The martingale part in the semimartingale decomposition of a Brownian motion with respect to an enlargement of its filtration, is an anticipative mapping of the given Brownian motion. In analogy to optimal transport theory, we define causal transport plans in the context of enlargement of filtrations, as the Kantorovich counterparts of the aforementioned non-adapted mappings. We provide a necessary and sufficient condition for a Brownian motion to remain a semimartingale in an enlarged filtration, in terms of certain minimization problems over sets of causal transport plans. The latter are also used in order to give robust transport-based estimates for the value of having additional information, as well as model sensitivity with respect to the reference measure, for the classical stochastic optimization problems of utility maximization and optimal stopping. Our results have natural extensions to the case of general multidimensional continuous semimartingales.

Abstract. Loosely speaking, causal transport plans are a relaxation of adapted processes in the same sense as Kantorovich transport plans extend Monge-type transport maps. The corresponding causal version of the transport problem has recently been introduced by Lassalle. Working in a discrete time setup, we establish a dynamic programming principle that links the causal transport problem to the transport problem for general costs recently considered by Gozlan et al. Based on this recursive principle, we give conditions under which the celebrated Knothe-Rosenblatt rearrangement can be viewed as a causal analogue to the Brenier map. Moreover, these considerations provide transport-information inequalities for the nested distance between stochastic processes pioneered by Pflug and Pichler, and so serve to gauge the discrepancy between stochastic programs driven by different noise distributions.Key words. Optimal transport, causality, nested distance, general transport costs, Knothe-Rosenblatt rearrangement, transport inequalities.AMS subject classifications. 90C15,60G70,39B621. Introduction. In this article we consider the optimal transport problem between two discretetime stochastic processes under the so-called causality constraint, highlighted recently by the work of Lassalle in [Las15] in a more general setting. A transport plan between two processes is said to be causal if, from an observed trajectory of the first process, the "mass" can be split at each moment of time into the second process only based on the information available up to that time. It is illustrative to think of the deterministic case (i.e. when there is no splitting of mass); such a causal plan is then an actual mapping which is further adapted, and so the relationship between causal plans and adapted processes is the same as between classical transport plans (Kantorovich) and transport maps (Monge).The idea of imposing a "causality" constraint on a transport plan between laws of processes seems to go back to the Yamada-Watanabe criterion for stochastic differential equations [YW71]. Under the name "compatibility" the same type of constraint was introduced by Kurtz [Kur07]. In this article we will also link these objects to the notion of nested distance, whose systematic investigation was initiated by Pflug [Pfl09] and Pflug-Pichler [PP12, PP14, PP15], and had a precursor in the "Markov-constructions" studied by Rüschendorf [Rüs85]. Roughly, the nested distance is defined through a problem of optimal transport over plans which are bicausal, this notion being the symmetrized analogue of causality. Interestingly, [Rüs85] and [PP14] established a recursive formulation for the problem, and [PP12, PP14] further obtained a dual formulation for the nested distance. Moreover, Pflug-Pichler [PP12] applied these considerations to the practical problem of reducing the complexity of multistage stochastic programs, by showing that the difference between the optimal value of a program w.r.t. two different noise distributions is dominated by the nested dist...

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