Functionals of a Lévy Process on Canonical and Generic Probability Spaces

Abstract

We develop an approach to Malliavin calculus for Lévy processes from the perspective of expressing a random variable $Y$ by a functional $F$ mapping from the Skorohod space of càdlàg functions to $\mathbb{R}$, such that $Y=F(X)$ where $X$ denotes the Lévy 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 Solé et al.) associated to a Lévy process with triplet $(\gamma ,\sigma ,\nu )$ to an arbitrary probability space $(\Omega ,\mathcal{F},\mathbb{P})$ which carries a Lévy process with the same triplet.

19 pages, in version 2 and 3 Lemma 3.2 has been generalized. Version 3 has some minor changes and some typos corrected

References in corpus (2)

• Universal Malliavin Calculus in Fock and Lévy-Itô Spaces
• On Skorohod spaces as universal sample path spaces