Higher cumulants for multivariate aggregate claim models

The joint aggregated claim models for the multivariate claims is of the fundamental importance for risk assessment in actuarial sciences. In fact, insurance claims usually arrive from contracts that are insuring against accidents of different nature but dependent in their occurrences. For example, it is natural to expect that weather conditions affect forest fires and draught in farms and thus inducing dependence for claim size arriving from such accidents. Dependencies can be present in the severities of the claims of different types as well as in the numbers of claims of different types occurring during particular types of accidents. Actuaries need tools that would facilitate the analysis of models that account for multivariate dependence structure.

The project aims at deriving theoretical results for higher cumulants for the aggregated multivariate claim models. Two important special cases of claim models are specifically considered. In the first one, multivariate skewed normal claims are considered and aggregated by a Poisson variable. The second case is dealing with multivariate asymmetric generalized Laplace and aggregation is made by a negative binomial variable. There is a well established relation between asymmetric Laplace motion and negative binomial process that corresponds to the invariance principle of the aggregating claims for the generalized asymmetric Laplace distribution. We explore this relation and provide multivariate continuous time version of the results. It is discussed how these results that deals only with dependence in the claim sizes can be used to obtain a formula for higher cumulants for more complex aggregate models of multivariate claims in which the dependence is also in the aggregating variables.