
Journal of Convex Analysis 26 (2019), No. 1, 245267 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 s.d.jacka@warwick.ac.uk Seb Armstrong Dept. of Statistics, University of Warwick, Coventry CV4 7AL, United Kingdom seb.armstrong@gmail.com Abdelkarem Berkaoui College of Sciences, AlImam Mohammed Ibn Saud Islamic University, P. O. Box 84880, Riyadh 11681, Saudi Arabia berkaoui@yahoo.fr [Abstractpdf] \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 mstability condition [{\it The structure of mstable sets and in particular of the set of risk neutral measures}, in: In Memoriam PaulAndr{\'e} Meyer, Springer, Berlin et al. (2006) 215258] for the problem of reserving in a collection of num\'eraires {\bf V}, called {\bf V}mstability, 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) 6871; {\it Coherent measures of risk}, Math. Finance 9(3) (1999) 203228]. We also prove that {\bf V}mstability is equivalent to timeconsistency when reserving in portfolios of {\bf V}, which is of particular interest to insurers. Keywords: Coherent risk measures, mstability, timeconsistency, Fatou property, reserving, hedging, representation, pricing mechanism, average value at risk. MSC: 91B24, 46N10, 91B30, 46E30, 91G80, 60E05, 60G99, 90C48. [ Fulltextpdf (191 KB)] for subscribers only. 