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 α-stable process into the unit ball. Specifically, we are interested in establishing the same law, but now for a one dimensional α-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 α-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 α-stable processes.

Publication
Annals of Probability 42, no. 1, 398–430
Date