How to quantise probabilities while preserving their convex order

TL;DR

This paper presents an algorithm for approximating probabilities in convex order with finitely-supported measures, ensuring the convex order is preserved, and analyzes its convergence and applications in Martingale Optimal Transport.

Contribution

The paper introduces a novel algorithm for constructing finitely-supported convex order approximations with convergence bounds and explores its applications in discretizing Martingale Optimal Transport.

Findings

Algorithm preserves convex order in approximations

Provides convergence speed bounds in Wasserstein distance

Demonstrates numerical implementation and applications

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 $\zeta \leq_{\text{cx}} \gamma$.