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.
Keywords
Cite
@article{arxiv.1112.3690,
title = {Hitting distributions of alpha-stable processes via path censoring and self-similarity},
author = {Andreas E. Kyprianou and Alex Watson and Juan Carlos Pardo},
journal= {arXiv preprint arXiv:1112.3690},
year = {2021}
}