  We study the sequential testing problem of two alternative hypotheses
regarding an unknown parameter in an exponential family when observations are
costly. In a Bayesian setting, the problem can be embedded in a Markovian
framework. Using the conditional probability of one of the hypotheses as the
underlying spatial variable, we show that the cost function is concave and that
the posterior distribution becomes more concentrated as time goes on. Moreover,
we study time monotonicity of the value function. For a large class of model
specifications, the cost function is non-decreasing in time, and the optimal
stopping boundaries are thus monotone.
