The Annals of Probability

Functional Itô calculus and stochastic integral representation of martingales

Rama Cont and David-Antoine Fournié

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We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Itô formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative admits a suitable extension to the space of square-integrable martingales. This extension defines a weak derivative which is shown to be the inverse of the Itô integral and which may be viewed as a nonanticipative “lifting” of the Malliavin derivative.

These results lead to a constructive martingale representation formula for Itô processes. By contrast with the Clark–Haussmann–Ocone formula, this representation only involves nonanticipative quantities which may be computed pathwise.

Article information

Ann. Probab. Volume 41, Number 1 (2013), 109-133.

First available in Project Euclid: 23 January 2013

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Mathematical Reviews number (MathSciNet)

Primary: 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus 60G44: Martingales with continuous parameter 60H25: Random operators and equations [See also 47B80]

Stochastic calculus functional calculus functional Itô formula Malliavin derivative martingale representation semimartingale Wiener functionals Clark–Ocone formula


Cont, Rama; Fournié, David-Antoine. Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab. 41 (2013), no. 1, 109--133. doi:10.1214/11-AOP721.

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