On convex risk measures on L p -spaces

Much of the recent literature on risk measures is concerned with essentially bounded risks in L (az). In this paper we investigate in detail continuity and representation properties of convex risk measures on L (p) spaces. This frame for risks is natural from the point of view of applications since risks are typically modelled by unbounded random variables. The various continuity properties of risk measures can be interpreted as robustness properties and are useful tools for approximations. As particular examples of risk measures on L (p) we discuss the expected shortfall and the shortfall risk. In the final part of the paper we consider the optimal risk allocation problem for L (p) risks.