Excursions of a spectrally negative L\'evy process from a two-point set

TL;DR

This paper derives explicit formulas for the behavior of spectrally negative Lévy processes when they exit a two-point set, including resolvents and excursion measures, with applications to last visit times.

Contribution

It provides new explicit expressions for resolvents and excursion measures of spectrally negative Lévy processes from a two-point set, extending fluctuation theory results.

Findings

Explicit resolvent formulas for the process killed on hitting two points.

Laplace transforms of excursion entrance laws are derived.

Application to last visit times between points in the process.

Abstract

Let $a\in (0,\infty)$. For a spectrally negative L\'evy process $X$ with infinite variation paths the resolvent of the process killed on hitting the two-point set $V=\{-a,a\}$ is identified. When further $X$ has no diffusion component the Laplace transforms of the entrance laws of the excursion measures of $X$ from $V$ are determined. This is then applied to establishing the Laplace transform of the amount of time that elapses between the last visit of $X$ to a given point $x$, before hitting some other point $y>x$, and the hitting time of $y$. All the expressions are explicit and tractable in the standard fluctuation quantities associated to $X$.