Computing the volume of an high-dimensional semi-unsupervised Hierarchical copula

We propose an algorithm for the computation of the volume of a multivariate copula function (and the probability distribution of the counting variable linked to this multidimensional copula function), which is very complex for large dimensions. As is common practice for large dimensional problem, we restrict ourselves to positive orthant dependence and we construct a Hierarchical copula which describes the joint distribution of random variables accounting for dependence among them. This approach approximates a multivariate distribution function of heterogenous variables with a distribution of a fixed number of homogenous clusters, organized through a semi-unsupervised clustering method. These clusters, representing the second-level sectors of hierarchical copula function, are characterized by an into-sector dependence parameter determined by a method which is very similar to the Diversity Score method. The algorithm, implemented in MatLab™ code, is particularly efficient allowing us to treat cases with a large number of variables, as can be seen in our scalability analysis. As an application, we study the problem of valuing the risk exposure of an insurance company, given the marginals i.e. the risks of each policy.