α-stable Lévy processes entering the half space or a slab

Abstract

Recently a series of publications, including e.g. (Kyprianou, 2016 [1]; Kyprianou et al., 2018 [2]; Kyprianou et al., 2019; Kyprianou et al., 2014; Kyprianou and Pardo, 2022), considered a number of new fluctuation identities for $\alpha$-stable Lévy processes in one and higher dimensions by appealing to underlying Lamperti-type path decompositions. In the setting of $d$-dimensional isotropic processes, (Kyprianou et al., 2019) in particular, developed so called $n$-tuple laws for first entrance and exit of balls. Fundamental to these works is the notion that the paths can be decomposed via generalised spherical polar coordinates revealing an underlying Markov Additive Process (MAP) for which a more advanced form of excursion theory (in the sense of Maisonneuve (1975)) can be exploited.

Inspired by this approach, we give a different decomposition of the $d$-dimensional isotropic $\alpha$-stable Lévy processes in terms of orthogonal coordinates. Accordingly we are able to develop a number of $n$-tuple laws for first entrance into a half-space bounded by an $R^{d-1}$ hyperplane, expanding on existing results of (Byczkowski et al., 2009; Tamura and Tanaka, 2008). This gives us the opportunity to numerically construct the law of first entry of the process into a slab of the form $(-1,1)\times R^{d-1}$ using a ‘walk-on-half-spaces’ Monte Carlo approach in the spirit of the ‘walk-on-spheres’ Monte Carlo method given in Kyprianou et al. (2018).