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Denseness of certain smooth Lévy functionals in D1;2

Geiss C., Laukkarinen E.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 1

1--15

The Malliavin derivative for a Lévy process (Xt) can be defined on the space D1;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(Xt1 ; : : : ;Xtn); where f is a smooth function with compact support. We show that the closure of L2(P) ⊇ S D→ L2(m⊗P) yields to the space D1;2: As an application we conclude that Lipschitz functions operate on D1;2: