Ludger Rüschendorf

Abstract:

Generalized Hoeffding-Fréchet functionals aiming at describing the possible influence of dependence on functionals of a statistical experiment, where the marginal distributions are fixed, have a long history. The problem of mass transportation can be seen as a particular case of this problem with two marginals and a linear functional induced by a distance. In the first part we give a review of some basic developments of this topic. We also describe several results for the solution for nonlinear functionals in the context of the analysis of worst case risk distributions. We show that these problems can be reduced to a variational problem and the solution of a finite class of (linear) mass transportation problems.

In the second part we review several approaches to improve risk bounds for aggregated portfolios of risks based on marginal information only. This endevour is motivated by the fact that the dependence uncertainty on the aggregated risks based on marginal information only is typically too wide to be acceptable in applications. Several methods have been developed in recent years to include structural and partial dependence information in order to reduce the model uncertainty. These include higher order marginals (method of reduced bounds), global variance or higher order moment bounds, partial positive or negative dependence restrictions and structural information given by common risk factors (partially specified risk factor models) or given by models with subgroup structures. Also an effective two-sided variant of the method of improved standard bounds has been developed.

The third part is devoted to some recent more detailed ordering results w.r.t. dependence orderings of relevant risk models making essential use of structural properties (like subgroup structure, graph structure or factor models) and on dependence properties of the models. The dependence structure of these models is given by *-products of copulas. Comparison results for *-products then allow to derive (sharp) risk bounds in various subclasses of risk models induced by additional restrictions.

Several applications show that these improved risk bounds may lead to results acceptable in praxis.