Capturing non-exchangeable dependence in multivariate loss processes with nested Archimedean Levy copulas

The class of spectrally positive Levy processes is a frequent choice for modelling loss processes in areas such as insurance or operational risk. Dependence between such processes (e. g. between different lines of business) can be modelled with Levy copulas. This approach is a parsimonious, efficient and flexible method which provides many of the advantages akin to distributional copulas for random variables. Literature on Levy copulas seems to have primarily focussed on bivariate processes. When multivariate settings are considered, these usually exhibit an exchangeable dependence structure (whereby all subset of the processes have an identical marginal Levy copula). In reality, losses are not always associated in an identical way, and models allowing for non-exchangeable dependence patterns are needed. In this paper, we present an approach which enables the development of such models. Inspired by ideas and techniques from the distributional copula literature we investigate the procedure of nesting Archimedean Levy copulas. We provide a detailed analysis of this construction, and derive conditions under which valid multivariate (nested) Levy copulas are obtained. Our results are discussed and illustrated, notably with an example of model fitting to data.