Bayesian sequential composite hypothesis testing in discrete time

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