Numerical computation of first-passage times of increasing Lévy processes
Files
First author draft
Date
2010-12-01
Authors
Veillette, Mark
Taqqu, Murad S.
Version
First author draft
OA Version
Citation
Mark Veillette, Murad S Taqqu. 2010. "Numerical Computation of First-Passage Times of Increasing Levy Processes." METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY, Volume 12, Issue 4, pp. 695 - 729 (35). https://doi.org/10.1007/s11009-009-9158-y
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)=𝔼𝐸(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 𝑈˜(𝜆)=(𝜆𝜙(𝜆))−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.