Functional Ito calculus and stochastic integral representation of martingales
Résumé
We develop a non-anticipative calculus for functionals of a continuous semimartingale, using a notion of pathwise functional derivative. A functional extension of the Ito formula is derived and used to obtain a constructive martingale representation theorem for a class of continuous martingales verifying a regularity property. By contrast with the Clark-Haussmann-Ocone formula, this representation involves non-anticipative quantities which can be computed pathwise. These results are used to construct a weak derivative acting on square-integrable martingales, which is shown to be the inverse of the Ito integral, and derive an integration by parts formula for Ito stochastic integrals. We show that this weak derivative may be viewed as a non-anticipative ``lifting" of the Malliavin derivative. Regular functionals of an Ito martingale which have the local martingale property are characterized as solutions of a functional differential equation, for which a uniqueness result is given.
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