Denseness of biadapted Monge mappings
Mathias Beiglböck, Gudmund Pammer, Stefan Schrott
TL;DR
The paper extends classical transport to the stochastic-process setting by enforcing temporal causality through biadapted mappings and bicausal couplings. It provides a representation result that every coupling can be obtained as a projection of a Monge coupling on an extended space and proves that, under absolute continuity of marginals, biadapted Monge couplings are dense in the bicausal Kantorovich set, bridging Monge and Kantorovich viewpoints for processes. The authors establish static and time-dependent denseness results, derive an adapted Wasserstein distance $ ext{AW}_p$, and discuss regularity assumptions and the strengths/limits of these adapted topologies. Together, these results yield a robust framework for process-aware optimal transport with stability under Doob decomposition, stochastic control, and finance applications. The work strengthens the theoretical foundation for temporally consistent transport and provides tools for practical computation via extended-space representations and dense approximations.
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.