Hitting distributions of α-stable processes via path censoring and self-similarity

Andreas Kyprianou, J. C. Pardo, Alexander Watson

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Abstract

In this paper we return to the problem of Blumenthal-Getoor-Ray, published in 1961, which gave the law of the position of first entry of a symmetric alpha-stable process into the unit ball. Specifically, we are interested in establishing the same law, but now for a one dimensional alpha-stable process which enjoys two-sided jumps, and which is not necessarily symmetric. Our method is modern in the sense that we appeal to the relationship between alpha-stable processes and certain positive self-similar Markov processes. However there are two notable additional innovations. First, we make use of a type of path censoring. Second, we are able to describe in explicit analytical detail a non-trivial Wiener-Hopf factorisation of an auxiliary Levy process from which the desired solution can be sourced. Moreover, as a consequence of this approach, we are able to deliver a number of additional, related identities in explicit form for alpha-stable processes.
Original languageEnglish
Pages (from-to)398-430
Number of pages33
JournalAnnals of Probability
Volume42
Issue number1
DOIs
Publication statusPublished - Jan 2014

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Stable Process
Self-similarity
Censoring
Path
Wiener-Hopf Factorization
Self-similar Processes
Appeal
Lévy Process
Unit ball
Markov Process
Half line
Jump

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Hitting distributions of α-stable processes via path censoring and self-similarity. / Kyprianou, Andreas; Pardo, J. C. ; Watson, Alexander.

In: Annals of Probability, Vol. 42, No. 1, 01.2014, p. 398-430.

Research output: Contribution to journalArticle

Kyprianou, Andreas ; Pardo, J. C. ; Watson, Alexander. / Hitting distributions of α-stable processes via path censoring and self-similarity. In: Annals of Probability. 2014 ; Vol. 42, No. 1. pp. 398-430.
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