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Regularizing mappings of Lévy measures

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  • Institut for Matematiske Fag
In this paper we introduce and study a regularizing one-to-one mapping from the class of one-dimensional Lévy measures into itself. This mapping appeared implicitly in our previous paper [O.E. Barndorff-Nielsen, S. Thorbjørnsen, A connection between free and classical infinite divisibility, Inf. Dim. Anal. Quant. Probab. 7 (2004) 573–590], where we introduced a one-to-one mapping from the class of one-dimensional infinitely divisible probability measures into itself. Based on the investigation of in the present paper, we deduce further properties of . In particular it is proved that maps the class of selfdecomposable laws onto the so called Thorin class . Further, partly motivated by our previous studies of infinite divisibility in free probability, we introduce a one-parameter family of one-to-one mappings , which interpolates smoothly between (α=0) and the identity mapping on (α=1). We prove that each of the mappings shares many of the properties of . In particular, they are representable in terms of stochastic integrals with respect to associated Levy processes.
OriginalsprogEngelsk
TidsskriftStochastic Processes and Their Applications
Vol/bind116
Nummer3
Sider (fra-til)423-446
Antal sider24
ISSN0304-4149
DOI
StatusUdgivet - 2006

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