Refinements of Kusuoka representations on L ^∞

We study Kusuoka representations of law-invariant coherent risk measures on the space of bounded random variables, which says that any law-invariant coherent risk measure is the supremum of integrals of Average-Value-at-Risk measures. We refine this representation by showing that the supremum in Kusuoka representation is attained for some probability measure in the unit interval. Namely, we prove that any law-invariant coherent risk measure on the space of bounded random variables can be written as an integral of the Average-Value-at-Risk measures on the unit interval with respect to some probability measure. This representation gives a numerically constructive way to bound any law-invariant coherent risk measure on the space of essentially bounded random variables from above and below. The results are illustrated on specific law-invariant coherent risk measures along with numerical simulations.