Optimal Stopping with a Probabilistic Constraint

We present an efficient method for solving optimal stopping problems with a probabilistic constraint. The goal is to optimize the expected cumulative cost, but constrained by an upper bound on the probability that the cost exceeds a specified threshold. This probabilistic constraint causes optimal policies to be time-dependent and randomized, however, we show that an optimal policy can always be selected with “piecewise-monotonic” timedependence and “nearly-deterministic” randomization. We prove these properties using the Bellman optimality equations for a Lagrangian relaxation of the original problem. We present an algorithm that exploits these properties for computational efficiency. Its performance and the structure of optimal policies are illustrated on two numerical examples.