On confidence intervals for Brownian motion changepoint times

We consider a sequential problem of finding the best confidence interval for a changepoint time of a Brownian motion. Namely, let B= (B sub()t sub()t> or =, slanted]0be a standard Brownian motion defined on a probability space [Omega, [scriptF], P) and let [straighttheta] be an unobservable non-negative random variable which does not depend on B and has a known distribution. Observable is the process X sub()t [mu](t- [straighttheta]) super(+)+ B sub()t t[> or =, slanted] 0, where [mu] [not =] 0 is a known parameter. Thus, at time [straighttheta] a 'disorder' occurs, which is manifested through the change of the drift coefficient from zero to [mu].