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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
- 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.
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