turkmath.org

Set-valued risk measures are defined naturally in environments where it is necessary to measure the risk of a random vector. In this talk, we consider set-valued risk measures that are defined as compositions of two set-valued functions. This class of risk measures is rich enough to cover many examples of the systemic risk measures studied in the recent literature. We pay special attention to the properties of the constituent set-valued functions that guarantee the quasiconvexity of the composite risk measure. It turns out that 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. The main results provide dual representations for quasiconvex, naturally quasiconvex and convex compositions in terms of the dual functions of the constituents.