On a problem of optimal transport under marginal martingale constraints

TL;DR

This paper investigates a variant of the optimal transport problem where the joint distribution must form a martingale, providing a variational principle and identifying a specific optimal martingale coupling with properties similar to classical monotone couplings.

Contribution

It introduces a variational framework for martingale-constrained optimal transport and characterizes the structure of optimal solutions, including a monotone martingale coupling analogous to Brenier's theorem.

Findings

Established a variational principle for martingale optimal transport.

Identified a monotone martingale coupling supported on graphs of two functions.

Connected the structure of solutions to classical monotone transport theory.

Abstract

The basic problem of optimal transportation consists in minimizing the expected costs $\mathbb {E}[c(X_1,X_2)]$ by varying the joint distribution $(X_1,X_2)$ where the marginal distributions of the random variables $X_1$ and $X_2$ are fixed. Inspired by recent applications in mathematical finance and connections with the peacock problem, we study this problem under the additional condition that $(X_i)_{i=1,2}$ is a martingale, that is, $\mathbb {E}[X_2|X_1]=X_1$. We establish a variational principle for this problem which enables us to determine optimal martingale transport plans for specific cost functions. In particular, we identify a martingale coupling that resembles the classic monotone quantile coupling in several respects. In analogy with the celebrated theorem of Brenier, the following behavior can be observed: If the initial distribution is continuous, then this "monotone martingale" is supported by the graphs of two functions $T_1,T_2:\mathbb {R}\to \mathbb {R}$.