  We study the problem of bounding path-dependent expectations (within any
finite time horizon $d$) over the class of discrete-time martingales whose
marginal distributions lie within a prescribed tolerance of a given collection
of benchmark marginal distributions. This problem is a relaxation of the
martingale optimal transport (MOT) problem and is motivated by applications to
super-hedging in financial markets. We show that the empirical version of our
relaxed MOT problem can be approximated within $O\left( n^{-1/2}\right)$ error
where $n$ is the number of samples of each of the individual marginal
distributions (generated independently) and using a suitably constructed
finite-dimensional linear programming problem.
