Multivariate risk measures as quasiconvex compositions

We consider multivariate risk measures that are defined as compositions of set-valued functions. Such class of risk measures is rich enough to cover many examples of systemic risk measures studied recently as well as... [ view full abstract ]

We consider multivariate risk measures that are defined as compositions of set-valued functions. Such class of risk measures is rich enough to cover many examples of systemic risk measures studied recently as well as their scalarizations. We pay special attention to properties of the constituent set-valued functions that guarantee quasiconvexity of the composite risk measure. The so-called natural quasiconvexity property, an old but not so well-known property between convexity and quasiconvexity, plays a key role in the study of these risk measures. Our main results provide dual representations for compositions in terms of the dual representations of the constituents.