Wasserstein distance induces a natural Riemannian structure for the probabilities on the Euclidean space. This insight of optimal transport theory is fundamental for tremendous applications in various fields of pure and applied mathematics. We believe that an appropriate dynamic variant of optimal transport, adapted transport, can play a similar role for stochastic processes. By imposing a causality constraint on couplings, the flow of information which is encoded in the filtration is adequately taken into account. Notably, the resulting adapted Wasserstein distance is a geodesic distance for stochastic processes and suitable for sensitivity analysis in finance as well as for dynamic stochastic optimization problems. In this talk we will explore the connection of adapted transport with a robust hedging problem, along with further recent developments.

The talk is based on joint works with Beatrice Acciaio, Daniel Bartl, Mathias Beiglbo ̈ck, Stephan Eckstein and Daniel Krˇsek.