backward stochastic differential equations

2009/10/26 by Zhongmin Qian, Qian, Zhongmin, ; Jiangang Ying +1
Mathematics · #60H10 #60H30 #60J45 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60H10 #msc:60H30 #msc:60J45

paper · pdf · doi:10.48550/arxiv.0910.4911

28 pages

arxiv created 2009/10/26 · arxiv updated 2009/12/01

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.

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