Brownian meander

In the mathematical theory of probability, Brownian meander [math]\displaystyle{ W^+ = \{ W_t^+, t \in [0,1] \} }[/math] is a continuous non-homogeneous Markov process defined as follows:

Let [math]\displaystyle{ W = \{ W_t, t \geq 0 \} }[/math] be a standard one-dimensional Brownian motion, and [math]\displaystyle{ \tau := \sup \{ t \in [0,1] : W_t = 0 \} }[/math], i.e. the last time before t = 1 when [math]\displaystyle{ W }[/math] visits [math]\displaystyle{ \{ 0 \} }[/math]. Then the Brownian meander is defined by the following:

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

In words, let [math]\displaystyle{ \tau }[/math] be the last time before 1 that a standard Brownian motion visits [math]\displaystyle{ \{ 0 \} }[/math]. ([math]\displaystyle{ \tau \lt 1 }[/math] almost surely.) We snip off and discard the trajectory of Brownian motion before [math]\displaystyle{ \tau }[/math], 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 [math]\displaystyle{ \{ 0 \} }[/math].

The transition density [math]\displaystyle{ 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]\displaystyle{ 0 \lt s \lt t \leq 1 }[/math] and [math]\displaystyle{ x, y \gt 0 }[/math], and writing

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

we have

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

and

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

In particular,

[math]\displaystyle{ P(W^+_1 \in dy ) = y \exp \{ -y^2/2 \} \, dy, \quad y \gt 0, }[/math]

i.e. [math]\displaystyle{ W^+_1 }[/math] has the Rayleigh distribution with parameter 1, the same distribution as [math]\displaystyle{ \sqrt{2 \mathbf{e}} }[/math], where [math]\displaystyle{ \mathbf{e} }[/math] 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. doi:10.1214/aop/1176995895. https://projecteuclid.org/journals/annals-of-probability/volume-5/issue-1/Weak-Convergence-to-Brownian-Meander-and-Brownian-Excursion/10.1214/aop/1176995895.full.
Revuz, Daniel; Yor, Marc (1999). Continuous Martingales and Brownian Motion (2nd ed.). New York: Springer-Verlag. ISBN 3-540-57622-3.

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