On a problem of optimal transport under marginal martingale constraints

The basic problem of optimal transportation consists in minimizing the expected costs E [c(X₁,X₂)] by varying the joint distribution (X₁,X₂) where the marginal distributions of the random variables X₁ and X₂ are fixed. Inspired by recent applications in mathematical finance and connections with the peacock problem, we study this problem under the additional condition that (Xᵢ)i=1,2 is a martingale, that is, E [X₂|X₁]=X₁. 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 T₁,T₂:R.