Denseness of certain smooth L\'evy functionals in $\DD_{1,2}$

TL;DR

This paper establishes the density of certain smooth Le9vy functionals in the Sobolev space _{1,2} using Malliavin calculus, and shows Lipschitz functions preserve this space.

Contribution

It defines the Malliavin derivative on smooth functionals of Le9vy processes and proves their denseness in _{1,2}, extending the applicability of Malliavin calculus.

Findings

Smooth Le9vy functionals are dense in _{1,2}.

Lipschitz functions map _{1,2} into itself.

The Malliavin derivative can be extended from smooth to general functionals.

Abstract

The Malliavin derivative for a L\'evy process $(X_t)$ can be defined on the space $\DD_{1,2}$ using a chaos expansion or in the case of a pure jump process also via an increment quotient operator \cite{sole-utzet-vives}. In this paper we define the Malliavin derivative operator $\D$ on the class $\mathcal{S}$ of smooth random variables $f(X_{t_1}, ..., X_{t_n}),$ where $f$ is a smooth function with compact support. We show that the closure of $L_2(\Om) \supseteq \mathcal{S} \stackrel{\D}{\to} L_2(\m\otimes \mass)$ yields to the space $\DD_{1,2}.$ As an application we conclude that Lipschitz functions map from $\DD_{1,2}$ into $\DD_{1,2}.$