This project aims at contributing to the mathematical foundations of risk management in finance and robust generative models. The first major focus is the design of dynamic stochastic models subject to domain and/or distribution constraints. Such models will play a key role for a robust representation of the underlying uncertainties and will allow for a better generation of risk scenarios thus improving the back testing abilities of financial risk management. The second key component addresses model risk and hedging by developing sensitivity analysis tools in the context of distributionally robust optimization. This project contributes new mathematical methods for optimizing systems of interacting agents which plays a crucial role in the analysis of financial risks. More specifically, the project investigates ergodic optimal semimartingale transport problems to model multidimensional stochastic processes under both domain (support) and distributional constraints. These results can be applied to diffusion-based generative methods in artificial intelligence and are expected to outperform standard score-based procedures. A second major part addresses model risk assessment and hedging through the so-called distributional robust optimization. This involves defining model deviations within small Wasserstein balls around a reference martingale model--a novel concept for quantifying and mitigating model risk. When volatility surface calibration gives access to marginals, the proje