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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 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.
- Revuz, Daniel; Yor, Marc (1999). Continuous Martingales and Brownian Motion (2nd ed.). New York: Springer-Verlag. ISBN 3-540-57622-3.