
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 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 (497 KB)] for subscribers only. 