Optimal stopping with expectation constraints

Abstract

We analyze an optimal stopping problem with a series of inequality-type and equality-type expectation constraints in a general non-Markovian framework. We show that the optimal stopping problem with expectation constraints (OSEC) in an arbitrary probability setting is equivalent to the constrained problem in weak formulation (an optimization over joint laws of stopping rules with Brownian motion and state dynamics on an enlarged canonical space), and thus the OSEC value is independent of a specific probabilistic setup. Using a martingale-problem formulation, we make an equivalent characterization of the probability classes in weak formulation, which implies that the OSEC value function is upper semianalytic. Then we exploit a measurable selection argument to establish a dynamic programming principle in weak formulation for the OSEC value function, in which the conditional expected costs act as additional states for constraint levels at the intermediate horizon.