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Journal of Convex Analysis 26 (2019), No. 1, [final page numbers not yet available]
Copyright Heldermann Verlag 2019

On Representing and Hedging Claims for Coherent Risk Measures

Saul Jacka
Dept. of Statistics, University of Warwick, Coventry CV4 7AL, United Kingdom

Seb Armstrong
Dept. of Statistics, University of Warwick, Coventry CV4 7AL, United Kingdom

Abdelkarem Berkaoui
College of Sciences, Al-Imam Mohammed Ibn Saud Islamic University, P. O. Box 84880, Riyadh 11681, Saudi Arabia


\def\cF{\mathcal{F}} We provide a dual characterisation of the weak$^*$-closure of a finite sum of cones in $L^\infty$ adapted to a discrete time filtration $\cF_t$: the $t^{th}$ cone in the sum contains bounded random variables that are $\cF_t$-measurable. Hence we obtain a generalisation of F. Delbaen's m-stability condition [{\it The structure of m-stable sets and in particular of the set of risk neutral measures}, in: In Memoriam Paul-Andr{\'e} Meyer, Springer, Berlin et al. (2006) 215--258] for the problem of reserving in a collection of num\'eraires {\bf V}, called {\bf V}-m-stability, provided these cones arise from acceptance sets of a dynamic coherent measure of risk [see P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath: {\it Thinking coherently}, Risk 10 (1997) 68--71; {\it Coherent measures of risk}, Math. Finance 9(3) (1999) 203--228]. We also prove that {\bf V}-m-stability is equivalent to time-consistency when reserving in portfolios of {\bf V}, which is of particular interest to insurers.

Keywords: Coherent risk measures, m-stability, time-consistency, Fatou property, reserving, hedging, representation, pricing mechanism, average value at risk.

MSC: 91B24, 46N10, 91B30, 46E30, 91G80, 60E05, 60G99, 90C48.

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