The Annals of Probability

Explicit Stochastic Integral Representations for Historical Functionals

Steven N. Evans and Edwin A. Perkins

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It is known from previous work of the authors that any square-integrable functional of a superprocess may be represented as a constant plus a stochastic integral against the associated orthogonal martingale measure. Here we give, for a large class of such functionals, an explicit description of the integrand that is analogous to Clark's formula for the representation of certain Brownian functionals. As a consequence, we develop a partial analogue of the Wiener chaos expansion in the superprocess setting. Rather than work with superprocesses per se, our results are stated and proved in the richer and more natural context of the associated historical process.

Article information

Ann. Probab., Volume 23, Number 4 (1995), 1772-1815.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60H05: Stochastic integrals
Secondary: 60G57: Random measures 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Superprocess historical process martingale measure stochastic integral predictable representation Clark's formula Wiener chaos


Evans, Steven N.; Perkins, Edwin A. Explicit Stochastic Integral Representations for Historical Functionals. Ann. Probab. 23 (1995), no. 4, 1772--1815. doi:10.1214/aop/1176987803.

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