More on hypergeometric Levy processes

TL;DR

This paper extends the parameter range of hypergeometric Lévy processes, providing new explicit calculations of their Lévy measures and Wiener--Hopf factor potentials, thereby deepening understanding of their fluctuation properties.

Contribution

It broadens the parameter scope of hypergeometric Lévy processes and derives explicit forms for their Lévy and potential measures.

Findings

Extended the parameter range for hypergeometric Lévy processes.

Calculated explicit Lévy measures for the extended processes.

Derived potential measures of the Wiener--Hopf factors.

Abstract

Kuznetsov et al. (2011) and Kuznetsov and Pardo (2013) introduced the family of Hypergeometric L\'evy processes. They appear naturally in the study of fluctuations of stable processes when one analyses stable processes through the theory of positive self-similar Markov processes. Hypergeometric L\'evy processes are defined through their characteristic exponent, which, as a complex-valued function, has four independent parameters. Kyprianou et al. (2014) showed that the definition of a Hypergeometric L\'evy process could be taken to include a greater range of the aforesaid parameters than originally specified. In this short article, we push the parameter range even further. In particular, we calculate the underlying L\'evy measure and potential measures of the Wiener--Hopf factors.