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Minimal Representation of Insurance Prices

Alois Pichler (alois.pichler***at***univie.ac.at)
Alexander Shapiro (ashapiro***at***isye.gatech.edu)

Abstract: This paper addresses law invariant coherent risk measures and their Kusuoka representations. By elaborating the existence of a minimal representation we show that every Kusuoka representation can be reduced to its minimal representation. Uniqueness -- in a sense specified in the paper -- of the risk measure's Kusuoka representation is derived from this initial result. Further, stochastic order relations are employed to identify the minimal Kusuoka representation. It is shown that measures in the minimal representation are extremal with respect to the order relations. The tools are finally employed to provide the minimal representation for important practical examples. Although the Kusuoka representation is usually given only for nonatomic probability spaces, this presentation closes the gap to spaces with atoms.

Keywords: Law invariant coherent measure of risk, Fenchel-Moreau theorem, Kusuoka representation, stochastic order relations

Category 1: Applications -- OR and Management Sciences (Finance and Economics )

Category 2: Applications -- Science and Engineering (Statistics )

Category 3: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: University of Vienna, Universitaetsstrasse 5, 1010 Vienna and School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA 30332-0205

Download: [PDF]

Entry Submitted: 10/27/2012
Entry Accepted: 10/27/2012
Entry Last Modified: 02/02/2015

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