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Optimal Stopping Time on Semi-Markov Processes with Finite Horizon

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  • Fang Chen

    (Sun Yat-Sen University)

  • Xianping Guo

    (Sun Yat-Sen University)

  • Zhong-Wei Liao

    (Beijing Normal University)

Abstract

In this paper, we consider the optimal stopping problems on semi-Markov processes (sMPs) with finite horizon and aim to establish the existence and algorithm of optimal stopping times. The key method is the equivalence between optimal stopping problems on sMPs and a special class of semi-Markov decision processes (sMDPs). We first introduce the optimality equation and show the existence of the optimal policies of finite-horizon sMDPs with additional terminal costs. Based on the optimal stopping problems on sMPs, we give an explicit construction of sMDPs such that the optimal stopping times of sMPs are equivalent to the optimal policies of the constructed sMDPs. Then, using the results of sMDPs developed here, we not only prove the existence of the optimal stopping times of sMPs, but also provide an algorithm for computing the optimal stopping times of sMPs. Moreover, we show that the optimal and $$\varepsilon $$ ε -optimal stopping time can be characterized by the hitting time of some special sets. Finally, we give an example to illustrate the effectiveness of our conclusions.

Suggested Citation

  • Fang Chen & Xianping Guo & Zhong-Wei Liao, 2022. "Optimal Stopping Time on Semi-Markov Processes with Finite Horizon," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 408-439, August.
    • Handle: RePEc:spr:joptap:v:194:y:2022:i:2:d:10.1007_s10957-022-02026-x
    DOI: 10.1007/s10957-022-02026-x
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    References listed on IDEAS

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    1. Huang, Yonghui & Guo, Xianping, 2011. "Finite horizon semi-Markov decision processes with application to maintenance systems," European Journal of Operational Research, Elsevier, vol. 212(1), pages 131-140, July.
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    3. Dinah Rosenberg & Eilon Solan & Nicolas Vieille, 1999. "Stopping Games with Randomized Strategies," Discussion Papers 1258, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    4. Christensen, Sören & Lindensjö, Kristoffer, 2020. "On time-inconsistent stopping problems and mixed strategy stopping times," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2886-2917.
    5. Benoîte Saporta & François Dufour & Christophe Nivot, 2017. "Partially observed optimal stopping problem for discrete-time Markov processes," 4OR, Springer, vol. 15(3), pages 277-302, September.
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