Research Project: The Wasserstein distance has become a major tool in modern probability theory. Both the topological as well as the geometrical structure that it gives the space of probability measures are widely applied and celebrated.
However, utilizing the Wasserstein distance for distributions of time series often behaves oddly: barycenters of martingale measures do not remain martingales, basic optimization problems like optimal stopping problems are not continuous, etc. The reason is that the Wasserstein distance is indifferent to the underlying information structure of a time series. Recently, an adapted version of the Wasserstein distance has been introduced which does incorporate the underlying information structure. Among other desirable properties, this adapted Wasserstein distance remedies the above mentioned oddities.

This project aims to quantify certain aspects of the adapted Wasserstein distance, like stability of optimization problems in mathematical finance, and further make the corresponding calculations tractable numerically. Qualitative notions of stability, like continuity of optimal hedging problems or optimal stopping, have recently been obtained in a 2020 paper by Backhoff, Bartl, Beiglböck and Eder from the University of Vienna. Beyond these qualitative notions, we aim to calculate certain steepest descent directions and derivative values explicitly. Methodologically, this builds on the geometric structure of the space of stochastic processes under the adapted Wasserstein distance, which has been made tangible by Bartl, Beiglböck and Pammer (2021). Thus, making general geometrical calculations within this space (like interpolation, barycenters, etc.) tractable numerically is a key goal of the project. The step towards steepest descent directions is in relation to distributionally robust optimization, and here we aim at numerically suitable reformulations, which builds on established analogues for the normal Wasserstein distance.