On a problem of optimal transport under marginal martingale constraints

International audienceThe basic problem of optimal transportation consists in minimizing the expected costs 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, 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 : R → R