Coherent conditional measures of risk defined by the Choquet integral with respect to Hausdorff outer measure and stochastic independence in risk management

Abstract

Coherent conditional measures of risk are defined, in a metric space, by the Choquet integral with respect to Hausdorff outer measures; they allow the evaluation of a risk conditioned to a fractal set, that is a set with non-integer Hausdorff dimension. The notions of s -irrelevance and s -independence for risks defined on fractal sets are given to capture dependence. Sufficient conditions for s -irrelevance are given and random variables which are surjective and strictly monotone are proven to be s -dependent. • Coherent conditional measures of risk are defined by the Choquet integral with respect to Hausdorff outer measure. • A new definition of stochastic independence for random variables is given. • Independence of the indicator functions of two events does not imply independence of the indicator functions of their complements. • The results given in the paper could be used when financial market is studied by means fractal models.