On the modelling of multivariate counts with Cox processes and dependent shot noise intensities

In this paper, we develop a method to model and estimate several, _dependent_ count processes, using granular data. Specifically, we develop a multivariate Cox process with shot noise intensities to jointly model the arrival process of counts (e.g. insurance claims). The dependency structure is introduced via multivariate shot noise _intensity_ processes which are connected with the help of L\'evy copulas. In aggregate, our approach allows for (i) over-dispersion and auto-correlation within each line of business; (ii) realistic features involving time-varying, known covariates; and (iii) parsimonious dependence between processes without requiring simultaneous primary (e.g. accidents) events. The explicit incorporation of time-varying, known covariates can accommodate characteristics of real data and hence facilitate implementation in practice. In an insurance context, these could be changes in policy volumes over time, as well as seasonality patterns and trends, which may explain some of the relationship (dependence) between multiple claims processes, or at least help tease out those relationships. Finally, we develop a filtering algorithm based on the reversible-jump Markov Chain Monte Carlo (RJMCMC) method to estimate the latent stochastic intensities and illustrate model calibration using real data from the AUSI data set.