Day 11 : Brownian Meander#

A Brownian Meander is yet another stochastic process which arises from a Weiner process (standard Brownian motion) with the additional condition that it stays positive over a given interval.

Definition#

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 zero. Then the Brownian meander is defined by the following expression

\[X(t) = \frac{1}{1 - \tau} | W(\tau + t(1-\tau)) |, \qquad t \in \[0,1\].\]

That is, given the last time before 1 that a standard Brownian motion visits zero (note that \(\tau <1\) almost surely), we snip off and discard the trajectory 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.

đź”” Random Facts đź””#

  • Both the Brownian Meander and the Brownian Excursion arise as the limit process in the study of the behaviour of random walks conditioned to stay positive.

  • Brownian meanders emerge in path decompositions of the Brownian motion. In particular Denisov (1984) showed that a Brownian motion around a maximum point can be represented (in law) by means of a two-sided Brownian meander, which is constructed by gluing together two meanders.

  • If \(X\) is a Brownian meander then \(X(t)\) has probability density  $\( 2 t^{-\frac{3}{2}} x e^{-\frac{x^2}{2t}} \Phi\_{1-t}(x), \qquad x>0, t \in \[0,1\].\)$

  • In a similar fashion as the Brownian Bridge, the Brownian Meander can be restricted to end on a particular point. Such process is the so-called a Tied Brownian Meander. The Tied Brownian Meander has more complicated distributional properties compared to the unconstrained Brownian motion or the regular meander.

More to Read 📚#

  • Jim Pitman. “Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times.” Electron. J. Probab. 4 1 - 33, 1999. https://doi.org/10.1214/EJP.v4-48

  • I. V. Denisov. A random walk and a wiener process near a maximum. Theor. Prob. Appl., 28:821–824, 1984.

  • Richard T. Durrett. Donald L. Iglehart. Douglas R. Miller. “Weak Convergence to Brownian Meander and Brownian Excursion.” Ann. Probab. 5 (1) 117 - 129, February, 1977.https://doi.org/10.1214/aop/1176995895

P.s. If you are curious about probability distributions visit the Advent Calendar 2023 ✨