The Annals of Probability

Explicit Stochastic Integral Representations for Historical Functionals

Steven N. Evans and Edwin A. Perkins

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176987803

Digital Object Identifier
doi:10.1214/aop/1176987803

Mathematical Reviews number (MathSciNet)
MR1379168

Zentralblatt MATH identifier
0852.60062

JSTOR
links.jstor.org

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

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

Citation

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. https://projecteuclid.org/euclid.aop/1176987803


Export citation