Causal transport plans and their Monge–Kantorovich problems

This paper investigates causal optimal transport problems. Within this framework, primal attainments and dual formulations are obtained under standard hypothesis, for the related variational problems. Causal transport plans are intrinsically related to martingales by a preserving property. Specific concretizations yield primal problems equivalent to several classical problems of stochastic control, and of stochastic calculus; trivial filtrations yield usual problems of optimal transport.