Abstract: In this talk, we study the causal distributionally robust optimization (DRO) in both discrete- and continuous- time settings. The framework captures model uncertainty, with potential models penalized in function of their adapted Wasserstein distance to a given reference model. The strength of model uncertainty is parameterized via a penalization parameter, and we compute the first-order sensitivity of the value of causal DRO with respect to this parameter. Moreover, we investigate the case where a martingale constraint is imposed on the underlying model, as is the case for pricing measures in mathematical finance. We introduce different scaling regimes, which allow us to obtain the continuous-time sensitivities as nontrivial limits of their discrete-time counterparts.

Our proofs rely on novel methods. In particular, we introduce a pathwise Malliavin derivative, which agrees with its classical counterpart under the Wiener measure, and we extend the adjoint operator, the Skorokhod integral, to regular martingale integrators and show it satisfies a stochastic Fubini theorem.