Numerical Computation of First-Passage Times of Increasing Levy Processes

TL;DR

This paper develops two numerical methods to compute the mean first-hitting time of increasing Lévy processes, facilitating practical calculations for applications involving inverse subordinators.

Contribution

It introduces and compares Bromwich integral and Post-Widder inversion methods for numerically inverting Laplace transforms of first-hitting times.

Findings

Two effective numerical methods are proposed for Laplace inversion.

Software implementation provided for practical computation.

Methods are demonstrated with illustrative examples.

Abstract

Let $\{D(s), s \geq 0\}$ be a non-decreasing L\'evy process. The first-hitting time process $\{E(t) t \geq 0\}$ (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 $\phi$ is the L\'evy 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.