Complete Duality for Martingale Optimal Transport on the Line

TL;DR

This paper establishes a comprehensive duality theory for martingale optimal transport on the real line, addressing classical formulation limitations and providing geometric insights into optimal plans.

Contribution

It introduces a quasi-sure dual formulation that ensures no duality gap and the existence of optimizers for general marginals and costs.

Findings

Complete duality theory for martingale optimal transport established

Duality gap and optimizer existence issues resolved

Geometric principle of cyclical monotonicity derived

Abstract

We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to fail in the classical formulation. As a consequence of the duality result, we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports.