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The Transformation Method for Continuous-Time Markov Decision Processes

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  • Alexey Piunovskiy

    (University of Liverpool)

  • Yi Zhang

    (University of Liverpool)

Abstract

In this paper, we show that a discounted continuous-time Markov decision process in Borel spaces with randomized history-dependent policies, arbitrarily unbounded transition rates and a non-negative reward rate is equivalent to a discrete-time Markov decision process. Based on a completely new proof, which does not involve Kolmogorov’s forward equation, it is shown that the value function for both models is given by the minimal non-negative solution to the same Bellman equation. A verifiable necessary and sufficient condition for the finiteness of this value function is given, which induces a new condition for the non-explosion of the underlying controlled process.

Suggested Citation

  • Alexey Piunovskiy & Yi Zhang, 2012. "The Transformation Method for Continuous-Time Markov Decision Processes," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 691-712, August.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:2:d:10.1007_s10957-012-0015-8
    DOI: 10.1007/s10957-012-0015-8
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    References listed on IDEAS

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    1. Richard F. Serfozo, 1979. "Technical Note—An Equivalence Between Continuous and Discrete Time Markov Decision Processes," Operations Research, INFORMS, vol. 27(3), pages 616-620, June.
    2. Xianping Guo & Alexei Piunovskiy, 2011. "Discounted Continuous-Time Markov Decision Processes with Constraints: Unbounded Transition and Loss Rates," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 105-132, February.
    3. Xianping Guo, 2007. "Continuous-Time Markov Decision Processes with Discounted Rewards: The Case of Polish Spaces," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 73-87, February.
    4. Alexey Piunovskiy & Yi Zhang, 2011. "Accuracy of fluid approximations to controlled birth-and-death processes: absorbing case," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(2), pages 159-187, April.
    5. Eugene A. Feinberg, 2004. "Continuous Time Discounted Jump Markov Decision Processes: A Discrete-Event Approach," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 492-524, August.
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    Cited by:

    1. F. M. Spieksma, 2016. "Kolmogorov forward equation and explosiveness in countable state Markov processes," Annals of Operations Research, Springer, vol. 241(1), pages 3-22, June.
    2. Ping Cao & Jingui Xie, 2016. "Optimal control of a multiclass queueing system when customers can change types," Queueing Systems: Theory and Applications, Springer, vol. 82(3), pages 285-313, April.
    3. Guo, Xianping & Zhang, Wenzhao, 2014. "Convergence of controlled models and finite-state approximation for discounted continuous-time Markov decision processes with constraints," European Journal of Operational Research, Elsevier, vol. 238(2), pages 486-496.

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