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Stochastic approximations for finite-state Markov chains

Author

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  • Ma, D.-J.
  • Makowski, A.M.
  • Shwartz, A.

Abstract

This paper develops an a.s. convergence theory for a class of projected stochastic approximations driven by finite-state Markov chains. The conditions are mild and are given explicitly in terms of the model data, mainly the Lipschitz continuity of the one-step transition probabilities. The approach used here is a version of the ODE method as proposed by Métivier and Priouret. It combines the Kushner-Clark Lemma with properties of the Poisson equation associated with the underlying family of Markov chains. The class of algorithms studied here was motivated by implementation issues for constrained Markov decision problems, where the policies of interest often depend on quantities not readily available due either to insufficient knowledge of the model parameters or to computational difficulties. This naturally leads to the on-line estimation (or computation) problem investigated here. Several examples from the area of queueing systems are discussed.

Suggested Citation

  • Ma, D.-J. & Makowski, A.M. & Shwartz, A., 1990. "Stochastic approximations for finite-state Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 35(1), pages 27-45, June.
    • Handle: RePEc:eee:spapps:v:35:y:1990:i:1:p:27-45
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    Citations

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    Cited by:

    1. Rydén, Tobias, 1997. "On recursive estimation for hidden Markov models," Stochastic Processes and their Applications, Elsevier, vol. 66(1), pages 79-96, February.
    2. Prasenjit Karmakar & Shalabh Bhatnagar, 2018. "Two Time-Scale Stochastic Approximation with Controlled Markov Noise and Off-Policy Temporal-Difference Learning," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 130-151, February.
    3. Beggs, Alan, 2022. "Reference points and learning," Journal of Mathematical Economics, Elsevier, vol. 100(C).
    4. Liu, Z. & Almhana, J. & Choulakian, V. & McGorman, R., 2006. "Online EM algorithm for mixture with application to internet traffic modeling," Computational Statistics & Data Analysis, Elsevier, vol. 50(4), pages 1052-1071, February.

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