Duality of Causal Distributionally Robust Optimization

Abstract. We study distributionally robust optimization (DRO) in a dynamic context, where model uncertainty is captured by penalizing potential models based on their adapted Wasserstein distance to a reference model. We consider both discrete- and continuous-time settings and derive dynamic duality formulas that reformulate the worst-case expectation as a tractable minimax problem. The inner maximization admits a recursive representation in discrete time, while in continuous time, it is characterized by a path-dependent Hamilton–Jacobi–Bellman equation. We further extend these duality results from the worst-case expectation to the worst-case expected shortfall, a nonlinear expectation. Finally, we apply this framework to optimal stopping problems in discrete time. We recast the original problem as a classical Wasserstein DRO on a nested space by introducing a novel relaxation that considers stopping times with respect to general filtrations.