Methodology and Computing in Applied Probability

, Volume 16, Issue 1, pp 31–52

Numerical Techniques in Lévy Fluctuation Theory

Authors

  • Naser M. Asghari
    • Korteweg-de Vries Institute for MathematicsUniversity of Amsterdam
  • Peter den Iseger
    • ABN-Amro
    • Korteweg-de Vries Institute for MathematicsUniversity of Amsterdam
    • EurandomEindhoven University of Technology
    • CWI
Open AccessArticle

DOI: 10.1007/s11009-012-9296-5

Cite this article as:
M. Asghari, N., den Iseger, P. & Mandjes, M. Methodol Comput Appl Probab (2014) 16: 31. doi:10.1007/s11009-012-9296-5

Abstract

This paper presents a framework for numerical computations in fluctuation theory for Lévy processes. More specifically, with \(\bar X_t:= \sup_{0\le s\le t} X_s\) denoting the running maximum of the Lévy process X t , the aim is to evaluate \({\mathbb P}(\bar X_t \le x)\) for t,x > 0. We do so by approximating the Lévy process under consideration by another Lévy process for which the double transform \({\mathbb E} e^{-\alpha \bar X_{\tau(q)}}\) is known, with τ(q) an exponentially distributed random variable with mean 1/q; then we use a fast and highly accurate Laplace inversion technique (of almost machine precision) to obtain the distribution of \(\bar X_t\). A broad range of examples illustrates the attractive features of our approach.

Keywords

Lévy processes Fluctuation theory Wiener–Hopf Phase-type distributions Mathematical finance

AMS 2000 Subject Classifications

60G51 65T99

Copyright information

© The Author(s) 2012