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CHARACTERIZATIONS OF OPTIMAL POLICIES IN A GENERAL STOPPING PROBLEM AND STABILITY ESTIMATING

Published online by Cambridge University Press:  05 June 2014

Evgueni Gordienko  and
Andrey Novikov
Evgueni Gordienko
Affiliation:
Department of Mathematics, UAM-Iztapalapa, San Rafael Atlixco 186, col. Vicentina, C.P. 11320, Mexico City, Mexico E-mail: [email protected] and [email protected]
Andrey Novikov
Affiliation:
Department of Mathematics, UAM-Iztapalapa, San Rafael Atlixco 186, col. Vicentina, C.P. 11320, Mexico City, Mexico E-mail: [email protected] and [email protected]
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Abstract

We consider an optimal stopping problem for a general discrete-time process X1, X2, …, Xn, … on a common measurable space. Stopping at time n (n = 1, 2, …) yields a reward Rn(X1, …, Xn) ≥ 0, while if we do not stop, we pay cn(X1, …, Xn) ≥ 0 and keep observing the process. The problem is to characterize all the optimal stopping times τ, i.e., such that maximize the mean net gain:

$$E(R_\tau(X_1,\dots,X_\tau)-\sum_{n=1}^{\tau-1}c_n(X_1,\dots,X_n)).$$
We propose a new simple approach to stopping problems which allows to obtain not only sufficient, but also necessary conditions of optimality in some natural classes of (randomized) stopping rules.

In the particular case of Markov sequence X1, X2, … we estimate the stability of the optimal stopping problem under perturbations of transition probabilities.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 28 , Issue 3 , July 2014 , pp. 335 - 352
Copyright
Copyright © Cambridge University Press 2014 

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