Some Numerical Aspects of (Martingale) Optimal Transportation

Martingale optimal transport (MOT), a version of the optimal transport (OT) with an additional martingale constraint on transport’s dynamics, is an optimisation problem motivated by, and contributing to model-independent pricing problems in quantitative finance. Compared to the OT, numerical solution techniques for MOT problems are close to non-existent, relative to the theory and applications. In fact, the martingale constraint destroys the continuity of the value function, and thus renders any of the usual OT approximation techniques unusable. With Obloj, we proved that the MOT value could be approximated by a sequence of linear programming (LP) problems to which we apply the entropic regularisation. Further, we obtain in dimension one the convergence rate, which, to the best of our knowledge, is the first estimation of convergence rate in the literature. In the second part, we consider a semi-discrete Wasserstein distance of order 2, which could be solved by means of Voronoi diagram -- which is a static object in computational geometry. Inspired by a criterion in statistic physics, we may construct a sequence of probability distributions and we aim to show its convergence to some limit related to the minimal energy.