A note on robust convex risk measures

In this paper, we refine and generalize closed forms for worst-case law invariant convex risk measures with uncertainty sets based on: i) closed balls under $p$-norms and Wasserstein distance; and ii) moment constraints involving mean and variance. We also characterize the argmax of the worst-case problem in both settings. From such general results, we illustrate our framework by developing explicit closed forms for concrete examples of convex risk measures. Furthermore, we use extensive numerical simulations in order to assess the impact of robustness on capital determination and portfolio optimization.