A theory for combinations of risk measures

Marcelo Brutti Righi

Abstract

We study combinations of risk measures under no restrictive assumptions on the set of alternatives. We obtain and discuss results regarding the preservation of properties and acceptance sets for these combinations of risk measures. One main result is the representation of risk measures resulting from the properties of both alternative functionals and combination functions. We build on a dual representation for an arbitrary mixture of convex risk measures and obtain a penalty that recalls the notion of inf-convolution under theoretical measure integration. We develop results related to this specific context. We also explore features generated by our frameworks that are of interest in themselves, such as the preservation of continuity properties and the representation of worst-case risk measures.