We study the causal optimal transport problem between laws of stochastic processes.
Causality here means to add a physically relevant time constraint into the classical optimal
transport problem. Intuitively, one can say that the relationship between causal plans and
adapted processes is the same as that between classical transport plans (Kantorovich) and
transport maps (Monge).
Following the roadmap for classical transport, we tackle the questions of duality and
characterization of optimizers in the nite-dimensional case. Under appropriate assump-
tions, the causal analogue to the Brenier map of the classical case is identied as the
so-called Knothe-Rosenblatt rearrangement. Furthermore, we present a connection be-
tween transport-information inequalities and stochastic optimization.
As the next step, we show suitable extensions of the discrete-time results to the innite-
dimensional/continuous-time setup. Most importantly, the projections/limits arguments
enable to recover the equality between causal transports and the relative entropy for a
specic class of diusions. Finally, the developed techniques allow to provide a novel point
of view on various problems related to enlargement of ltrations and Girsanov theory in