  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 (optimization over joint laws of stopping rules with Brownian
motion and state dynamics on an enlarged canonical space) and thus the OSEC
value. Using a martingale-problem formulation, we make an equivalent
characterization of the probability classes in weak formulation, which implies
that the OSEC value function s upper semi-analytic. 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.
