In this talk, I will introduce the adapted Wasserstein (AW) distance — an extension of the classical Wasserstein distance to stochastic processes. It captures the filtration generated by the underlying processes and plays a fundamental role in the study of stochastic analysis and optimal control problems. I will present an explicit formula for the 2-AW distance between Gaussian processes and show that the synchronous coupling is optimal between real-valued fractional SDEs.

We then turn to applications in distributionally robust optimization (DRO) problems in a dynamic context. This framework addresses decision-making under model uncertainty by optimizing against the worst-case scenario, where the potential model lies in an adapted Wasserstein ball around a given reference model. I will discuss tractable reformulations of the worst-case performance via duality and sensitivity approaches. Both discrete- and continuous-time results will be included.