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Optimal detection of a hidden target: The median rule

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  • Peskir, Goran

Abstract

We show that in the absence of any information about the ‘hidden’ target in terms of the observed sample path, and irrespectively of the distribution law of the observed process, the ‘median’ rule is optimal in both the space domain and the time domain. While the fact that the median rule minimises the spatial expectation can be seen as a direct extension of the well-known median characterisation dating back to Boscovich, the fact that this also holds for the temporal expectation seems to have stayed unnoticed until now. Building on this observation we derive new classes of median/quantile rules having a dynamic character.

Suggested Citation

  • Peskir, Goran, 2012. "Optimal detection of a hidden target: The median rule," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2249-2263.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:5:p:2249-2263
    DOI: 10.1016/j.spa.2012.02.004
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    References listed on IDEAS

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    1. Jacques du Toit & Goran Peskir, 2009. "Selling a stock at the ultimate maximum," Papers 0908.1014, arXiv.org.
    2. Cohen, Albert, 2010. "Examples of optimal prediction in the infinite horizon case," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 950-957, June.
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    Cited by:

    1. Gapeev, Pavel V., 2020. "Optimal stopping problems for running minima with positive discounting rates," LSE Research Online Documents on Economics 105849, London School of Economics and Political Science, LSE Library.
    2. Gapeev, Pavel V. & Rodosthenous, Neofytos & Chinthalapati, V.L Raju, 2019. "On the Laplace transforms of the first hitting times for drawdowns and drawups of diffusion-type processes," LSE Research Online Documents on Economics 101272, London School of Economics and Political Science, LSE Library.
    3. T. De Angelis & G. Peskir, 2016. "Optimal prediction of resistance and support levels," Applied Mathematical Finance, Taylor & Francis Journals, vol. 23(6), pages 465-483, November.
    4. Gapeev, Pavel V. & Rodosthenous, Neofytos, 2016. "Perpetual American options in diffusion-type models with running maxima and drawdowns," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 2038-2061.
    5. Gapeev, Pavel V., 2020. "Optimal stopping problems for running minima with positive discounting rates," Statistics & Probability Letters, Elsevier, vol. 167(C).
    6. Pavel V. Gapeev & Neofytos Rodosthenous & V. L. Raju Chinthalapati, 2019. "On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes," Risks, MDPI, vol. 7(3), pages 1-15, August.

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