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 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.
Highlights
Random evolutions were introduced over 40 years ago, see, e.g., in [1,2] and for its asymptotic theory in [3,4,5,6] and references therein
Compared with our previous work [20], where we studied random evolutions of semi-Markov chains, here we considered a control on the random evolution, which we call controlled discrete-time semi-Markov random evolution (CDTSMRE) in a Banach space, and we presented timerescaled convergence theorems
We present here the rates of convergence of CDTSMRE in the averaging, diffusion approximation and diffusion approximation with equilibrium schemes and, as corollaries, we give the rates of convergence for controlled additive functionals (CAF) and CGMRP in the corresponding limits
Summary
Random evolutions were introduced over 40 years ago, see, e.g., in [1,2] and for its asymptotic theory in [3,4,5,6] and references therein. Compared with our previous work [20], where we studied random evolutions of semi-Markov chains, here we considered a control on the random evolution, which we call controlled discrete-time semi-Markov random evolution (CDTSMRE) in a Banach space, and we presented timerescaled convergence theorems. The last section concludes the paper and indicates some future works
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