Two-sided disorder problem for a Brownian motion in a Bayesian setting

A two-sided disorder problem for a Brownian motion in a Bayesian setting is considered. It is shown how to reduce this problem to the standard optimal stopping problem for a posterior probability process. Qualitative properties of a solution are analyzed; namely, the concavity, continuity, and the smooth-fit principle for the risk function are proved. Optimal stopping boundaries are characterized as a unique solution to some integral equation.