Paper detail

Functionals of a Lévy Process on Canonical and Generic Probability Spaces

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 $f(ω,Y(ω))$. 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 $(γ,σ,ν)$ to an arbitrary probability space $(Ω,\mathcal{F},\mathbb{P})$ which carries a Lévy process with the same triplet.