## The Annals of Mathematical Statistics

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

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

#### 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

#### 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.Project Euclid: euclid.aoms/1177693186