Stochastic orders for functional dependence

The project brings together current research in the fields of dependence measures, optimal transport, and the robustness of portfolio risks. The focus is on novel, order-based methods for analyzing dependencies between random variables. These methods contribute significantly to a better understanding of modern correlation concepts and dependence measures. Classical correlations describe the degree of linear dependence or the strength of a positive or negative relationship between two random variables. Recently developed correlation concepts, in contrast, capture the overall strength of dependence ranging from independence to complete dependence. A central goal of the project is to systematically study the behavior of such dependence measures. This involves analyzing the underlying structures of different models to derive mathematical theorems. To better reflect the complexity of real-world situations, researchers increasingly rely on robust classes of models. Typically, the focus is on either the average behavior or the worst- case scenario across a range of plausible models to obtain realistic or conservative evaluations. To quantify such scenarios, we analyze risks and the behavior of optimal transport plans. We expect similar structural properties that govern both the transport-based quantities and dependence measures. The developed methods have broad applications in economics, financial and risk mathematics, statistics, and computer science, particularly in image processing and artificial intelligence.