Bi-martingale optimal transport and its applications

We introduce a new non-linear optimal transport formulation for a pair of probability measures on $\mathbb{R}^d$ sharing a common barycentre, in which admissible transference plans satisfy two martingale-type constraints. This bi-martingale framework underlies and interconnects several variational problems on the space of probability measures. For the quadratic cost, it provides an optimal transport interpretation of the second Zolotarev distance on $\mathrm{P}_2(\mathbb{R}^d)$. For a broader class of convex costs, it leads to optimization problems under convex order constraints, encompassing in particular the Zolotarev projection onto the cone of dominating probability measures. As a main application, we construct a $\Gamma$-convergent bi-martingale approximation of the classical martingale optimal transport problem. This scheme robustly accommodates deviations from convex order between the marginal distributions and overcomes the well-known instability of MOT with respect to variations of the marginals in higher dimensions.