Walter Schachermeyer

University of Vienna

Law invariant risk measures on L^∞(R^d)

Abstract: Kusuoka (2001) has obtained explicit representation theorems for comonotone risk measures and, more generally, for law invariant risk measures. These theorems pertain, like most of the previous literature, to the case of scalar-valued risks. Jouini-Meddeb-Touzi (2004) and Burgert-Ruschendorf (2006) extended the notion of risk measures to the vector-valued case. Recently Ekeland-Galichon-Henry (2009) obtained extensions of the above theorems of Kusuoka to this setting. Their results were conned to the regular case. In general, Kusuoka's representation theorem for comonotone risk measures also involves a singular part. In the present work we give a full generalization of Kusuoka's theorems to the vector-valued case. The singular component turns out to have a richer structure than in the scalar case.

Joint work with Ivar Ekeland.