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On Asymptotics of Optimal Stopping Times

Author

Listed:
  • Hugh N. Entwistle

    (Department of Mathematics and Statistics, Macquarie University, Sydney, NSW 2109, Australia)

  • Christopher J. Lustri

    (Department of Mathematics and Statistics, Macquarie University, Sydney, NSW 2109, Australia)

  • Georgy Yu. Sofronov

    (Department of Mathematics and Statistics, Macquarie University, Sydney, NSW 2109, Australia)

Abstract

We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density. The objective of such problems is to find a procedure which maximizes the expected reward. In this analysis, we obtained asymptotic expressions for the expectation and variance of the optimal stopping time as the number of drawn variables became large. In the case of distributions with infinite upper bound, the asymptotic behaviour of these statistics depends solely on the algebraic power of the probability distribution decay rate in the upper limit. In the case of densities with finite upper bound, the asymptotic behaviour of these statistics depends on the algebraic form of the distribution near the finite upper bound. Explicit calculations are provided for several common probability density functions.

Suggested Citation

  • Hugh N. Entwistle & Christopher J. Lustri & Georgy Yu. Sofronov, 2022. "On Asymptotics of Optimal Stopping Times," Mathematics, MDPI, vol. 10(2), pages 1-18, January.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:2:p:194-:d:720597
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    References listed on IDEAS

    as
    1. Stein, William E. & Seale, Darryl A. & Rapoport, Amnon, 2003. "Analysis of heuristic solutions to the best choice problem," European Journal of Operational Research, Elsevier, vol. 151(1), pages 140-152, November.
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    3. Georgy Sofronov, 2020. "An Optimal Decision Rule for a Multiple Selling Problem with a Variable Rate of Offers," Mathematics, MDPI, vol. 8(5), pages 1-11, May.
    4. Seale, Darryl A. & Rapoport, Amnon, 1997. "Sequential Decision Making with Relative Ranks: An Experimental Investigation of the "Secretary Problem">," Organizational Behavior and Human Decision Processes, Elsevier, vol. 69(3), pages 221-236, March.
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