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January 2013 Functional Itô calculus and stochastic integral representation of martingales
Rama Cont, David-Antoine Fournié
Ann. Probab. 41(1): 109-133 (January 2013). DOI: 10.1214/11-AOP721


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.


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Rama Cont. David-Antoine Fournié. "Functional Itô calculus and stochastic integral representation of martingales." Ann. Probab. 41 (1) 109 - 133, January 2013.


Published: January 2013
First available in Project Euclid: 23 January 2013

zbMATH: 1272.60031
MathSciNet: MR3059194
Digital Object Identifier: 10.1214/11-AOP721

Primary: 60G44 , 60H05 , 60H07 , 60H25

Keywords: Clark–Ocone formula , functional calculus , functional Itô formula , Malliavin derivative , Martingale representation , Semimartingale , stochastic calculus , Wiener functionals

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 1 • January 2013
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