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March 2012 Law of the absorption time of some positive self-similar Markov processes
P. Patie
Ann. Probab. 40(2): 765-787 (March 2012). DOI: 10.1214/10-AOP638

Abstract

Let X be a spectrally negative self-similar Markov process with 0 as an absorbing state. In this paper, we show that the distribution of the absorption time is absolutely continuous with an infinitely continuously differentiable density. We provide a power series and a contour integral representation of this density. Then, by means of probabilistic arguments, we deduce some interesting analytical properties satisfied by these functions, which include, for instance, several types of hypergeometric functions. We also give several characterizations of the Kesten’s constant appearing in the study of the asymptotic tail distribution of the absorbtion time. We end the paper by detailing some known and new examples. In particular, we offer an alternative proof of the recent result obtained by Bernyk, Dalang and Peskir [Ann. Probab. 36 (2008) 1777–1789] regarding the law of the maximum of spectrally positive Lévy stable processes.

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P. Patie. "Law of the absorption time of some positive self-similar Markov processes." Ann. Probab. 40 (2) 765 - 787, March 2012. https://doi.org/10.1214/10-AOP638

Information

Published: March 2012
First available in Project Euclid: 26 March 2012

zbMATH: 1241.60020
MathSciNet: MR2952091
Digital Object Identifier: 10.1214/10-AOP638

Subjects:
Primary: 33E30 , 60E07 , 60G18 , 60G51

Keywords: Absorption time , exponential functional , generalized hypergeometric functions , Lévy processes , Self-similar processes

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 2 • March 2012
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