Functional Itô calculus and stochastic integral representation of martingales

TLDR

The resulting representation differs from the Clark–Haussmann–Ocone formula by involving only nonanticipative, pathwise computable quantities. The paper develops a nonanticipative calculus for functionals of continuous semimartingales by extending the Itô formula to path‑dependent functionals with directional derivatives. This is achieved by employing Dupire’s pathwise derivative on the space of càdlàg functions, yielding a nonanticipative extension of the Itô formula. The extension defines a weak derivative that is the inverse of the Itô integral, lifts the Malliavin derivative, and provides a constructive martingale representation for Itô processes using only nonanticipative, pathwise computable terms.

Abstract

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.