Brownian meander explained

In the mathematical theory of probability, Brownian meander

W ^{+} = \{ W ^{+ ,} _{t} t \in [ 0 , 1 ] \} is a continuous non-homogeneous Markov process defined as follows:

Let

W = \{ W _{t,} t \geq 0 \} be a standard one-dimensional Brownian motion, and \tau : = \sup \{ t \in [ 0 , 1 ] : W _{t} = 0 \} , i.e. the last time before t = 1 when W visits \{ 0 \} . Then the Brownian meander is defined by the following:W ^{+} _{t} : = (1)/(\sqrt{ 1- \tau) } | W_ |, \quad t \in [0,1].

In words, let

\tau be the last time before 1 that a standard Brownian motion visits \{ 0 \} . (\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 \{ 0 \} .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):=} (\exp \{ - x ^{2/(2t)} \)/() } \quad \text \quad \Phi_t(x,y):= \int^y_x\varphi_t(w) \, dw,

we have

\begin{align} p(s,x,t,y) dy : = {} & P(W ^{+} _{t} \in dy \mid W ^{+} _{s} = x) \\ = {} & l ( \varphi _{t-s} (y - x) - \varphi _{t-s} (y + x) l ) (\Phi _{1-t} ( 0 ,y))/(\Phi _{1-s} ( 0 ,x)) dy \end{align}

and

p( 0 , 0 ,t,y) dy : = P(W ^{+} _{t} \in dy ) = 2 \sqrt{ 2 \pi} (y)/(t) \varphi _{t(y)\Phi} _{1-t} ( 0 ,y) dy.

In particular,

P(W ^{+} _{1} \in dy ) = y \exp \{ - y ^{2/2} \} dy, y > 0 ,

i.e.

W ^{+} _{1} has the Rayleigh distribution with parameter 1, the same distribution as \sqrt{ 2 e }, where e is an exponential random variable with parameter 1.

References

• Durett, Richard . Iglehart, Donald . Miller, Douglas . 1977. Weak convergence to Brownian meander and Brownian excursion . The Annals of Probability . 5 . 1. 117–129 . 10.1214/aop/1176995895. free .
• Book: Revuz, Daniel . Yor, Marc . Continuous Martingales and Brownian Motion . 2nd . 3-540-57622-3 . Springer-Verlag . New York . 1999.