Lévy Processes with Two-sided ReflectionL. N. Andersen, S. Asmussen, P. W. Glynn, and M. Pihlsgård Lévy Matters, Springer Lecture Notes in Mathematics (Editors: O.E. Barndorff-Nielsen, J. Bertoin, J. Jacod, C. Klüppelberg) pp.67-182. Let X be a Lévy process and V the reflection at boundaries 0 and b > 0. A number of properties of V are studied, with particular emphasis on the behaviour at the upper boundary b. The process V can be represented as solution of a Skorokhod problem V(t) = V(0)+X(t)+L(t)-U(t) where L, U are the local times (regulators) at the lower and upper barriers. Explicit forms of V in terms of X are surveyed as well as more pragmatic approaches to the construction of V, and the stationary distribution p is characterised in terms of a two-barrier first passage problem. A key quantity in applications is the loss rate l^b at b, defined as E_\pi U(1). Various forms of l^b and various derivations are presented, and the asymptotics as b tends to infinity is exhibited in both the light-tailed and the heavy-tailed regime. The drift zero case EX(1) = 0 plays a particular role, with Brownian or stable functional limits being a key tool. Further topics include studies of the first hitting time of b, central limit theorems and large deviations results for U, and a number of explicit calculations for Lévy processes where the jump part is compound Poisson with phase-type jumps. |