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Teoriya Veroyatnostei i ee Primeneniya, 2000, Volume 45, Issue 1, Pages 125–136
DOI: https://doi.org/10.4213/tvp327
(Mi tvp327)
 

This article is cited in 60 scientific papers (total in 60 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.
Received: 21.10.1999
English version:
Theory of Probability and its Applications, 2001, Volume 45, Issue 1, Pages 41–50
DOI: https://doi.org/10.1137/S0040585X97978075
Bibliographic databases:
Document Type: Article
Language: English
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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\by S.~E.~Graversen, G.~Peskir, A.~N.~Shiryaev
\paper Stopping Brownian motion without anticipation as close as possible to its ultimate maximum
\jour Teor. Veroyatnost. i Primenen.
\yr 2000
\vol 45
\issue 1
\pages 125--136
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\crossref{https://doi.org/10.4213/tvp327}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1810977}
\zmath{https://zbmath.org/?q=an:0982.60082}
\transl
\jour Theory Probab. Appl.
\yr 2001
\vol 45
\issue 1
\pages 41--50
\crossref{https://doi.org/10.1137/S0040585X97978075}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000167428900003}
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  • https://www.mathnet.ru/eng/tvp327
  • https://doi.org/10.4213/tvp327
  • https://www.mathnet.ru/eng/tvp/v45/i1/p125
  • This publication is cited in the following 60 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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