Distribution of the Brownian motion on its way to hitting zero

P. Chigansky, F. C. Klebaner

Published 2008-11-06, updated 2008-12-18Version 2

For the one-dimensional Brownian motion $B=(B_t)_{t\ge 0}$, started at $x>0$, and the first hitting time $\tau=\inf\{t\ge 0:B_t=0\}$, we find the probability density of $B_{u\tau}$ for a $u\in(0,1)$, i.e. of the Brownian motion on its way to hitting zero.