On Kusuoka Representation of Law Invariant Risk Measures

In this paper we discuss representations of law invariant coherent risk measures in a form of integrals of the average value-at-risk measures. We show that such an integral representation exists iff the dual set of the considered risk measure is generated by one of its elements, and this representation is uniquely defined. On the other hand, representation of risk measures as a maximum of such integral forms is not unique. The suggested approach gives a constructive way for writing such representations.