Loss Rates for Lévy Processes with Two Reflecting Barriers

Let {Xt} be a Lévy process which is reflected at 0 andK > 0. The reflected process {V K t } is constructed as V K t = V K 0 +Xt+Lt −Lt where {Lt} and {Lt } are the local times at 0 and K, respectively. We consider the loss rate ` , defined by ` = EπKL1 , where EπK is the expectation under the stationary measure πK . The main result of the paper is the identification of ` in terms of πK and the characteristic triplet of {Xt}. We also derive asymptotics of ` as K → ∞ when EX1 < 0 and the Lévy measure of {Xt} is light-tailed.