On a problem of optimal transport under marginal martingale constraints

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 $T_{1},T_{2}:\mathbb{R} \to\mathbb{R}$.