On the time consistent solution to optimal stopping problems with expectation constraint

We study the (weak) equilibrium problem arising from the problem of optimally stopping a one-dimensional diffusion subject to an expectation constraint on the time until stopping. The weak equilibrium problem is realized with a set of randomized but purely state dependent stopping times as admissible strategies. We derive a verification theorem and necessary conditions for equilibria, which together basically characterize all equilibria. Furthermore, additional structural properties of equilibria are obtained to feed a possible guess-and-verify approach, which is then illustrated by an example.