On a problem of optimal transport under marginal martingale constraints

Understand

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.