Numerical Computation of First-Passage Times of Increasing Lévy Processes

Mark Veillette () and Murad S. Taqqu ()
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Mark Veillette: Boston University
Murad S. Taqqu: Boston University

Methodology and Computing in Applied Probability, 2010, vol. 12, issue 4, 695-729

Abstract: Abstract Let {D(s), s ≥ 0} be a non-decreasing Lévy process. The first-hitting time process {E(t), t ≥ 0} (which is sometimes referred to as an inverse subordinator) defined by $E(t) = \inf \{s: D(s) > t \}$ is a process which has arisen in many applications. Of particular interest is the mean first-hitting time $U(t)=\mathbb{E}E(t)$ . This function characterizes all finite-dimensional distributions of the process E. The function U can be calculated by inverting the Laplace transform of the function $\widetilde{U}(\lambda) = (\lambda \phi(\lambda))^{-1}$ , where ϕ is the Lévy exponent of the subordinator D. In this paper, we give two methods for computing numerically the inverse of this Laplace transform. The first is based on the Bromwich integral and the second is based on the Post-Widder inversion formula. The software written to support this work is available from the authors and we illustrate its use at the end of the paper.

Keywords: Lévy subordinators; First-hitting times; Post-Widder method; Numerical inversion of transforms; Anomalous diffusion; Jump processes; Primary 60G40, 60G51; Secondary 60J75, 60E07 (search for similar items in EconPapers)
Date: 2010
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s11009-009-9158-y

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