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The Theory of Scale Functions for Spectrally Negative Lévy Processes

  • Alexey Kuznetsov
  • Andreas E. Kyprianou
  • Victor Rivero
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2061)

Abstract

The purpose of this review article is to give an up to date account of the theory and applications of scale functions for spectrally negative Lévy 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 Lévy processes, in particular a reasonable understanding of the Lévy–Khintchine formula and its relationship to the Lévy–Itô 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 Lévy processes; (Bertoin, Lévy Processes (1996); Sato, Lévy Processes and Infinitely Divisible Distributions (1999); Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes and Their Applications (2006); Doney, Fluctuation Theory for Lévy Processes (2007)), Applebaum Lévy Processes and Stochastic Calculus (2009).

Keywords

Applied probability Excursion theory First passage problem Fluctuation theory Laplace transform Scale functions Spectrally negative Lévy processes 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alexey Kuznetsov
    • 1
  • Andreas E. Kyprianou
    • 2
  • Victor Rivero
    • 3
  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK
  3. 3.Centro de Investigación en Matemáticas A.C.GuanajuatoGto. Mexico

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