Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes

  • Authors:

  • A. Kuznetsov;J. C. Pardo

  • Affiliations:

  • Department of Mathematics and Statistics, York University, Toronto, Canada M3J 1P3;Centro de Investigación en Matemáticas, Guanajuato, México 36240

  • Venue:
  • Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
  • Year:
  • 2013

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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.