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Acta Applicandae Mathematicae

, Volume 123, Issue 1, pp 113–139 | Cite as

Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes

  • A. Kuznetsov
  • J. C. Pardo
Article

Abstract

We study the distribution and various properties of exponential functionals of hypergeometric Lévy processes. We derive an explicit formula for the Mellin transform of the exponential functional and give both convergent and asymptotic series expansions of its probability density function. As applications we present a new proof of some of the results on the density of the supremum of a stable process, which were recently obtained in Hubalek and Kuznetsov (Electron. Commun. Probab. 16:84–95, 2011) and Kuznetsov (Ann. Probab. 39(3):1027–1060, 2011). We also derive several new results related to (i) the entrance law of a stable process conditioned to stay positive, (ii) the entrance law of the excursion measure of a stable process reflected at its past infimum, (iii) the distribution of the lifetime of a stable process conditioned to hit zero continuously and (iv) the entrance law and the last passage time of the radial part of a multidimensional symmetric stable process.

Keywords

Hypergeometric Lévy processes Lamperti-stable processes Exponential functional Double gamma function Lamperti transformation Extrema of stable processes 

Mathematics Subject Classification

60G51 60G52 

Notes

Acknowledgements

A. Kuznetsov’s research is supported by the Natural Sciences and Engineering Research Council of Canada.

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada
  2. 2.Centro de Investigación en MatemáticasGuanajuatoMéxico

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