Canonical Supermartingale Couplings

TL;DR

This paper introduces two canonical supermartingale couplings as optimal solutions to constrained transport problems, revealing their unique order-theoretic and geometric properties, and showing how supermartingale transport decomposes into classical and martingale parts.

Contribution

It constructs and characterizes two canonical supermartingale couplings using order-theoretic and geometric properties, advancing the understanding of supermartingale optimal transport.

Findings

Couplings are characterized by minimality and no-crossing support conditions.

Supermartingale optimal transport decomposes into classical and martingale transport.

The couplings have asymmetric geometries and are optimal for certain reward functions.

Abstract

Two probability distributions $\mu$ and $\nu$ in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are allowed as transports. Much like the Hoeffding-Fr\'echet coupling of classical transport and its symmetric counterpart, the antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward (or cost) functions, and through no-crossing conditions on their supports; however, our two couplings have asymmetric geometries. Remarkably, supermartingale optimal transport decomposes into classical and martingale transport in several ways.