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Applications of Borovkov's Renovation Theory to Non-Stationary Stochastic Recursive Sequences and Their Control

Part of: Stochastic systems and control Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Eitan Altman  and
Arie Hordijk
Eitan Altman*
Affiliation:
INRIA
Arie Hordijk*
Affiliation:
Leiden University
*
∗ Postal address: INRIA, 2004 Route des Lucioles, BP93, 06902 Sophia-Antipolis Cedex, France.
∗∗ Postal address: Department of Mathematics and Computer Science, Leiden University, P.O. Box 9512, 2300RA Leiden, The Netherlands.
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Abstract

We investigate in this paper the stability of non-stationary stochastic processes, arising typically in applications of control. The setting is known as stochastic recursive sequences, which allows us to construct on one probability space stochastic processes that correspond to different initial states and even different control policies. It does not require any Markovian assumptions. A natural criterion for stability for such processes is that the influence of the initial state disappears after some finite time; in other words, starting from different initial states, the process will couple after some finite time to the same limiting (not necessarily stationary nor ergodic) stochastic process. We investigate this as well as other types of coupling, and present conditions for them to occur uniformly in some class of control policies. We then use the coupling results to establish new theoretical aspects in the theory of non-Markovian control.

Keywords

COUPLINGRENOVATING EVENTS

MSC classification

Primary: 60G10: Stationary processes
Secondary: 93E20: Optimal stochastic control
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 2 , June 1997 , pp. 388 - 413
Copyright
Copyright © Applied Probability Trust 1997 

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