From isotonic Banach functionals to coherent risk measures

Zbigniew Dudek¹

¹ Faculty of Mathematics and Information Science Warsaw University of Technology Pl. Politechniki 1 00-661 Warszawa, Poland

Applicationes Mathematicae, Tome 28 (2001) no. 4, pp. 427-436

Résumé

Coherent risk measures [ADEH], introduced to study both market and nonmarket risks, have four characteristic properties that lead to the term “coherent” present in their name. Coherent risk measures regarded as functionals on the space $L^{\infty }({\mit \Omega } , {\cal F},{\Bbb P})$ have been extensively studied [De] with respect to these four properties. In this paper we introduce CRM functionals, defined as isotonic Banach functionals [Al], and use them to characterize coherent risk measures on the space $L^{\infty }({\mit \Omega } ,{\cal F},{\Bbb P})$ as order opposites of CRM functionals. The characterization involves only three axioms and leaves room for a larger class of functionals that can be related to a larger class of possible risks. We show that every CRM functional, when restricted to constant functions, is represented by a convex real function on ${\Bbb R}$ which is linear for nonnegative and nonpositive arguments separately. Next, we show that those CRM functionals which are extensions of the map $ {\Bbb R}\ni t\mapsto \beta t\in {\Bbb R}$, with $\beta >0$, are represented as maxima over a set of positive linear extensions.

Détail

DOI : 10.4064/am28-4-4

Keywords: coherent risk measures adeh introduced study market nonmarket risks have characteristic properties lead term coherent present their name coherent risk measures regarded functionals space infty mit omega cal bbb have extensively studied respect these properties paper introduce crm functionals defined isotonic banach functionals characterize coherent risk measures space infty mit omega cal bbb order opposites crm functionals characterization involves only three axioms leaves room larger class functionals related larger class possible risks every crm functional restricted constant functions represented convex real function bbb which linear nonnegative nonpositive arguments separately those crm functionals which extensions map bbb mapsto beta bbb beta represented maxima set positive linear extensions