, Volume 16, Issue 1, pp 31-52,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 24 Aug 2012

Numerical Techniques in Lévy Fluctuation Theory

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.