On $q$-scale functions of spectrally negative Lévy processes

Abstract

We obtain series expansions of the $q$-scale functions of arbitrary spectrally negative Lévy 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 Lévy processes with infinite jump activity. This complements previous results of Chan et al. [7] for spectrally negative Lévy processes with Gaussian component or bounded variation.