On moments of downward passage times for spectrally negative L\'evy processes

TL;DR

This paper investigates the conditions under which moments of first downward passage times exist for spectrally negative Lévy processes, revealing how the process's drift influences the finiteness of these moments.

Contribution

It generalizes existing results by establishing moment existence criteria based on the Lévy process's drift and jump measure properties.

Findings

Moments exist if the process drifts to +∞ and the jump measure's higher moments are finite.

All moments exist if the process drifts to -∞.

No integer moments exist for oscillating Lévy processes.

Abstract

The existence of moments of first downward passage times of a spectrally negative L\'evy process is governed by the general dynamics of the L\'evy process, i.e. whether the L\'evy process is drifting to $+\infty$, $-\infty$ or oscillates. Whenever the L\'evy process drifts to $+\infty$, we prove that the $\kappa$-th moment of the first passage time (conditioned to be finite) exists if and only if the $(\kappa+1)$-th moment of the L\'evy jump measure exists. This generalises a result shown earlier by Delbaen for Cram\'er-Lundberg risk processes \cite{Delbaen1990}. Whenever the L\'evy process drifts to $-\infty$, we prove that all moments of the first passage time exist, while for an oscillating L\'evy process we derive conditions for non-existence of the moments and in particular we show that no integer moments exist.