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JOURNALS // Teoriya Veroyatnostei i ee Primeneniya // Archive

Teor. Veroyatnost. i Primenen., 1999 Volume 44, Issue 1, Pages 3–13 (Mi tvp594)

This article is cited 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

DOI: 10.4213/tvp594


 English version:
Theory of Probability and its Applications, 2000, 44:1, 29–38

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