On the optimal stopping of Gauss-Markov bridges with random pinning points

We consider the optimal stopping problem for a Gauss-Markov process conditioned to adopt a prescribed terminal distribution. By applying a time-space transformation, we show it is equivalent to stopping a Brownian bridge pinned at a random endpoint with a time-dependent payoff. We prove that the optimal rule is the first entry into the stopping region, and establish that the value function is Lipschitz continuous on compacts via a coupling of terminal pinning points across different initial conditions. A comparison theorems then order value functions according to likelihood-ratio ordering of terminal densities, and when these densities have bounded support, we bound the optimal boundary by that of a Gauss-Markov bridge. Although the stopping boundary need not be the graph of a function in general, we provide sufficient conditions under which this property holds, and identify strongly log-concave terminal densities that guarantee this structure. Numerical experiments illustrate representative boundary shapes.