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Teor. Veroyatnost. i Primenen., 2000, Volume 45, Issue 1, Pages 125–136 (Mi tvp327)  

This article is cited in 41 scientific papers (total in 41 papers)

Stopping Brownian motion without anticipation as close as possible to its ultimate maximum

S. E. Graversena, G. Peskira, A. N. Shiryaevb

a Institute of Mathematics, University of Aarhus, Denmark
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Let $B=(B_t)_{0 \le t \le 1}$ be the standard Brownian motion started at 0, and let $S_t=\max_{ 0 \le r \le t} B_r$ for $0 \le t \le 1$. Consider the optimal stopping problem
$$ V_*=\inf_\tau{\mathsf E}(B_\tau-S_1)^2, $$
where the infimum is taken over all stopping times of $B$ satisfying $0 \le \tau \le 1$. We show that the infimum is attained at the stopping time $\tau_*=\inf\{0\le t\le 1\mid S_t-B_t\ge z_*\sqrt{1-t}\}$, where $z_*=1.12 \ldots$ is a unique root of the equation $4\Phi(z_*)-2z_*\varphi(z_*)-3=0$ with $\varphi(x)=(1/\sqrt{2 \pi }) e^{-x^2/2}$ and $ \Phi (x)=\int_{-\infty}^x \varphi(y) dy$. The value $V_*$ equals $2 \Phi (z_*)-1$. The method of proof relies upon a stochastic integral representation of $S_1$, time-change arguments, and the solution of a free-boundary (Stefan) problem.

Keywords: Brownian motion, optimal stopping, anticipation, ultimate maximum, free-boundary (Stefan) problem, Ito–Clark representation theorem, Markov process, diffusion.

DOI: https://doi.org/10.4213/tvp327

Full text: PDF file (471 kB)

English version:
Theory of Probability and its Applications, 2001, 45:1, 41–50

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Received: 21.10.1999
Language:

Citation: S. E. Graversen, G. Peskir, A. N. Shiryaev, “Stopping Brownian motion without anticipation as close as possible to its ultimate maximum”, Teor. Veroyatnost. i Primenen., 45:1 (2000), 125–136; Theory Probab. Appl., 45:1 (2001), 41–50

Citation in format AMSBIB
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\pages 125--136
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    Citing articles on Google Scholar: Russian citations, English citations
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