The robustness of risk measures to changes in underlying loss distributions (distributional uncertainty) is of crucial importance in making well‐informed decisions. In this paper, we quantify, for the class of distortion risk measures with an absolutely continuous distortion function, its robustness to distributional uncertainty by deriving its largest (smallest) value when the underlying loss distribution has a known mean and variance and, furthermore, lies within a ball—specified through the Wasserstein distance—around a reference distribution. We employ the technique of isotonic projections to provide for these distortion risk measures a complete characterization of sharp bounds on their value, and we obtain quasi‐explicit bounds in the case of Value‐at‐Risk and Range‐Value‐at‐Risk. We extend our results to account for uncertainty in the first two moments and provide applications to portfolio optimization and to model risk assessment.

We present a reinforcement learning (RL) approach for robust optimization of risk-aware performance criteria. To allow agents to express a wide variety of risk-reward profiles, we assess the value ...

We study the problem of active portfolio management where an investor aims to outperform a benchmark strategy’s risk profile while not deviating too far from it. Specifically, an investor considers alternative strategies whose terminal wealth lies within a Wasserstein ball surrounding a benchmark’s terminal wealth—being distributionally close—and that have a specified dependence/copula tying state-by-state outcomes to it. The investor then chooses the alternative strategy that minimizes a distortion risk measure of terminal wealth. In a general complete market model, we prove that an optimal dynamic strategy exists and provide its characterization through the notion of isotonic projections. We further propose a simulation approach to calculate the optimal strategy’s terminal wealth, making our approach applicable to a wide range of market models. Finally, we illustrate how investors with different copula and risk preferences invest and improve upon the benchmark using the Tail Value-at-Risk, inverse S-shaped, and lower- and upper-tail distortion risk measures as examples. We find that investors’ optimal terminal wealth distribution has larger probability masses in regions that reduce their risk measure relative to the benchmark while preserving the benchmark’s structure.