Numerical Techniques in Lévy Fluctuation Theory

Naser M. Asghari (), Peter Iseger () and Michael Mandjes ()
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Naser M. Asghari: University of Amsterdam
Peter Iseger: ABN-Amro
Michael Mandjes: University of Amsterdam

Methodology and Computing in Applied Probability, 2014, vol. 16, issue 1, 31-52

Abstract: 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; 60G51; 65T99 (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s11009-012-9296-5

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