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Characterization, Robustness, and Aggregation of Signed Choquet Integrals

This article contains various results on a class of nonmonotone, law-invariant risk functionals called the signed Choquet integrals. A functional characterization via comonotonic additivity is established along with some theoretical properties, including six equivalent conditions for a signed Choquet integral to be convex. We proceed to address two practical issues currently popular in risk management, namely robustness (continuity) issues and risk aggregation with dependence uncertainty, for signed Choquet integrals. Our results generalize in several directions those in the literature of risk functionals. From the results obtained in this paper, we see that many profound and elegant mathematical results in the theory of risk measures hold for the general class of signed Choquet integrals; thus, they do not rely on the assumption of monotonicity.