Deep factorisation of the stable process II; potentials and applications

TL;DR

This paper introduces a novel approach to deep factorisation of stable processes using potential theory and the Lamperti-Kiu transform, providing explicit formulas and new identities for stable process behaviors.

Contribution

It presents a new, independent factorisation method that yields explicit potential densities and identities for stable processes, advancing fluctuation theory and potential analysis.

Findings

Explicit potential densities for ladder MAPs of Lamperti-stable MAP

Identities for points of closest and furthest reach of stable processes

Explicit stationary distribution for multiplicatively reflected stable processes

Abstract

Here we propose a different perspective of the deep factorisation in Kyprianou (2015) based on determining potentials. Indeed, we factorise the inverse of the MAP-exponent associated to a stable process via the Lamperti-Kiu transform. Here our factorisation is completely independent from the derivation in Kyprianou (2015) , moreover there is no clear way to invert the factors in Kyprianou (2015) to derive our results. Our method gives direct access to the potential densities of the ascending and descending ladder MAP of the Lamperti-stable MAP in closed form. In the spirit of the interplay between the classical Wiener-Hopf factorisation and fluctuation theory of the underlying Levy process, our analysis will produce a collection of of new results for stable processes. We give an identity for the point of closest reach to the origin for a stable process with index $\alpha\in (0,1)$ as well as and identity for the point of furthest reach before absorption at the origin for a stable process with index $\alpha\in (1,2)$. Moreover, we show how the deep factorisation allows us to compute explicitly the stationary distribution of stable processes multiplicatively reflected in such a way that it remains in the strip [-1,1].