Multiperiod Martingale Transport

Consider a multiperiod optimal transport problem where distributions ${\mu }_0,\dots ,{\mu }_n$ are prescribed and a transport corresponds to a scalar martingale $X$ with marginals $X_t\sim {\mu }_t$. 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 ${\mu }_0$ 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 ${\mu }_0,\dots ,{\mu }_n$ 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 ${\mu }_1,\dots ,{\mu }_{n-1}$ are not prescribed.