Denseness of biadapted Monge mappings

Abstract

Adapted or causal transport theory aims to extend classical optimal transport from probability measures to stochastic processes. On a technical level, the novelty is to restrict to couplings which are bicausal, i.e. satisfy a property which reflects the temporal evolution of information in stochastic processes. We show that in the case of absolutely continuous marginals, the set of bicausal couplings is obtained precisely as the closure of the set of (bi-) adapted processes. That is, we obtain an analogue of the classical result on denseness of Monge couplings in the set of Kantorovich transport plans: bicausal transport plans represent the relaxation of adapted mappings in the same manner as Kantorovich transport plans are the appropriate relaxation of Monge-maps.