An optimal sequential procedure for determining the drift of a Brownian motion among three values

We consider a one-dimensional Brownian motion, having a random and unobservable drift which can take one of three known values. Assuming that we monitor the position of the process in real time, the problem is to determine as soon as possible and with minimal probabilities of the wrong terminal decisions, which value the drift has taken. We derive the exact solution to the problem in the Bayesian formulation, under any prior probability distribution on the three values that the drift can assume, when the cost of observation is linear. Remarkably, the optimal stopping boundaries of the present problem are non-monotone.