Loss Rates for Lévy Processes with Two Reflecting Barriers

Abstract

Let {X t } be a Lévy process which is reflected at 0 and K > 0. The reflected process {V K t } is constructed as V K t = V K 0 + X t + L 0 t − L K t where {L 0 t } and {L K t } are the local times at 0 and K, respectively. We consider the loss rate K , defined by K = E π K L K 1 , where E π K is the expectation under the stationary measure π K. The main result of the paper is the identification of K in terms of π K and the characteristic triplet of {X t }. We also derive asymptotics of K as K → ∞ when EX 1 < 0 and the Lévy measure of {X t } is light-tailed.

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