Bayesian sequential testing problem for a Brownian bridge

The present paper gives a solution to the Bayesian sequential testing problem of two simple hypotheses about the mean of a Brownian bridge. The method of the proof is based on reducing the sequential analysis problem to the optimal stopping problem for a strong Markov posterior probability process. The key idea in solving the above problem is the application of the one-to-one Kolmogorov time-space transformation, which enables one to consider, instead of the optimal stopping problem on a finite time horizon for a time-inhomogeneous diffusion process, an optimal stopping problem on an infinite time horizon for a homogeneous diffusion process with a slightly more complicated risk functional. The continuation and stopping sets are determined by two continuous boundaries, which constitute a unique solution of a system of two nonlinear integral equations.