LAW OF THE ABSORPTION TIME OF SOME POSITIVE SELF-SIMILAR MARKOV PROCESSES

• 成果类型: Article
• 署名作者: Patie, P.
• 署名单位: Universite Libre de Bruxelles
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP638
• 发表日期: 2012
• 页码: 765-787
• 关键词: exponential functionals; EQUATIONS; supremum

摘要

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 Levy stable processes.