Article

Methodology and Computing in Applied Probability

, Volume 16, Issue 1, pp 31-52

Open Access This content is freely available online to anyone, anywhere at any time.

Numerical Techniques in Lévy Fluctuation Theory

  • Naser M. AsghariAffiliated withKorteweg-de Vries Institute for Mathematics, University of Amsterdam
  • , Peter den IsegerAffiliated withABN-Amro
  • , Michael MandjesAffiliated withKorteweg-de Vries Institute for Mathematics, University of AmsterdamEurandom, Eindhoven University of TechnologyCWI Email author 

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