  We solve the martingale optimal transport problem for cost functionals
represented by optimal stopping problems. The measure-valued martingale
approach developed in ArXiv: 1507.02651 allows us to obtain an equivalent
infinite-dimensional controller-stopper problem. We use the stochastic Perron's
method and characterize the finite dimensional approximation as a viscosity
solution to the corresponding HJB equation. It turns out that this solution is
the concave envelope of the cost function with respect to the atoms of the
terminal law. We demonstrate the results by finding explicit solutions for a
class of cost functions.
