On approximative solutions of multistopping problems

Abstract

In this paper, we consider multistopping problems for finite discrete time sequences X₁, …, X_n. m-stops are allowed and the aim is to maximize the expected value of the best of these m stops. The random variables are neither assumed to be independent not to be identically distributed. The basic assumption is convergence of a related imbedded point process to a continuous time Poisson process in the plane, which serves as a limiting model for the stopping problem. The optimal m-stopping curves for this limiting model are determined by differential equations of first order. A general approximation result is established which ensures convergence of the finite discrete time m-stopping problem to that in the limit model. This allows the construction of approximative solutions of the discrete time m-stopping problem. In detail, the case of i.i.d. sequences with discount and observation costs is discussed and explicit results are obtained.