Abstract:
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.