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August, 1970 The Representation of Functionals of Brownian Motion by Stochastic Integrals
J. M. C. Clark
Ann. Math. Statist. 41(4): 1282-1295 (August, 1970). DOI: 10.1214/aoms/1177696903


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.


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J. M. C. Clark. "The Representation of Functionals of Brownian Motion by Stochastic Integrals." Ann. Math. Statist. 41 (4) 1282 - 1295, August, 1970.


Published: August, 1970
First available in Project Euclid: 27 April 2007

zbMATH: 0213.19402
MathSciNet: MR270448
Digital Object Identifier: 10.1214/aoms/1177696903

Rights: Copyright © 1970 Institute of Mathematical Statistics

Vol.41 • No. 4 • August, 1970
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