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

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.

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• Censored Stable Subordinators and Fractional Derivatives
• The uniqueness of the Wiener-Hopf factorisation of Lévy processes and random walks
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• Markov additive friendships
• Representation of stationary and stationary increment processes via Langevin equation and self-similar processes
• Asian options and meromorphic Levy processes
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