How to quantise probabilities while preserving their convex order

Abstract

We introduce an algorithm which, given probabilities $\mu \leq_{\text{cx}} \nu$ in convex order and defined on a separable Banach space $B$, constructs finitely-supported approximations $\mu_n \to \mu, \nu_n\to \nu$ which are in convex order $\mu_n \leq_{\text{cx}} \nu_n$. We provide upper-bounds for the speed of convergence, in terms of the Wasserstein distance. We discuss the (dis)advantages of our algorithm and its link with the discretisation of the Martingale Optimal Transport problem, and we illustrate its implementation with numerical examples. We study the operation which, given $\mu$/$\nu$ and some (finite) partition of $B$, outputs $\mu_n$/$\nu_n$, showing that applied to a probability $\gamma$ and to all partitions it outputs the set of all probabilities $\zeta \leq_{\text{cx}} \gamma$.