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.
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