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


Reference:

Kyprianou, A., Pardo, J. C. and Watson, A., 2014. Hitting distributions of α-stable processes via path censoring and self-similarity. Annals of Probability, 42 (1), pp. 398-430.

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Official URL:

http://dx.doi.org/10.1214/12-AOP790

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.

Details

Item Type Articles
CreatorsKyprianou, A., Pardo, J. C. and Watson, A.
DOI10.1214/12-AOP790
DepartmentsFaculty of Science > Mathematical Sciences
RefereedYes
StatusPublished
ID Code32429

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