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Law of the absorption time of some positive self-similar Markov processes. (English) Zbl 1241.60020
Summary: 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 V. Bernyk, R. C. Dalang and G. Peskir [Ann. Probab. 36, No. 5, 1777–1789 (2008; Zbl 1185.60051)] regarding the law of the maximum of spectrally positive Lévy stable processes.

MSC:
60G18 Self-similar stochastic processes
60E07 Infinitely divisible distributions; stable distributions
60G51 Processes with independent increments; Lévy processes
33E30 Other functions coming from differential, difference and integral equations
60J25 Continuous-time Markov processes on general state spaces
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