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Martingale Representations for Diffusion Processes and Backward Stochastic Differential Equations

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Séminaire de Probabilités XLIV

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2046))

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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 diffusion in a bounded domain with a continuous boundary. Together with an approach put forward in (Liang et al., Ann. Probab.), 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.

AMS Classification: 60H10, 60H30, 60J45

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Acknowledgements

The research of the first author was supported in part by EPSRC grant EP/F029578/1, and by the Oxford-Man Institute. The second author’s research was supported in part by the National Basic Research Program of China (973 Program) under grant No. 2007CB814904, and a Royal Society Visiting grant.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, OX1 3LB, England

    Zhongmin Qian

  2. Institute of Mathematics, Fudan University, Shanghai, China

    Jiangang Ying

Authors
  1. Zhongmin Qian
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  2. Jiangang Ying
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Correspondence to Zhongmin Qian .

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Editors and Affiliations

  1. , Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des États-Unis 45, Versailles, 78035, France

    Catherine Donati-Martin

  2. IECN, Campus scientifique, BP 239, Nancy-Université, INRIA, Vandoeuvre-lès-Nancy, 54506, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des Etats-Unis 45, Versailles, 78035, France

    Alain Rouault

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Qian, Z., Ying, J. (2012). Martingale Representations for Diffusion Processes and Backward Stochastic Differential Equations. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIV. Lecture Notes in Mathematics(), vol 2046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27461-9_4

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