On the quantiles of the Brownian motion and their hitting times
Dassios, A.
(2005).
On the quantiles of the Brownian motion and their hitting times.
Bernoulli,
11(1), 29-36.
(2005).
On the quantiles of the Brownian motion and their hitting times.
Bernoulli,
11(1), 29-36.
The distribution of the Æ-quantile of a Brownian motion on an interval [0, t] has been obtained motivated by a problem in financial mathematics. In this paper we generalize these results by calculating an explicit expression for the joint density of the Æ-quantile of a standard Brownian motion, its first and last hitting times and the value of the process at time t. Our results can easily be generalized to a Brownian motion with drift. It is shown that the first and last hitting times follow a transformed arcsine law.