The Wasserstein distance is an important instance of an optimal transport cost. Its numerous mathematical properties as well as applications to various fields such as mathematical finance and statistics have been well studied in recent years. The adapted Wasserstein distance extends this theory to laws of stochastic processes, making it particularly well suited for analyzing stochastic optimization problems in mathematical finance.

In this talk, I will present our study on the quantitative and structural aspects of adapted optimal transport. We provide quantitative comparisons between adapted and classical Wasserstein distances, characterizing compact sets in the adapted Wasserstein space. I will then discuss how the concentration of measure phenomenon is connected to transport-entropy inequalities in the adapted setting. These results yield explicit quantitative bounds and stability estimates that are useful for applications in mathematical finance. I will also discuss some ongoing and prospective research directions.

This talk is based on joint works with B.Acciaio, J.Blanchet, M.Larsson and J.Wiesel.