On probability characteristics of ``downfalls'' in a standard Brownian motion
Teoriâ veroâtnostej i ee primeneniâ, Tome 44 (1999) no. 1, pp. 3-13
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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.
@article{TVP_1999_44_1_a0,
author = {R. Douady and M. Yor and A. N. Shiryaev},
title = {On probability characteristics of ``downfalls'' in a standard {Brownian} motion},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {3--13},
publisher = {mathdoc},
volume = {44},
number = {1},
year = {1999},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1999_44_1_a0/}
}
TY - JOUR AU - R. Douady AU - M. Yor AU - A. N. Shiryaev TI - On probability characteristics of ``downfalls'' in a standard Brownian motion JO - Teoriâ veroâtnostej i ee primeneniâ PY - 1999 SP - 3 EP - 13 VL - 44 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TVP_1999_44_1_a0/ LA - ru ID - TVP_1999_44_1_a0 ER -
R. Douady; M. Yor; A. N. Shiryaev. On probability characteristics of ``downfalls'' in a standard Brownian motion. Teoriâ veroâtnostej i ee primeneniâ, Tome 44 (1999) no. 1, pp. 3-13. http://geodesic.mathdoc.fr/item/TVP_1999_44_1_a0/