Nonconcave Stochastic Optimal Control in Finite Discrete Time Under Model Uncertainty

ABSTRACT In this article, we present a general framework for nonconcave robust stochastic control problems under model uncertainty in a discrete time finite horizon setting. Our framework allows to consider a variety of different path‐dependent ambiguity sets of probability measures comprising, as a natural example, the ambiguity set defined via Wasserstein balls around path‐dependent reference measures with path‐dependent radii, as well as parametric classes of probability distributions. We establish a dynamic programming principle, which allows to derive both optimal control and worst‐case measure by solving recursively a sequence of one‐step optimization problems. Moreover, we derive upper bounds for the difference of the values of the robust and nonrobust stochastic control problem in the Wasserstein uncertainty and parameter uncertainty case. As a concrete application, we study the robust hedging problem of financial derivatives under an asymmetric (and nonconvex) loss function accounting for different preferences of sell and buy side when it comes to the hedging of financial derivatives. As our entirely data‐driven ambiguity set of probability measures, we consider Wasserstein balls around the empirical measure derived from real financial data. We demonstrate that during adverse scenarios such as a financial crisis, our robust approach outperforms typical model‐based hedging strategies, such as the classical Delta‐hedging strategy as well as the hedging strategy obtained in the nonrobust setting with respect to the empirical measure and therefore overcomes the problem of model misspecification in such critical periods.