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Article

Controlled Discrete-Time Semi-Markov Random Evolutions and Their Applications

by 1,† and 2,*,†
1
Department of Mathematics and Statistics, Faculty of Science, University of Calgary, Calgary, AB T2N 1N4, Canada
2
Applied Mathematics Lab, Université de Technologie de Compiègne, Sorbonne University Alliance, 60203 Compiègne, France
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2021, 9(2), 158; https://doi.org/10.3390/math9020158
Received: 9 December 2020 / Revised: 4 January 2021 / Accepted: 11 January 2021 / Published: 13 January 2021
(This article belongs to the Special Issue New Trends in Random Evolutions and Their Applications)
In this paper, we introduced controlled discrete-time semi-Markov random evolutions. These processes are random evolutions of discrete-time semi-Markov processes where we consider a control. applied to the values of random evolution. The main results concern time-rescaled weak convergence limit theorems in a Banach space of the above stochastic systems as averaging and diffusion approximation. The applications are given to the controlled additive functionals, controlled geometric Markov renewal processes, and controlled dynamical systems. We provide dynamical principles for discrete-time dynamical systems such as controlled additive functionals and controlled geometric Markov renewal processes. We also produce dynamic programming equations (Hamilton–Jacobi–Bellman equations) for the limiting processes in diffusion approximation such as controlled additive functionals, controlled geometric Markov renewal processes and controlled dynamical systems. As an example, we consider the solution of portfolio optimization problem by Merton for the limiting controlled geometric Markov renewal processes in diffusion approximation scheme. The rates of convergence in the limit theorems are also presented. View Full-Text
Keywords: semi-Markov chain; controlled discrete-time semi-Markov random evolutions; averaging; diffusion approximation; diffusion approximation with equilibrium; rates of convergence; controlled additive functional; controlled dynamical systems; controlled geometric Markov renewal processes; HJB equation; Merton problem; Banach space semi-Markov chain; controlled discrete-time semi-Markov random evolutions; averaging; diffusion approximation; diffusion approximation with equilibrium; rates of convergence; controlled additive functional; controlled dynamical systems; controlled geometric Markov renewal processes; HJB equation; Merton problem; Banach space
MDPI and ACS Style

Swishchuk, A.; Limnios, N. Controlled Discrete-Time Semi-Markov Random Evolutions and Their Applications. Mathematics 2021, 9, 158. https://doi.org/10.3390/math9020158

AMA Style

Swishchuk A, Limnios N. Controlled Discrete-Time Semi-Markov Random Evolutions and Their Applications. Mathematics. 2021; 9(2):158. https://doi.org/10.3390/math9020158

Chicago/Turabian Style

Swishchuk, Anatoliy, and Nikolaos Limnios. 2021. "Controlled Discrete-Time Semi-Markov Random Evolutions and Their Applications" Mathematics 9, no. 2: 158. https://doi.org/10.3390/math9020158

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