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

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.

Published in at http://dx.doi.org/10.1214/10-AOP638 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• The law of the supremum of a stable Lévy process with no negative jumps
• Law of the exponential functional of one-sided Lévy processes and Asian options

Cited by in corpus (4)

• Asymptotics of self-similar growth-fragmentation processes
• Extinction time of non-Markovian self-similar processes, persistence, annihilation of jumps and the Fréchet distribution
• Bivariate Bernstein-gamma functions, potential measures, and asymptotics of exponential functionals of Lévy processes
• The first passage time of a stable process conditioned to not overshoot