  This article presents a new continuous-time modelling framework for
multivariate time series of counts which have an infinitely divisible marginal
distribution. The model is based on a mixed moving average process driven by
L\'{e}vy noise - called a trawl process - where the serial correlation and the
cross-sectional dependence are modelled independently of each other. Such
processes can exhibit short or long memory. We derive a stochastic simulation
algorithm and a statistical inference method for such processes. The new
methodology is then applied to high frequency financial data, where we
investigate the relationship between the number of limit order submissions and
deletions in a limit order book.
