Functional It\^{o} calculus and stochastic integral representation of martingales

TL;DR

This paper introduces a nonanticipative calculus for path-dependent functionals of continuous semimartingales, extending Ito's formula and deriving a martingale representation that is computable pathwise without anticipation.

Contribution

It develops a new pathwise derivative extension that provides a constructive, nonanticipative martingale representation formula for Ito processes.

Findings

Introduces a pathwise derivative extending Dupire's concept.

Derives a nonanticipative martingale representation formula.

Provides a constructive, pathwise computable approach to stochastic integration.

Abstract

We develop a nonanticipative 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 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 nonanticipative "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 nonanticipative quantities which may be computed pathwise.