scholarly journals Discrete-Time Semi-Markov Random Evolutions in Asymptotic Reduced Random Media with Applications

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 963
Author(s):  
Nikolaos Limnios ◽  
Anatoliy Swishchuk
Keyword(s):  
Discrete Time ◽  
Diffusion Approximation ◽  
Random Media ◽  
Asymptotic Approximations ◽  
Asymptotic Results ◽  
Additive Functionals ◽  
Markov Random ◽  
Random Evolutions ◽  
Convergence Method ◽  
Weak Convergence Method

This paper deals with discrete-time semi-Markov random evolutions (DTSMRE) in reduced random media. The reduction can be done for ergodic and non ergodic media. Asymptotic approximations of random evolutions living in reducible random media (random environment) are obtained. Namely, averaging, diffusion approximation and normal deviation or diffusion approximation with equilibrium by martingale weak convergence method are obtained. Applications of the above results to the additive functionals and dynamical systems in discrete-time produce the above tree types of asymptotic results.

scholarly journals Discrete-Time Semi-Markov Random Evolutions and their Applications

2013 ◽  
Vol 45 (1) ◽  
pp. 214-240 ◽  
Author(s):  
Nikolaos Limnios ◽  
Anatoliy Swishchuk
Keyword(s):  
Discrete Time ◽  
Diffusion Approximation ◽  
Asymptotic Properties ◽  
Rates Of Convergence ◽  
Bellman Equations ◽  
Additive Functionals ◽  
Markov Random ◽  
Random Evolutions ◽  
Convergence Method ◽  
Hamilton Jacobi Bellman

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.

scholarly journals Discrete-Time Semi-Markov Random Evolutions and their Applications

2013 ◽  
Vol 45 (01) ◽  
pp. 214-240 ◽  
Author(s):  
Nikolaos Limnios ◽  
Anatoliy Swishchuk
Keyword(s):  
Discrete Time ◽  
Diffusion Approximation ◽  
Asymptotic Properties ◽  
Rates Of Convergence ◽  
Bellman Equations ◽  
Additive Functionals ◽  
Markov Random ◽  
Random Evolutions ◽  
Convergence Method ◽  
Hamilton Jacobi Bellman

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.

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

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 158
Author(s):  
Anatoliy Swishchuk ◽  
Nikolaos Limnios
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Diffusion Approximation ◽  
Limit Theorems ◽  
Stochastic Systems ◽  
Renewal Processes ◽  
Additive Functionals ◽  
Markov Random ◽  
Markov Renewal Processes ◽  
Random Evolutions

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.

scholarly journals Asymptotic results for the first and second moments and numerical computations in discrete-time bulk-renewal process

2019 ◽  
Vol 29 (1) ◽  
pp. 135-144
Author(s):  
James Kim ◽  
Mohan Chaudhry ◽  
Abdalla Mansur
Keyword(s):  
Discrete Time ◽  
Renewal Process ◽  
Generating Functions ◽  
Continuous Time ◽  
Constant Term ◽  
Numerical Results ◽  
Asymptotic Results ◽  
Numerical Computations ◽  
Second Moments ◽  
Simplified Solution

This paper introduces a simplified solution to determine the asymptotic results for the renewal density. It also offers the asymptotic results for the first and second moments of the number of renewals for the discrete-time bulk-renewal process. The methodology adopted makes this study distinguishable compared to those previously published where the constant term in the second moment is generated. In similar studies published in the literature, the constant term is either missing or not clear how it was obtained. The problem was partially solved in the study by Chaudhry and Fisher where they provided a asymptotic results for the non-bulk renewal density and for both the first and second moments using the generating functions. The objective of this work is to extend their results to the bulk-renewal process in discrete-time, including some numerical results, give an elegant derivation of the asymptotic results and derive continuous-time results as a limit of the discrete-time results.

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