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Title:聽Robust Distortion Risk Measures

Abstract: In the presence of uncertainty, robustness of risk measures is of crucial importance. Distributional uncertainty may be accounted for by providing bounds on the values of a risk measure, so-called worst- and best-case risk measures. Worst (best)-case risk measures are determined as the maximal (minimal) value a risk measure can attain when the underlying distribution is unknown - usually up to its first moments. However, these bounds as well as the (worst- and best-case) distributions that attain the worst- and best-case values are too large, respectively 鈥渦nrealistic鈥�, to be practically relevant.

We provide sharp bounds for the class of distortion risk measures with constraints on the first two moments combined with a constraint on the Wasserstein distance with respect to a reference distribution. Adding the Wasserstein distance constraint, leads to significantly improved bounds and more 鈥渞ealistic鈥� worst-case distributions. Specifically, the worst-case distribution of the two most widely used risk measures, the Value-at-Risk and the Tail-Value-at-Risk, depend on the reference distribution and thus, are no longer two-point distributions.