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Denseness of certain smooth L\'evy functionals in $\DD_{1,2}$ by Geiss, Christel; Laukkarinen, Eija is a scholarly article available to read on EtoBox.

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

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 $\DD_{1,2}.$

Author

Geiss, Christel; Laukkarinen, Eija

Published

2008

Language

EN