Malliavin calculus and Clark-Ocone formula for functionals of a square-integrable L\'evy process

TL;DR

This paper develops a Malliavin calculus framework for functionals of square-integrable Lévy processes, deriving a Clark-Ocone formula and illustrating its application with explicit martingale representations.

Contribution

It introduces a Malliavin derivative for Lévy process functionals using chaos expansions and derives a Clark-Ocone formula, extending stochastic calculus tools to Lévy processes.

Findings

Constructed a Malliavin derivative for Lévy process functionals.

Derived a Clark-Ocone formula for these functionals.

Provided explicit martingale representation for the maximum of a Lévy process.

Abstract

In this paper, we construct a Malliavin derivative for functionals of square-integrable L\'evy processes and derive a Clark-Ocone formula. The Malliavin derivative is defined via chaos expansions involving stochastic integrals with respect to Brownian motion and Poisson random measure. As an illustration, we compute the explicit martingale representation for the maximum of a L\'evy process.