The Annals of Probability

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

P. Patie

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

First available in Project Euclid: 26 March 2012

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Zentralblatt MATH identifier

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

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


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.

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