The theory of scale functions for spectrally negative Le vy processes

TL;DR

This review provides a comprehensive overview of the theory, applications, and numerical methods related to scale functions for spectrally negative Levy processes, highlighting recent developments and key ideas.

Contribution

It offers the first extensive overview of numerical methods for working with scale functions and synthesizes recent theoretical advances in the field.

Findings

Detailed account of scale function theory and applications

Introduction to numerical techniques for scale functions

Connections to excursion theory and semi-martingale calculus

Abstract

The purpose of this review article is to give an up to date account of the theory and application of scale functions for spectrally negative Levy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Levy processes, in particular a reasonable understanding of the Levy-Khintchine formula and its relationship to the Levy-Ito decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Le\'vy processes; Bertoin (1996), Sato (1999), Applebaum (2004), Kyprianou (2006) and Doney (2007).