Arthur Chiragiev, Zinoviy Landsman Multivariate Families with Mixture Dependence

In this thesis we consider the problem of capital allocation, based on tail conditional expectation (TCE), for the class of the dependent multivariate family of distributions that essentially generalizes the classical multivariate Pareto distribution. This class can be obtained from independent exponential distributions, by a mixture of their common scale parameter. The distribution of mixture parameter belongs to the general class of distributions and, in particular, to the rich class of the exponential dispersion family (EDF). Special attention is paid to the important subclass of EDF, Tweedie family. We show that TCE-based portfolio allocation for the considered multivariate dependency structure can be represented by the tool of divided differences, actually known in numerical analysis. The results are illustrated with examples of multivariate Pareto, Weibull, and other distributions.