  In this article we present a general framework for non-concave
distributionally robust stochastic control problems 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, 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. As a concrete application, we study the robust
hedging problem of a financial derivative under an asymmetric (and non-convex)
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 non-robust setting
with respect to the empirical measure and therefore overcomes the problem of
model misspecification in such critical periods.
