Brownian meander

In the mathematical theory of probability, Brownian meander $W^+ = \$ is a continuous non-homogeneous Markov process defined as follows:

Let $W = \$ be a standard one-dimensional Brownian motion, and $\tau := \sup \$, i.e. the last time before ''t'' = 1 when $W$ visits $\$. Then the Brownian meander is defined by the following:

:W^+_t := \frac 1 , W_ , , \quad t \in ,1

In words, let $\tau$ be the last time before 1 that a standard Brownian motion visits $\$. ($\tau < 1$ almost surely.) We snip off and discard the trajectory of Brownian motion before $\tau$, and scale the remaining part so that it spans a time interval of length 1. The scaling factor for the spatial axis must be square root of the scaling factor for the time axis. The process resulting from this snip-and-scale procedure is a Brownian meander. As the name suggests, it is a piece of Brownian motion that spends all its time away from its starting point $\$.

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

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

: $\varphi_t(x):= \frac \quad \text \quad \Phi_t(x,y):= \int^y_x\varphi_t(w) \, dw,$

we have

:$\begin p(s,x,t,y) \, dy := & P(W^+_t \in dy \mid W^+_s = x) \\ = & \bigl( \varphi_(y-x) - \varphi_(y+x) \bigl) \frac \, dy \end$

and

: $p(0,0,t,y) \, dy := P(W^+_t \in dy ) = 2\sqrt \frac y t \varphi_t(y)\Phi_(0,y) \, dy.$

In particular,

: $P(W^+_1 \in dy ) = y \exp \ \, dy, \quad y > 0,$

i.e. $W^+_1$ has the Rayleigh distribution with parameter 1, the same distribution as $\sqrt$, where $\mathbf$ is an exponential random variable with parameter 1.

References

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Wiener process
Markov processes

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