On $q$-scale functions of spectrally negative L\'evy processes

TL;DR

This paper derives series expansions for the $q$-scale functions of spectrally negative Lévy processes, including those with infinite jump activity, and investigates their smoothness properties, expanding the class of explicit examples available.

Contribution

It provides new series expansions and smoothness analysis for $q$-scale functions of spectrally negative Lévy processes with infinite jump activity, extending previous results.

Findings

Derived series expansions for $q$-scale functions

Identified explicit examples of $q$-scale functions

Analyzed smoothness properties of the functions

Abstract

We obtain series expansions of the $q$-scale functions of arbitrary spectrally negative L\'evy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit $q$-scale functions. Moreover, we study smoothness properties of the $q$-scale functions of spectrally negative L\'evy processes with infinite jump activity. This complements previous results of Chan et al. [7] for spectrally negative L\'evy processes with Gaussian component or bounded variation.