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

TL;DR

This paper investigates the distribution of absorption times for spectrally negative self-similar Markov processes, revealing it has a smooth density with explicit representations and exploring its analytical properties and asymptotic behavior.

Contribution

It provides new explicit representations and analytical characterizations of the absorption time distribution, including hypergeometric functions and properties of Kesten's constant.

Findings

Absorption time density is infinitely differentiable and absolutely continuous.

Provides power series and contour integral representations of the density.

Includes new proofs and examples related to spectrally positive Lévy 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 L\'{e}vy stable processes.