Optimal Stopping of a random Gauss-Markov bridge

We study an Optimal Stopping Problem (OSP) for Gauss–Markov processes conditioned to hit a prescribed terminal distribution. Using a time-space transformation, we show that this OSP is equivalent to that of a Brownian bridge with a random pinning point. This allows us to characterize the optimal stopping time as the first hitting time into the stopping region, and establish continuity and higher smoothness of the value function, as well as to obtain the free-boundary formulation of the problem. We also derive comparison results showing that value functions are ordered by the likelihood-ratio order of terminal densities. When terminal densities are Gaussian or degenerate at a point, the transformed drift simplifies, and optimal stopping boundaries are uniquely characterized by Volterra-type integral equations.