The Annals of Mathematical Statistics

The Representation of Functionals of Brownian Motion by Stochastic Integrals

J. M. C. Clark

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Abstract

It is known that any functional of Brownian motion with finite second moment can be expressed as the sum of a constant and an Ito stochastic integral. It is also known that homogeneous additive functionals of Brownian motion with finite expectations have a similar representation. This paper extends these results in several ways. It is shown that any finite functional of Brownian motion can be represented as a stochastic integral. This representation is not unique, but if the functional has a finite expectation it does have a unique representation as a constant plus a stochastic integral in which the process of indefinite integrals is a martingale. A corollary of this result is that any martingale (on a closed interval) that is measurable with respect to the increasing family of $\sigma$-fields generated by a Brownian motion is equal to a constant plus an indefinite stochastic integral. Sufficiently well-behaved Frechet-differentiable functionals have an explicit representation as a stochastic integral in which the integrand has the form of conditional expectations of the differential.

Article information

Source
Ann. Math. Statist., Volume 41, Number 4 (1970), 1282-1295.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177696903

Digital Object Identifier
doi:10.1214/aoms/1177696903

Mathematical Reviews number (MathSciNet)
MR270448

Zentralblatt MATH identifier
0213.19402

JSTOR
links.jstor.org

Citation

Clark, J. M. C. The Representation of Functionals of Brownian Motion by Stochastic Integrals. Ann. Math. Statist. 41 (1970), no. 4, 1282--1295. doi:10.1214/aoms/1177696903. https://projecteuclid.org/euclid.aoms/1177696903


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Corrections

  • Correction: J. M. C. Clark. Correction to "The Representation of Functionals of Brownian Motion by Stochastic Integrals" . Ann. Math. Statist., Vol. 42, Iss. 5 (1971), 1778.