Using Differential Equations to Obtain Joint Moments of First-Passage Times of Increasing Levy Processes

Mark S. Veillette; Murad S. Taqqu

paper

Using Differential Equations to Obtain Joint Moments of First-Passage Times of Increasing Levy Processes

arXiv:0906.5083

Abstract

Let ${D (s), s \geq 0}$ be a Lévy subordinator, that is, a non-decreasing process with stationary and independent increments and suppose that $D (0) = 0$ . We study the first-hitting time of the process $D$ , namely, the process $E (t) = in f {s : D (s) > t}$ , $t \geq 0$ . The process $E$ is, in general, non-Markovian with non-stationary and non-independent increments. We derive a partial differential equation for the Laplace transform of the $n$ -time tail distribution function $P [E (t_{1}) > s_{1}, ..., E (t_{n}) > s_{n}]$ , and show that this PDE has a unique solution given natural boundary conditions. This PDE can be used to derive all $n$ -time moments of the process $E$ .

13 pages, one figure