A Bayesian sequential soft classification problem for a Brownian motion's drift

In this note we introduce and solve a soft classification version of the famous Bayesian sequential testing problem for a Brownian motion's drift. We establish that the value function is the unique non-trivial solution to a free boundary problem, and that the continuation region is characterized by two boundaries which may coincide if the observed signal is not strong enough. By exploiting the solution structure we are able to characterize the functional dependence of the stopping boundaries on the signal-to-noise ratio. We illustrate this relationship and compare our stopping boundaries to those derived in the classical setting.