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Contents of PMS, Vol. 31, Fasc. 1,
pages 1 - 15
 

DENSENESS OF CERTAIN SMOOTH LÉVY FUNCTIONALS IN D1,2

Christel Geiss
Eija Laukkarinen

Abstract: The Malliavin derivative for a Lévy process (X ) t can be defined on the space D 1,2 using a chaos expansion or in the case of a pure jump process also via an increment quotient operator. In this paper we define the Malliavin derivative operator D on the class S of smooth random variables f(X ,...,X ), t1 tn where f is a smooth function with compact support. We show that the closure of D L2(ℙ ) ⊇ S → L2(m ⊗ ℙ) yields to the space D1,2. As an application we conclude that Lipschitz functions operate on D1,2.

2000 AMS Mathematics Subject Classification: Primary: 60H07; Secondary: 60G51.

Keywords and phrases: Malliavin calculus, Lévy processes.

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