  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.
