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Law of the absorption time of some positive self-similar Markov processes
- Source :
- Ann. Probab. 40, no. 2 (2012), 765-787
- Publication Year :
- 2012
- Publisher :
- Institute of Mathematical Statistics, 2012.
-
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.<br />Comment: 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)
- Subjects :
- Statistics and Probability
Power series
Markov process
33E30
symbols.namesake
60E07
FOS: Mathematics
Hypergeometric function
Mathematics
absorption time
Probability (math.PR)
exponential functional
Self-similar processes
State (functional analysis)
Absolute continuity
Methods of contour integration
Distribution (mathematics)
Lévy processes
60G18
Law
generalized hypergeometric functions
symbols
Statistics, Probability and Uncertainty
Constant (mathematics)
60G51
Mathematics - Probability
Subjects
Details
- ISSN :
- 00911798
- Volume :
- 40
- Database :
- OpenAIRE
- Journal :
- The Annals of Probability
- Accession number :
- edsair.doi.dedup.....f580a776f401ace59b2fe31776a94a66