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Keywords = controlled geometric Markov renewal processes

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26 pages, 350 KiB  
Article
Controlled Discrete-Time Semi-Markov Random Evolutions and Their Applications
by Anatoliy Swishchuk and Nikolaos Limnios
Mathematics 2021, 9(2), 158; https://doi.org/10.3390/math9020158 - 13 Jan 2021
Cited by 1 | Viewed by 1885
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue New Trends in Random Evolutions and Their Applications)
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