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A general "bang-bang" principle for predicting the maximum of a random walk

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  • Pieter C. Allaart
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    Abstract

    Let $(B_t)_{0\leq t\leq T}$ be either a Bernoulli random walk or a Brownian motion with drift, and let $M_t:=\max\{B_s: 0\leq s\leq t\}$, $0\leq t\leq T$. This paper solves the general optimal prediction problem \sup_{0\leq\tau\leq T}\sE[f(M_T-B_\tau)], where the supremum is over all stopping times $\tau$ adapted to the natural filtration of $(B_t)$, and $f$ is a nonincreasing convex function. The optimal stopping time $\tau^*$ is shown to be of "bang-bang" type: $\tau^*\equiv 0$ if the drift of the underlying process $(B_t)$ is negative, and $\tau^*\equiv T$ is the drift is positive. This result generalizes recent findings by S. Yam, S. Yung and W. Zhou [{\em J. Appl. Probab.} {\bf 46} (2009), 651--668] and J. Du Toit and G. Peskir [{\em Ann. Appl. Probab.} {\bf 19} (2009), 983--1014], and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good ones as long as possible.

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    File URL: http://arxiv.org/pdf/0910.0545
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0910.0545.

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    Date of creation: Oct 2009
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    Publication status: Published in J. Appl. Probab. 47, no. 4, 1072-1083 (2010)
    Handle: RePEc:arx:papers:0910.0545

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    1. Hlynka, M. & Sheahan, J. N., 1988. "The secretary problem for a random walk," Stochastic Processes and their Applications, Elsevier, vol. 28(2), pages 317-325, June.
    2. Albert Shiryaev & Zuoquan Xu & Xun Yu Zhou, 2008. "Thou shalt buy and hold," Quantitative Finance, Taylor & Francis Journals, vol. 8(8), pages 765-776.
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