On representing and hedging claims for coherent risk measures

Abstract

We provide a dual characterisation of the weak${}^{\ast }$-closure of a finite sum of cones in $L^{\infty }$ adapted to a discrete time filtration ${\mathcal{F}}_t$: the $t^{th}$ cone in the sum contains bounded random variables that are ${\mathcal{F}}_t$-measurable. Hence we obtain a generalisation of Delbaen's m-stability condition for the problem of reserving in a collection of numéraires $\mathbf{V}$, called $\mathbf{V}$-m-stability, provided these cones arise from acceptance sets of a dynamic coherent measure of risk. We also prove that $\mathbf{V}$-m-stability is equivalent to time-consistency when reserving in portfolios of $\mathbf{V}$, which is of particular interest to insurers.