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.
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M. Asghari, N., den Iseger, P. & Mandjes, M. Numerical Techniques in Lévy Fluctuation Theory. Methodol Comput Appl Probab 16, 31–52 (2014). https://doi.org/10.1007/s11009-012-9296-5
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DOI: https://doi.org/10.1007/s11009-012-9296-5