$\hat{I}\pm$-stable Lévy processes entering the half space or a slab

Abstract

Recent fluctuation identities for $\hat{I}\pm$-stable Lévy processes have decomposed paths using generalised spherical polar coordinates revealing an underlying Markov Additive Process (MAP) for which a more advanced form of excursion theory can be exploited. Inspired by this approach, we give a different decomposition of the $d$-dimensional isotropic $\hat{I}\pm$-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. We also numerically construct the law of first entry of the process into a slab of the form $(-1,1)\times {\mathbb{R}}^{d-1}$ using a walk-on-half-spaces Monte Carlo approach.

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