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Three upsilon transforms related to tempered stable distributions. (English) Zbl 1328.60042

Summary: We discuss the properties of three upsilon transforms, which are related to the class of \(p\)-tempered \(\alpha\)-stable (\(TS^p_\alpha\)) distributions. In particular, we characterize their domains and show how they can be represented as compositions of each other. Further, we show that if \(-\infty<\beta<\alpha<2\) and \(0<q<p<\infty\) then they can be used to transform the Lévy measures of \(TS^p_\beta\) distributions into those of \(TS^q_\alpha\).

MSC:

60E07 Infinitely divisible distributions; stable distributions
60G51 Processes with independent increments; Lévy processes
60H05 Stochastic integrals
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