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

TL;DR

This paper develops a differential equation approach to compute joint moments of first-passage times for increasing Lévy processes, addressing the non-Markovian nature of the inverse process.

Contribution

It derives a PDE for the Laplace transform of the joint tail distribution of inverse Lévy subordinator first-passage times, enabling calculation of all joint moments.

Findings

Derived a PDE with unique solutions for the Laplace transform of joint tail distributions.

Provided a method to compute all joint moments of the inverse process.

Addressed the non-Markovian and non-stationary challenges of the inverse process.

Abstract

Let $\{D(s), s \geq 0 \}$ be a L\'evy 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 $E(t) = \inf \{s: D(s) > t \}$, $t \geq 0$. 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 $E$.