Detection of disorder before an observable event

We consider a continuously observable process, which behaves like a standard Brownian motion up to a random time τ1 and as a Brownian motion with a known drift after τ1. At a stopping-time τ2 that happens after the time τ1, an observable event occurs. We address the problem of detecting the change in the system's behaviour prior to the occurrence of the observable event. In particular, our formulation takes into account the information provided by the non-occurrence of the observable event and where it is favourable to “raise the alarm” before this event. We show that this problem can be reduced to a one-dimensional optimal stopping problem, to which we derive an explicit solution.