Functional Itô calculus and stochastic integral representation of martingales

• Autores: Rama Cont, David-Antoine Fournié
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 1, 2013, págs. 109-133
• Idioma: inglés
• DOI: 10.1214/11-AOP721
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• Resumen

  • 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.