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On the Local Time of the Brownian Bridge

  • Lajos Takács
Chapter
  • 676 Downloads
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 19)

Abstract

Let {η(t), 0 ≤ t} ≤ 1} be a standard Brownian bridge. We have for 0 < t < 1, where
$$ \Phi (x) = \frac{1}{{\sqrt {{2\pi }} }}\int_{{ - \infty }}^{x} {{{e}^{{ - u \frac{2}{2} }}}} du $$
(4.1)
is the normal distribution function. We define
$$ \tau (\alpha ) = \mathop{{\lim }}\limits_{{\varepsilon \to 0}} \frac{1}{\varepsilon }{\text{measure\{ }}t:\alpha \leqslant \eta (t) < \alpha + \varepsilon ,0 \leqslant t \leqslant 1{\text{\} }} $$
(4.2)
for any real α. The limit (4.2) exists with probability one, and τ(α) is a nonnegative random variable that is called the local time at level α. We have
$$ P\{ \tau (\alpha ) \leqslant x\} = 1 - {{e}^{{ - (2|\alpha | + x) \frac{2}{2} }}} $$
(4.3)
for x ≥ 0. The notion of local time was introduced by P. Lévy [9,10] in 1939 (see also [15] and [6]).

Keywords

Random Walk Local Time Random Mapping Normal Distribution Function Empirical Distribution Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media New York 1999

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  • Lajos Takács

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