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Discrete-Time Semi-Markov Random Evolutions and their Applications

Part of: Special processes Probability theory on algebraic and topological structures Limit theorems

Published online by Cambridge University Press:  04 January 2016

Nikolaos Limnios  and
Anatoliy Swishchuk
Nikolaos Limnios*
Affiliation:
Université de Technologie de Compiègne
Anatoliy Swishchuk*
Affiliation:
University of Calgary
*
∗ Postal address: Laboratoire de Mathématiques Appliquées, Université de Technologie de Compiègne, BP 20529, 60205 Compiègne Cedex, France. Email address: nikolaos.limnios@utc.fr
∗∗ Postal address: Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada.
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Abstract

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In this paper we introduce discrete-time semi-Markov random evolutions (DTSMREs) and study asymptotic properties, namely, averaging, diffusion approximation, and diffusion approximation with equilibrium by the martingale weak convergence method. The controlled DTSMREs are introduced and Hamilton–Jacobi–Bellman equations are derived for them. The applications here concern the additive functionals (AFs), geometric Markov renewal chains (GMRCs), and dynamical systems (DSs) in discrete time. The rates of convergence in the limit theorems for DTSMREs and AFs, GMRCs, and DSs are also presented.

Keywords

Semi-Markov chainrandom evolutionaveragingdiffusion approximationdiffusion approximation with equilibriumrates of convergenceadditive functionalgeometric Markov renewal processdynamical systemcontrolled random evolutioncontrolled additive functional and geometric Markov renewal chainoptimal controlHamilton–Jacobi–Bellman equation

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes 60B10: Convergence of probability measures 90C40: Markov and semi-Markov decision processes 60F17: Functional limit theorems; invariance principles
Secondary: 60K37: Processes in random environments
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 1 , March 2013 , pp. 214 - 240
Copyright
© Applied Probability Trust 

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