Abstract

We provide a dual characterisation of the weak∗^*-closure of a finite sum of cones in L∞L^\infty adapted to a discrete time filtration Ft\cF_t: the ttht^{th} cone in the sum contains bounded random variables that are Ft\cF_t-measurable. Hence we obtain a generalisation of F. Delbaen's m-stability condition [The structure of m-stable sets and in particular of the set of risk neutral measures, in: In Memoriam Paul-Andr{é} Meyer, Springer, Berlin et al. (2006) 215–258] for the problem of reserving in a collection of numéraires V, called 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: Thinking coherently, Risk 10 (1997) 68–71; Coherent measures of risk, Math. Finance 9(3) (1999) 203–228]. We also prove that V-m-stability is equivalent to time-consistency when reserving in portfolios of V, which is of particular interest to insurers.

Author information

Contact details are reproduced from the original publication and may be historical.

Abdelkarem Berkaoui

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

berkaoui@yahoo.fr

Keywords

  • Coherent risk measures
  • m-stability
  • time-consistency
  • Fatou property
  • reserving
  • hedging
  • representation
  • pricing mechanism
  • average value at risk

Mathematics Subject Classification

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

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Suggested citation

S. Jacka, S. Armstrong, A. Berkaoui. “On Representing and Hedging Claims for Coherent Risk Measures.” Journal of Convex Analysis 26 (2019), No. 1, 245–267. https://doi.org/10.68381/jca26014

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.