On the infimum attained by a reflected L\'evy process

TL;DR

This paper characterizes the distribution of the minimum of a reflected Lévy process over a time interval and explores its asymptotic behavior under different tail conditions, providing explicit formulas and asymptotic results.

Contribution

It provides an explicit Laplace transform characterization of the minimum distribution for spectrally one-sided Lévy processes and analyzes its asymptotics in heavy-tailed and light-tailed cases.

Findings

Explicit Laplace transform for the minimum distribution.

Asymptotic behavior of tail probabilities for large thresholds.

Distinction between heavy-tailed and light-tailed scenarios.

Abstract

This paper considers a L\'evy-driven queue (i.e., a L\'evy process reflected at 0), and focuses on the distribution of $M(t)$, that is, the minimal value attained in an interval of length $t$ (where it is assumed that the queue is in stationarity at the beginning of the interval). The first contribution is an explicit characterization of this distribution, in terms of Laplace transforms, for spectrally one-sided L\'evy processes (i.e., either only positive jumps or only negative jumps). The second contribution concerns the asymptotics of $\prob{M(T_u)> u}$ (for different classes of functions $T_u$ and $u$ large); here we have to distinguish between heavy-tailed and light-tailed scenarios.