Causal transference plans and their Monge-Kantorovich problems

Abstract

This paper investigates causal optimal transportation problems, in the framework of two Polish spaces, both endowed with filtrations. 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. Within this framework, primal attainments and dual formulations are obtained, under standard hypothesis, for the related variational problems. These problems are intrinsically related to martingales. Finally, we investigate applications to stochastic frameworks. A straightforward equivalence between specific causal optimization problems, and problems of stochastic control, is obtained. Solutions to a class of stochastic differential equations are characterized, as optimum to specific causal Monge-Kantorovich problems ; the existence of a unique strong solution is related to corresponding Monge problems.