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We construct a Bayesian sequential test of two simple hypotheses about thevalue of the unobservable drift coefficient of a Brownian motion, with apossibility to change the initial decision at subsequent moments of time forsome penalty. Such a testing procedure allows to correct the initial decisionif it turns out to be wrong. The test is based on observation of the posteriormean process and makes the initial decision and, possibly, changes it later,when this process crosses certain thresholds. The solution of the problem isobtained by reducing it to joint optimal stopping and optimal switchingproblems.

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