Causal optimal transport and the related adapted Wasserstein distance have recently been popularized as a more appropriate alternative to the classical Wasserstein distance in the context of stochastic analysis and mathematical finance. In this talk, we establish some interesting consequences of causality for transports on the space of continuous functions between the laws of stochastic differential equations.

We characterize bicausal transport plans and maps between the laws of stochastic differential equations and provide some applications. Time permitting, we discuss about the stability and optimality of the (bi)causal transport problem.

This is a joint work with Rama Cont.