On the continuity of martingale optimal transport in higher dimensions

We investigate qualitative and quantitative continuity of martingale optimal transport under perturbations of both marginals in dimensions d≥2. We assume that each pair of marginals admits a reference martingale coupling whose density relative to the product of its marginals is bounded above and away from zero, uniformly along the sequence. If the limiting target measure is not carried by a hyperplane, we prove continuity under W₁-convergence for continuous costs of at most linear growth, assuming only finite first moments. For Lipschitz costs, a uniform p-moment bound on the targets, $p>1$, yields an explicit stability estimate with exponent $(1-1/p)/2$; uniformly bounded target supports give exponent $1/2$. The proof constructs a correction of couplings with small martingale bias while preserving both marginals. Truncated moment corrections allow the target measure to have unbounded support. The construction also shows that every martingale coupling of the limiting marginals can be approximated in W₁ by martingale couplings of the approximating marginals. We also discuss counterexamples showing that neither the uniform density condition nor non-degeneracy of the limiting target measure can be omitted in general.