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

• 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.

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.