Loss rate for a general L\'evy process with downward periodic barrier

TL;DR

This paper derives an explicit expression for the loss rate of a general Lévy process reflected at a downward periodic barrier and analyzes its asymptotic behavior as the upper barrier tends to infinity, under specific conditions.

Contribution

It provides a novel formula for the loss rate of a Lévy process with a periodic downward barrier and explores its asymptotic properties for large barriers.

Findings

Explicit expression for the loss rate $l^K$.

Asymptotic behavior of $l^K$ as $K o iginfty$.

Results hold for processes with light-tailed jumps and negative mean.

Abstract

In this paper we consider a general L\'{e}vy process $X$ reflected at downward periodic barrier $A_t$ and constant upper barrier $K$ giving a process $V^K_t=X_t+L^A_t-L^K_t$. We find the expression for a loss rate defined by $l^K=\mathbb{E} L^K_1$ and identify its asymptotics as $K\to\infty$ when $X$ has light-tailed jumps and $\mathbb{E}X_1<0$.