en>David Eppstein: {{abstract-algebra-stub}}

2013-10-03T21:30:50Z

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{{technical|date=July 2013}}

In the mathematical [[theory of probability]], '''Brownian meander''' <math>W^+ = \{ W_t^+, t \in [0,1] \}</math> is a continuous non-homogenous [[Markov process]] defined as follows:

Let <math>W = \{ W_t, t \geq 0 \}</math> be a standard one-dimensional [[Wiener process|Brownian motion]], <math> \tau := \sup \{ t \in [0,1] : W_t = 0 \} </math> the last time before ''t'' = 1 when <math>W</math> visits <math>\{ 0 \}</math>. Then

:<math>W^+_t := \frac{1}{\sqrt{1 - \tau}} | W_{\tau + t (1-\tau)} |, \quad t \in [0,1].</math>

The [[transition density]] <math>p(s,x,t,y) \, dy := P(W^+_t \in dy \mid W^+_s = x)</math> of Brownian meander is described as follows:

For <math>0 < s < t \leq 1</math> and <math>x, y > 0</math>, and writing

: <math>\phi_t(x):= \frac{\exp \{ -x^2/(2t) \}}{\sqrt{2 \pi t}} \quad \text{and} \quad \Phi_t(x,y):= \int^y_x\phi_t(w) \, dw,</math>

we have

:<math>
\begin{align}
p(s,x,t,y) \, dy &:= P(W^+_t \in dy \mid W^+_s = x) \\
&= \bigl( \phi_{t-s}(y-x) - \phi_{t-s}(y+x) \bigl) \frac{\Phi_{1-t}(0,y)}{\Phi_{1-s}(0,x)} \, dy
\end{align}
</math>

and

: <math>
p(0,0,t,y) \, dy := P(W^+_t \in dy ) = 2\sqrt{2 \pi} \frac{y}{t}\phi_t(y)\Phi_{1-t}(0,y) \, dy.
</math>

In particular,

: <math>P(W^+_1 \in dy ) = y \exp \{ -y^2/2 \} \, dy, \quad y > 0,</math>

i.e <math> W^+_1 </math> has the [[Rayleigh distribution]] with parameter 1, the same distribution as <math>\sqrt{2 \mathbf{e}}</math>, where <math>\mathbf{e}</math> is an [[Exponential distribution|exponential random variable]] with parameter 1.

== References ==

* {{Cite journal |author=Durett, Richard; Iglehart, Donald; Miller, Douglas |year= 1977|title= Weak convergence to Brownian meander and Brownian excursion | journal = The Annals of Probability | volume = 5 | issue = 1| pages = 117–129 |
url = http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aop/1176995895}}
* {{Cite book |author=Revuz, Daniel; Yor, Marc |title=Continuous Martingales and Brownian Motion |edition=2nd |isbn=3-540-57622-3 |publisher=Springer-Verlag |location=New York |year=1999}}

{{Stochastic processes}}

[[Category:Stochastic processes]]
[[Category:Markov processes]]

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en>David Eppstein: {{abstract-algebra-stub}}