Sensitivity of causal distributionally robust optimization

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. Strength of the penalty is controlled using a real-valued parameter which, in the special case of an indicator penalty, is simply the radius of the uncertainty ball. Our main results derive the first-order sensitivity of the value of causal DRO with respect to the penalization parameter, i.e., we compute the sensitivity to model uncertainty. 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. We illustrate our results with examples. The sensitivities are naturally expressed using optional projections of Malliavin derivatives. To establish our results we obtain several novel results which are of independent interest. In particular, we introduce pathwise Malliavin derivatives and show these extend the classical notion. We also establish a novel stochastic Fubini theorem.