Brownian meander

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

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

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

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

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

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

${\displaystyle \varphi _{t}(x):={\frac {\exp\{-x^{2}/(2t)\}}{\sqrt {2\pi t}}}\quad {\text{and}}\quad \Phi _{t}(x,y):=\int _{x}^{y}\varphi _{t}(w)\,dw,}$

we have

${\displaystyle {\begin{aligned}p(s,x,t,y)\,dy:={}&P(W_{t}^{+}\in dy\mid W_{s}^{+}=x)\\={}&{\bigl (}\varphi _{t-s}(y-x)-\varphi _{t-s}(y+x){\bigl )}{\frac {\Phi _{1-t}(0,y)}{\Phi _{1-s}(0,x)}}\,dy\end{aligned}}}$

and

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

In particular,

${\displaystyle P(W_{1}^{+}\in dy)=y\exp\{-y^{2}/2\}\,dy,\quad y>0,}$

i.e. ${\displaystyle W_{1}^{+}}$ has the Rayleigh distribution with parameter 1, the same distribution as ${\displaystyle {\sqrt {2\mathbf {e} }}}$, where ${\displaystyle \mathbf {e} }$ 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. doi:10.1214/aop/1176995895.
• Revuz, Daniel; Yor, Marc (1999). Continuous Martingales and Brownian Motion (2nd ed.). New York: Springer-Verlag. ISBN 3-540-57622-3.