On $q$-scale functions of spectrally negative compound Poisson processes

TL;DR

This paper derives new explicit representations for the $q$-scale functions of spectrally negative compound Poisson processes with positive drift, enhancing understanding of their properties and smoothness.

Contribution

It introduces novel formulas for $q$-scale functions and their derivatives based on process characteristics, advancing the analytical tools for fluctuation theory.

Findings

New representation formulas for $q$-scale functions

Explicit expressions for derivatives and primitives

Enhanced understanding of smoothness properties

Abstract

Scale functions play a central role in the fluctuation theory of spectrally negative L\'evy processes. For spectrally negative compound Poisson processes with positive drift, a new representation of the $q$-scale functions in terms of the characteristics of the process is derived. Moreover, similar representations of the derivatives and the primitives of the $q$-scale functions are presented. The obtained formulae for the derivatives allow for a complete exposure of the smoothness properties of the considered $q$-scale functions. Some explicit examples of $q$-scale functions are given for illustration.