A sequential test for the drift of a Brownian motion with a possibility to change a decision

We construct a Bayesian sequential test of two simple hypotheses about the value of the unobservable drift coefficient of a Brownian motion, with a possibility to change the initial decision at subsequent moments of time for some penalty. Such a testing procedure allows to correct the initial decision if it turns out to be wrong. The test is based on observation of the posterior mean process and makes the initial decision and, possibly, changes it later, when this process crosses certain thresholds. The solution of the problem is obtained by reducing it to joint optimal stopping and optimal switching problems.