## The Annals of Mathematical Statistics

### The Representation of Functionals of Brownian Motion by Stochastic Integrals

J. M. C. Clark

#### 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

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