Explicit identities for L\'evy processes associated to symmetric stable processes

TL;DR

This paper introduces hypergeometric-stable Lévy processes derived from symmetric stable processes, providing their Lévy measure, Wiener-Hopf factorization, characteristic exponent, and exit problem solutions, highlighting their mathematical properties.

Contribution

It presents a new class of Lévy processes linked to symmetric stable processes, with explicit formulas involving hypergeometric functions, expanding the understanding of Lévy process transformations.

Findings

Explicit Lévy measure for hypergeometric-stable processes

Wiener-Hopf factorization derived for the new class

Solutions to exit problems related to these processes

Abstract

In this paper we introduce a new class of L\'evy processes which we call hypergeometric-stable L\'evy processes, because they are obtained from symmetric stable processes through several transformations and where the Gauss hypergeometric function plays an essential role. We characterize the L\'evy measure of this class and obtain several useful properties such as the Wiener Hopf factorization, the characteristic exponent and some associated exit problems.