Multiperiod Martingale Transport

Consider a multiperiod optimal transport problem where distributions μ0,dots,μn are prescribed and a transport corresponds to a scalar martingale X with marginals Xt~μt. We introduce particular couplings called left-monotone transports; they are characterized equivalently by a no-crossing property of their support, as simultaneous optimizers for a class of bivariate transport cost functions with a Spence--Mirrlees property, and by an order-theoretic minimality property. Left-monotone transports are unique if μ0 is atomless, but not in general. In the one-period case n=1, these transports reduce to the Left-Curtain coupling of Beiglb"ock and Juillet. In the multiperiod case, the bivariate marginals for dates (0,t) are of Left-Curtain type, if and only if μ0,dots,μn have a specific order property. The general analysis of the transport problem also gives rise to a strong duality result and a description of its polar sets. Finally, we study a variant where the intermediate marginals μ1,dots,μn-1 are not prescribed.