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Some explicit identities associated with positive self-similar Markov processes. (English) Zbl 1170.60017
Some special classes of Lévy processes are considered with no Gaussian component whose Lévy measure is of the type \(\pi(dx)=e^{\gamma x}\nu(e^x-1)dx\), where \(\nu\) is the density of the stable Lévy measure and \(\gamma\) is a positive parameter which depends on its characteristics. These processes were introduced by M. E. Caballero and L. Chaumont [J. Appl. Probab. 43, 967–983 (2006; Zbl 1133.60316)] as the underlying Lévy processes in the Lamperti representation of conditioned stable Lévy processes. The law of these Lévy processes at their first exit time from a finite or semi-finite interval, the law of their exponential functional and the first hitting time probability of a pair of points are computed explicitly.

60G18 Self-similar stochastic processes
60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes
60G40 Stopping times; optimal stopping problems; gambling theory
Full Text: DOI
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