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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
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
Linking options:
https://www.mathnet.ru/eng/tvp594https://doi.org/10.4213/tvp594 https://www.mathnet.ru/eng/tvp/v44/i1/p3
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Abstract page: | 740 | Full-text PDF : | 212 | First page: | 33 |
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