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 $$Y$$ Y by a functional $$F$$ F mapping from the Skorohod space of càdlàg functions to $$\mathbb {R}$$ R , such that $$Y=F(X)$$ Y = F ( X ) where $$X$$ X denotes the Lévy process. We also present a chain-rule-type application for random variables of the form $$f(\omega ,Y(\omega ))$$ 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 $$(\gamma ,\sigma ,\nu )$$ ( γ , σ , ν ) to an arbitrary probability space $$(\varOmega ,\mathcal {F},\mathbb {P})$$ ( Ω , F , P ) which carries a Lévy process with the same triplet.