Functional Ito calculus and stochastic integral representation of martingales

Rama Cont (1) , David-Antoine Fournié (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 Department of Mathematics [New York]

We develop a non-anticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by B 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 Ito integral and which may be viewed as a non-anticipative ''lifting" of the Malliavin derivative. These results lead to a constructive martingale representation formula for Ito processes. By contrast with the Clark-Haussmann-Ocone formula, this representation only involves non-anticipative quantities which may be computed pathwise.

Mots clés: Stochastic calculus; Functional calculus; Ito formula; Integration by parts; Malliavin derivative; Martingale representation; Semimartingale; Wiener functionals; Clark-Ocone formula

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Rama Cont, David-Antoine Fournié. Functional Ito calculus and stochastic integral representation of martingales. Annals of Probability, 2013, 41 (1), pp.109-133. ⟨10.1214/11-AOP721⟩. ⟨hal-00455700v4⟩