MaPhySto
Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

MPS-RR 1999-22
June 1999

# Martingale Representation, Chaos Expansion and Clark-Ocone Formulas

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## Abstract

Ito's representation theorem gives the existence of a martingale representation of stochastic variables with respect to Brownian motion. Similar results exist for instance for compensated Poisson processes and Azema's martingale. We give sufficient conditions for predictable representation (in a weak sense), i.e. there exists predictable processes \phi^\alpha such that every F \in L^2(\cal F_\infty; P) can be represented F = E[F] + \sum_\alpha \phi^\alpha\cdot M^\alpha for some given collection of martingales {M^\alpha}_{\alpha\in\cal I}}. Thereafter we show how one can obtain explicit expressions for the representation using Malliavin calculus methods. The theory is then applied to Levy processes.

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