Control of a finite dam with a Lévy input

Abstract

Bae et al. [7] consider the problem of optimal control of a finite dam using P λ,τ policies, assuming that the input process is a compound Poisson process with a negative drift. Lam and Lou [9] treat the case where the input is a Wiener process with a reflecting boundary at its infimum, with drift term μ ≥ 0, using the long-run average and total discounted cost criteria. Attia [5] obtains results similar to those of Lam and Lou, through simpler and more direct methods. Zuckermann [12] considers P λ,0 policies when the input process is a Wiener process with drift term μ ≥ 0. The techniques used by the above mentioned authors involve solving systems of differential or integral equations. In this paper we use the theory and methods of scale functions of Levy processes to unify and extend the results of these authors.