Brownian meander

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In the mathematical theory of probability, Brownian meander is a continuous non-homogeneous Markov process defined as follows:

Let be a standard one-dimensional Brownian motion, and , i.e. the last time before t = 1 when visits . Then the Brownian meander is defined by the following:

In words, let be the last time before 1 that a standard Brownian motion visits . ( almost surely.) We snip off and discard the trajectory of Brownian motion before , 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 of Brownian meander is described as follows:

For and , and writing

we have

and

In particular,

i.e. has the Rayleigh distribution with parameter 1, the same distribution as , where is an exponential random variable with parameter 1.

References[edit]

  • 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.