Quantization and martingale couplings

TL;DR

This paper explores how quantization preserves convex order and enables approximation of martingale couplings between probability measures, facilitating numerical solutions for martingale optimal transport problems in any dimension.

Contribution

It demonstrates that quantization preserves convex order and allows approximation of martingale couplings, extending stability results of martingale optimal transport to higher dimensions.

Findings

Quantization preserves convex order between probability measures.

Martingale couplings can be approximated via quantized measures with convergence guarantees.

Numerical value functions for martingale optimal transport converge in any dimension as quantization improves.

Abstract

Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly supported, it is smaller than any of its dual quantizations while the dominated original measure is greater than any of its stationary (and therefore any of its optimal) quadratic primal quantization. Moreover, the quantization errors then correspond to martingale couplings between each original probability measure and its quantization. This permits to prove that any martingale coupling between the original probability measures can be approximated by a martingale coupling between their quantizations in Wassertein distance with a rate given by the quantization errors but also in the much finer adapted Wassertein distance. As a consequence, while the stability of (Weak) Martingale Optimal Transport problems with respect to the marginal distributions has only been established in dimension $1$ so far, their value function computed numerically for the quantized marginals converges in any dimension to the value for the original probability measures as the numbers of quantization points go to $\infty$.