A note on Malliavin smoothness on the L\'evy space

TL;DR

This paper explores Malliavin calculus on the Lévy space, establishing conditions under which random variables exhibit differentiability properties based on their measurability and weighted Lebesgue spaces.

Contribution

It introduces a measurability condition that links Malliavin differentiability to weighted Lebesgue spaces for Lévy space random variables.

Findings

Differentiability determined by weighted Lebesgue spaces

Measurability condition satisfied for compound Poisson processes

Results applicable to finite interval Lévy processes

Abstract

We consider Malliavin calculus based on the It\^o chaos decomposition of square integrable random variables on the L\'evy space. We show that when a random variable satisfies a certain measurability condition, its differentiability and fractional differentiability can be determined by weighted Lebesgue spaces. The measurability condition is satisfied for all random variables if the underlying L\'evy process is a compound Poisson process on a finite time interval.