Sensitivity of causal distributionally robust optimization

TL;DR

This paper analyzes the sensitivity of causal distributionally robust optimization models to model uncertainty, deriving first-order sensitivities and exploring continuous-time limits, with applications in finance and novel theoretical contributions.

Contribution

It introduces the first-order sensitivity analysis of causal DRO with respect to penalization parameters, including continuous-time limits and new theoretical tools like pathwise Malliavin derivatives.

Findings

Derived first-order sensitivities of causal DRO to model uncertainty.

Established continuous-time sensitivity limits from discrete models.

Introduced pathwise Malliavin derivatives and a new stochastic Fubini theorem.

Abstract

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.