Estimating processes in adapted Wasserstein distance

A number of researchers have independently introduced topologies on the set of laws of stochastic processes that extend the usual weak topology. Depending on the respective scientific background this was motivated by applications and connections to various areas (e.g., Plug–Pichler—stochastic programming, Hellwig—game theory, Aldous—stability of optimal stopping, Hoover–Keisler—model theory). Remarkably, all these seemingly independent approaches define the same adapted weak topology in finite discrete time. Our first main result is to construct an adapted variant of the empirical measure that consistently estimates the laws of stochastic processes in full generality. A natural compatible metric for the adapted weak topology is the given by an adapted refinement of the Wasserstein distance, as established in the seminal works of Pflug–Pichler. Specifically, the adapted Wasserstein distance allows to control the error in stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etcetera in a Lipschitz fashion. The second main result of this article yields quantitative bounds for the convergence of the adapted empirical measure with respect to adapted Wasserstein distance. Surprisingly, we obtain virtually the same optimal rates and concentration results that are known for the classical empirical measure wrt. Wasserstein distance.