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Loss Rates for Lévy Processes with Two Reflecting Barriers

Department of Theoretical Statistics, Department of Mathematical Sciences, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark
Codan Insurance, Gammel Kongevej 60, 1790 København V, Denmark
asmus{at}imf.au.dk, http://home.imf.au.dk/asmus
mpi{at}codan.dk

We consider a Lévy process that is reflected at 0 and at K < 0. The reflected process is obtained by adding the difference between the local time at 0 and the local time at K to the sum of the feeding Lévy process and an initial condition. We define the loss rate to be the expectation of the local time at K at time 1 under stationary conditions. The main result of the paper is the identification of the loss rate in terms of the stationary measure of the reflected process and the characteristic triplet of the Lévy process. We also derive asymptotics of the loss rate as K when the drift of the feeding process is negative and the Lévy measure is light tailed. Finally, we extend the results for Lévy processes to hold for Markov-modulated Lévy processes.

Key Words: Lévy process; reflection; Skorokhod problem; local time; loss rate; light tail; martingale; Lundberg equation; Markov-modulated Lévy process; Cramér-Lundberg approximation; asymptotics
History: Received: November 7, 2005; revision received: February 20, 2006;