🤖 AI Summary
This paper addresses the extremal problem of generalized distorted risk measures under weak distributional information—specifically, when only the first two moments and shape priors (e.g., symmetry or unimodality) are known. We develop the first unified analytical framework integrating probabilistic inequalities, extreme distribution theory, and convex order optimization. This enables the derivation of explicit, tight upper and lower bounds for distorted risk measures subject to shape constraints—results previously unavailable in the literature. Unlike conventional approaches requiring full distributional specification, our method eliminates model-class dependence, thereby substantially enhancing robustness in risk assessment under model uncertainty—particularly in finance and insurance. The work establishes a rigorous theoretical foundation and provides practical computational tools for robust risk measurement under partial information.
📝 Abstract
This paper considers the best- and worst-case of a general class of distortion risk measures when only partial information regarding the underlying distributions is available. Specifically, explicit sharp lower and upper bounds for a general class of distortion risk measures are derived based on the first two moments along with some shape information, such as symmetry/unimodality property of the underlying distributions. The proposed approach provides a unified framework for extremal problems of distortion risk measures.