  This paper focuses on martingale optimal transport problems when the
martingales are assumed to have bounded quadratic variation. First, we give a
result that characterizes the existence of a probability measure satisfying
some convex transport constraints in addition to having given initial and
terminal marginals. Several applications are provided: martingale measures with
volatility uncertainty, optimal transport with capacity constraints, and
Skorokhod embedding with bounded times. Next, we extend this result to
multi-marginal constraints. Finally, we consider an optimal transport problem
with constraints and obtain its Kantorovich duality. A corollary of this result
is a monotonicity principle which gives a geometric way of identifying the
optimizer.
