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

Author

Listed:
  • Anatoliy Swishchuk

    (Department of Mathematics and Statistics, Faculty of Science, University of Calgary, Calgary, AB T2N 1N4, Canada
    These authors contributed equally to this work.)

  • Nikolaos Limnios

    (Applied Mathematics Lab, Université de Technologie de Compiègne, Sorbonne University Alliance, 60203 Compiègne, France
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Anatoliy Swishchuk & Nikolaos Limnios, 2021. "Controlled Discrete-Time Semi-Markov Random Evolutions and Their Applications," Mathematics, MDPI, vol. 9(2), pages 1-26, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:2:p:158-:d:479781
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    References listed on IDEAS

    as
    1. Cohen, Joel E., 1979. "Random evolutions in discrete and continuous time," Stochastic Processes and their Applications, Elsevier, vol. 9(3), pages 245-251, December.
    2. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    3. Anatoliy Swishchuk & M. Shafiqul Islam, 2010. "Diffusion Approximations of the Geometric Markov Renewal Processes and Option Price Formulas," International Journal of Stochastic Analysis, Hindawi, vol. 2010, pages 1-21, December.
    4. Borkar, V. S., 2003. "Dynamic programming for ergodic control with partial observations," Stochastic Processes and their Applications, Elsevier, vol. 103(2), pages 293-310, February.
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