The time at which a L\'evy process creeps

TL;DR

This paper investigates the properties of the renewal function associated with the ladder process of a Lévy process that creeps, revealing its absolute continuity and differentiability, and connects these findings to existing fluctuation identities.

Contribution

It establishes the absolute continuity and differentiability of the renewal function for creeping Lévy processes and introduces a Laplace transform identity generalizing the second factorization identity.

Findings

Renewal function is absolutely continuous for creeping Lévy processes.

Left derivative of the renewal function relates to the creeping distribution.

Generalizes the second factorization identity for Lévy processes.

Abstract

We show that if a L\'evy process creeps then, as a function of $u$, the renewal function $V(t,u)$ of the bivariate ascending ladder process $(L^{-1},H)$ is absolutely continuous on $[0,\infty)$ and left differentiable on $(0,\infty)$, and the left derivative at $u$ is proportional to the (improper) distribution function of the time at which the process creeps over level $u$, where the constant of proportionality is $\rmd_H^{-1}$, the reciprocal of the (positive) drift of $H$. This yields the (missing) term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As an application, we derive a Laplace transform identity which generalises the second factorization identity. We also relate Doney and Kyprianou's extension of Vigon's \'equation amicale invers\'ee to creeping. Some results concerning the ladder process of $X$, including the second factorization identity, continue to hold for a general bivariate subordinator, and are given in this generality.