Loss rate for a general Lévy process with downward periodic barrier

Abstract

In this paper we consider a general Lévy process X reflected at a downward periodic barrier A t and a constant upper barrier K, giving a processWe find the expression for a loss rate defined by l K = E L K 1 and identify its asymptotics as K → ∞ when X has light-tailed jumps and E X 1 < 0.

On the Dynamics of Semimartingales with Two Reflecting Barriers

“…Proof. Equations (17) and (18) are immediate consequences of Theorem 1 and Proposition 5 (or Proposition 6).…”

“…Let 0 < ε < b. It follows from [17] that if we take X to be a Lévy process and the barrier T to be of the form…”

“…The main themes of earlier work are the identification of = EU(1) (under stationarity) (see [6], where b is fixed, and [17], where the upper barrier is dynamic), asymptotics of as R + b → ∞ under different assumptions on X (see, e.g. [1], [2], [6], and [17]), and the Markov transition kernel of V (if relevant) and the corresponding stationary distribution (see, e.g. [10]).…”

“…Markov modulated input (see [6]), dynamic barriers (see, e.g. [13] and [17]), and combinations of the aforementioned generalizations (see, e.g. [10]).…”

Local martingales with two reflecting barriers

“…As an appetizer we provide a few references together with short descriptions (this reference list is by no means complete) which contain treatments of different reflection-type problems under various assumptions about the input X: [1] reflected Levy processes, [2] reflected Levy processes, [3] reflected Levy processes with emphasis on the reflection mechanism itself, [6] reflected Levy processes with an emphasis on the case where the Levy measure is light-tailed, [8] reflection of a Markov modulated Brownian motion, [11] on fundamental issues concerning reflection, [12] reflection of Levy process with a functional, i.e. not constant, upper barrier, [13] discrete time reflection, [15] reflection of Levy process in a functional upper barrier, [16] discrete time reflection, and [21] discrete time reflection.…”