On approximative solutions of multistopping problems

TL;DR

This paper investigates multistopping problems in finite discrete time sequences, establishing a limiting continuous model via Poisson processes, and provides approximation methods for optimal stopping strategies, including explicit solutions for i.i.d. cases.

Contribution

It introduces a limiting Poisson process model for multistopping problems and develops approximation techniques for finite sequences, with explicit results for i.i.d. sequences with costs.

Findings

Convergence of discrete multistopping problems to a continuous Poisson model.

Explicit solutions for i.i.d. sequences with costs.

Approximation methods for finite discrete problems.

Abstract

In this paper, we consider multistopping problems for finite discrete time sequences $X_1,...,X_n$. $m$-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.