The Annals of Probability

Law of the absorption time of some positive self-similar Markov processes

P. Patie

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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.

Article information

Source
Ann. Probab., Volume 40, Number 2 (2012), 765-787.

Dates
First available in Project Euclid: 26 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1332772720

Digital Object Identifier
doi:10.1214/10-AOP638

Mathematical Reviews number (MathSciNet)
MR2952091

Zentralblatt MATH identifier
1241.60020

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions 60G18: Self-similar processes 60G51: Processes with independent increments; Lévy processes 33E30: Other functions coming from differential, difference and integral equations

Keywords
Self-similar processes absorption time Lévy processes exponential functional generalized hypergeometric functions

Citation

Patie, P. Law of the absorption time of some positive self-similar Markov processes. Ann. Probab. 40 (2012), no. 2, 765--787. doi:10.1214/10-AOP638. https://projecteuclid.org/euclid.aop/1332772720


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