Functionals of a L\'evy Process on Canonical and Generic Probability Spaces

TL;DR

This paper develops a Malliavin calculus framework for Lévy processes using functional representations on Skorohod spaces, enabling transfer of identities between canonical and arbitrary probability spaces.

Contribution

It introduces a novel approach to Malliavin calculus for Lévy processes via functional representations and a transfer technique for identities across probability spaces.

Findings

Established a functional representation framework for Lévy processes.

Developed a transfer technique for identities between probability spaces.

Provided a chain rule application for functionals of Lévy processes.

Abstract

We develop an approach to Malliavin calculus for L\'evy processes from the perspective of expressing a random variable $Y$ by a functional $F$ mapping from the Skorohod space of c\`adl\`ag functions to $\mathbb{R}$, such that $Y=F(X)$ where $X$ denotes the L\'evy process. We also present a chain-rule-type application for random variables of the form $f(\omega,Y(\omega))$. An important tool for these results is a technique which allows us to transfer identities proved on the canonical probability space (in the sense of Sol\'e et al.) associated to a L\'evy process with triplet $(\gamma,\sigma,\nu)$ to an arbitrary probability space $(\Omega,\mathcal{F},\mathbb{P})$ which carries a L\'evy process with the same triplet.