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On optimal stopping of a sequence of independent random variables -- probability maximizing approach

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  • Bojdecki, Tomasz

Abstract

Let [xi]1,[xi]2,... be a sequence of independent, identically distributed r.v. with a continuous distribution function. Optimal stopping problems are considered for: (1) a finite sequence [xi]1,...,[xi]N, (2) sequences ([xi]n-cn)n[epsilon]N and (max([xi]1,...,[xi]n) - cn)n[epsilon]N, where c is a fixed positive number, (3) the sequence ([xi]n)n[epsilon]N, where it is additionally assumed that [xi]1,[xi]2,... appear according to a Poisson process which is independent of {[xi]n}n[epsilon]N, and the decision about stopping must be made before some fixed moment T. The object of optimization is not (as it is in the classical formulation of optimal stopping problems) the expected value of the reward, but the probability that at the moment of stopping the reward attains its maximal value. It is proved that optimal stopping rules (in the above sense) for all problems exist, and their forms are found.

Suggested Citation

  • Bojdecki, Tomasz, 1978. "On optimal stopping of a sequence of independent random variables -- probability maximizing approach," Stochastic Processes and their Applications, Elsevier, vol. 6(2), pages 153-163, January.
  • Handle: RePEc:eee:spapps:v:6:y:1978:i:2:p:153-163
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    Cited by:

    1. Lazar Obradović, 2020. "Robust best choice problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(3), pages 435-460, December.
    2. Ferenstein, Elzbieta Z. & Krasnosielska, Anna, 2010. "No-information secretary problems with cardinal payoffs and Poisson arrivals," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 221-227, February.
    3. Marek Skarupski, 2019. "Full-information best choice game with hint," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(2), pages 153-168, October.
    4. Porosinski, Zdzislaw, 2003. "On optimal choosing of one of the k best objects," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 419-432, December.
    5. Gnedin, A.V.Alexander V., 2004. "Best choice from the planar Poisson process," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 317-354, June.
    6. You, Peng-Sheng, 2000. "Sequential buying policies," European Journal of Operational Research, Elsevier, vol. 120(3), pages 535-544, February.

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