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 Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 1, Pages 3–26 (Mi tvp156)

On optimality in probability and almost surely for processes with a communication property. I. The discrete time case

T. A. Belkinaa, V. I. Rotar'ab

a Central Economics and Mathematics Institute, RAS
b San Diego State University

Abstract: We establish conditions under which the strategy minimizing the expected value of a cost functional has a much stronger property; namely, it minimizes the random cost functional itself for all realizations of the controlled process belonging to a set, the probability of which is close to one for large time horizons. The main difference of the conditions mentioned from those obtained earlier is that the former do not deal with value function properties but concern a possibility of transition of the controlled process from one state to another in a time with a finite mean. It makes the verification of these conditions in a number of situations of the general form much easier. The first part of the paper concerns processes in discrete time, and second part will be devoted to processes in continuous time.

Keywords: controlled processes, controlled Markov chains, asymptotic in probability.

DOI: https://doi.org/10.4213/tvp156

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English version:
Theory of Probability and its Applications, 2006, 50:1, 16–33

Bibliographic databases:

Citation: T. A. Belkina, V. I. Rotar', “On optimality in probability and almost surely for processes with a communication property. I. The discrete time case”, Teor. Veroyatnost. i Primenen., 50:1 (2005), 3–26; Theory Probab. Appl., 50:1 (2006), 16–33

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp156
• https://doi.org/10.4213/tvp156
• http://mi.mathnet.ru/eng/tvp/v50/i1/p3

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This publication is cited in the following articles:
1. T. A. Belkina, M. S. Levochkina, “Stochastic optimality in the problem of a linear controller perturbed by a sequence of dependent random variables”, Discrete Math. Appl., 16:2 (2006), 135–153
2. Zaslavski A.J., “Existence and structure of solutions for an infinite horizon optimal control problem arising in economic dynamics”, Adv. Differential Equations, 14:5-6 (2009), 477–496
3. Zaslavski A.J., “A Turnpike Property of Approximate Solutions of An Optimal Control Problem Arising in Economic Dynamics”, Dynam Systems Appl, 20:2–3 (2011), 395–421
4. T. A. Belkina, E. S. Palamarchuk, “On stochastic optimality for a linear controller with attenuating disturbances”, Autom. Remote Control, 74:4 (2013), 628–641
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