Old and new examples of scale functions for spectrally negative Levy processes

TL;DR

This paper reviews the current state of scale functions for spectrally negative Levy processes, introduces a new method for generating families of these functions, and presents a novel family within the GTSC class with detailed analysis.

Contribution

It introduces a general method for creating new scale functions and specifically develops a new family within the GTSC class, expanding the theoretical framework.

Findings

New family of scale functions in the GTSC class

Analytical behavior of new scale functions compared to known cases

Method for generating scale functions from existing classes

Abstract

We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Levy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations.