On Probability Characteristics of "Downfalls" in a Standard Brownian Motion

Raphael Douady (), A.N. Shiryaev and Marc Yor ()
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A.N. Shiryaev: SMI | RAS - Steklov Mathematical Institute [Moscow] - RAS - Russian Academy of Sciences [Moscow]
Marc Yor: IUF - Institut universitaire de France - M.E.N.E.S.R. - Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche, LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique

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Abstract: For a Brownian motion $B=(B_t)_{t\le 1}$ with $B_0=0$, {\bf E}$B_t=0$, {\bf E}$B_t^2=t$ problems of probability distributions and their characteristics are considered for the variables $$ \begin{array}{c} {\mathbb D} =\displaystyle\sup_{0\le t\le t'\le 1}(B_t-B_{t'}),\qquad {\mathbb D}_1=B_\sigma-\inf_{\sigma\le t'\le 1}B_{t'}, \\ {\mathbb D}_2=\displaystyle\sup_{0\le t\le\sigma'}B_{t}-B_{\sigma'}, \end{array} $$ 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; downfall (search for similar items in EconPapers)
Date: 2000-11-14
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Citations: View citations in EconPapers (14)

Published in Theory of Probability and Its Applications c/c of Teoriia Veroiatnostei i Ee Primenenie, 2000, 44 (1), pp.29-38. ⟨10.1137/S0040585X97977306⟩

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Working Paper: On Probability Characteristics of "Downfalls" in a Standard Brownian Motion (2000)
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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-01477104

DOI: 10.1137/S0040585X97977306

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