A Bayesian sequential testing problem of three hypotheses for Brownian motion

Abstract We consider a sequential testing problem of three hypotheses that the unknown drift of a Brownian motion takes one of three values. We show that this problem can be solved by a reduction to an optimal stopping problem for local times of the observable process. For the case of “large” periods of observation, we derive integral equations for the optimal stopping boundaries and study their limit behaviour. Other cases will be considered in subsequent papers. The work can be regarded as a further step in the sequential testing problem of two hypotheses for Brownian motion, solved more than 30 years ago (see [4]).