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Teoriya Veroyatnostei i ee Primeneniya, 1999, Volume 44, Issue 1, Pages 3–13
DOI: https://doi.org/10.4213/tvp594
(Mi tvp594)
 

This article is cited in 46 scientific papers (total in 46 papers)

On probability characteristics of “downfalls” in a standard Brownian motion

R. Douady, M. Yora, A. N. Shiryaevb

a Laboratoire de Probabilités, Université Paris VI, France
b Steklov Mathematical Institute, Russian Academy of Sciences
Abstract: For a Brownian motion $B=(B_t)_{t\le 1}$ with $B_0=0$, $\mathbf{E}B_t=0$, $\mathbf{E}B_t^2=t$ problems of probability distributions and their characteristics are considered for the variables
\begin{align*} \mathbb D&=\sup_{0\le t\le t'\le1}(B_t-B_{t'}), \qquad \mathbb D_1=B_{\sigma}-\inf_{\sigma\le t'\le1}B_{t'}, \\ \mathbb D_2&=\sup_{0\le t\le \sigma'}B_t-B_{\sigma'}, \end{align*}
where $\sigma$ and $\sigma'$ are times (non-Markov) of the absolute maximum and absolute minimum of the Brownian motion on $[0,1]$ (i.e., $B_\sigma=\sup_{0\le t\le 1}B_t$, $B_{\sigma'}=\inf_{0\le t'\le 1}B_{t'}$).
Keywords: Brownian motion, “downfalls” and “range”, Lévy theorem, Brownian meander.
Received: 24.08.1998
English version:
Theory of Probability and its Applications, 2000, Volume 44, Issue 1, Pages 29–38
DOI: https://doi.org/10.1137/S0040585X97977306
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: R. Douady, M. Yor, A. N. Shiryaev, “On probability characteristics of “downfalls” in a standard Brownian motion”, Teor. Veroyatnost. i Primenen., 44:1 (1999), 3–13; Theory Probab. Appl., 44:1 (2000), 29–38
Citation in format AMSBIB
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\by R.~Douady, M.~Yor, A.~N.~Shiryaev
\paper On probability characteristics of ``downfalls'' in a standard Brownian motion
\jour Teor. Veroyatnost. i Primenen.
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\issue 1
\pages 3--13
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\zmath{https://zbmath.org/?q=an:0959.60073}
\transl
\jour Theory Probab. Appl.
\yr 2000
\vol 44
\issue 1
\pages 29--38
\crossref{https://doi.org/10.1137/S0040585X97977306}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000087555000003}
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  • https://www.mathnet.ru/eng/tvp/v44/i1/p3
  • This publication is cited in the following 46 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Теория вероятностей и ее применения Theory of Probability and its Applications
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