Consider a multiperiod optimal transport problem where distributions µ 0 , . . . , µ n are prescribed and a transport corresponds to a scalar martingale X with marginals X t ∼ µ 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 µ 0 is atomless, but not in general. In the one-period case n = 1, these transports reduce to the Left-Curtain coupling of Beiglböck and Juillet. In the multiperiod case, the bivariate marginals for dates (0, t) are of Left-Curtain type, if and only if µ 0 , . . . , µ 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 µ 1 , . . . , µ n−1 are not prescribed.

Two probability distributions µ and ν in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are allowed as transports. Much like the Hoeffding-Fréchet coupling of classical transport and its symmetric counterpart, the antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward (or cost) functions, and through no-crossing conditions on their supports; however, our two couplings have asymmetric geometries. Remarkably, supermartingale optimal transport decomposes into classical and martingale transport in several ways.

Abstract. We revisit Kellerer's Theorem, that is, we show that for a family of real probability distributions (µt) t∈ [0,1] which increases in convex order there exists a Markov martingaleTo establish the result, we observe that the set of martingale measures with given marginals carries a natural compact Polish topology. Based on a particular property of the martingale coupling associated to Root's embedding this allows for a relatively concise proof of Kellerer's theorem.We emphasize that many of our arguments are borrowed from Kellerer [12], Lowther [14], and Hirsch-Roynette-Profeta-Yor [5,6].

We study the problem of stopping a Brownian motion at a given distribution \nu while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set \mathcal{T}(\nu) of stopping times embedding \nu is weakly dense in the set \mathcal{R}(\nu) of randomized embeddings. In particular, the optimal Skorokhod embedding problem over \mathcal{T}(\nu) has the same value as the relaxed one over \mathcal{R}(\nu) when the reward function is semicontinuous, which parallels a fundamental result about Monge maps and Kantorovich couplings in optimal transport. A second part studies the dual optimization in the sense of linear programming. While existence of a dual solution failed in previous formulations, we introduce a relaxation of the dual problem that exploits a novel compactness property and yields existence of solutions as well as absence of a duality gap, even for irregular reward functions. This leads to a monotonicity principle which complements the key theorem of Beiglböck, Cox and Huesmann [Optimal transport and Skorokhod embedding, Invent. Math. 208, 327–400 (2017)]. We show that these results can be applied to characterize the geometry of optimal embeddings through a variational condition.

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