Malliavin smoothness on the L\'evy space with H\"older continuous or $BV$ functionals

TL;DR

This paper investigates the Malliavin smoothness of functionals of pure jump Lévy processes, showing how regularity of the functional and Lévy measure influence differentiability properties.

Contribution

It establishes new links between the regularity of functionals and the Lévy measure's Blumenthal-Getoor index in the context of Malliavin calculus.

Findings

Malliavin differentiability depends on the regularity of the functional and Lévy measure.

Fractional differentiability is characterized by the interplay of regularity and Blumenthal-Getoor index.

Results apply to bounded H"older continuous and BV functionals of Lévy processes.

Abstract

We consider Malliavin smoothness of random variables $f(X_1)$, where $X$ is a pure jump L\'evy process and $f$ is either bounded and H\"older continuous or of bounded variation. We show that Malliavin differentiability and fractional differentiability of $f(X_1)$ depend both on the regularity of $f$ and the Blumenthal-Getoor index of the L\'evy measure.