Functional Itô Calculus

25 Pages Posted: 25 Jul 2009 Last revised: 28 Aug 2009

Date Written: July 17, 2009


Itô calculus deals with functions of the current state whilst we deal with functions of the current path to acknowledge the fact that often the impact of randomness is cumulative. We express the differential of the functional in terms of adequately defined partial derivatives to obtain an Itô formula. We develop an extension of the Feynman-Kac formula to the functional case and an explicit expression of the integrand in the Martingale Representation Theorem, providing an alternative to the Clark-Ocone formula from Malliavin Calculus. We establish that under certain conditions, even path dependent options prices satisfy a partial differential equation in a local sense.

Keywords: Itô calculus, path dependent options, functionals

JEL Classification: G13

Suggested Citation

Dupire, Bruno, Functional Itô Calculus (July 17, 2009). Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS, Available at SSRN: or

Bruno Dupire (Contact Author)

Bloomberg L.P. ( email )

731 Lexington Avenue
New York, NY 10022
United States

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