  This paper develops asymptotics and approximations for ruin probabilities in
a multivariate risk setting. We consider a model in which the individual
reserve processes are driven by a common Markovian environmental process. We
subsequently consider a regime in which the claim arrival intensity and
transition rates of the environmental process are jointly sped up, and one in
which there is (with overwhelming probability) maximally one transition of the
environmental process in the time interval considered. The approximations are
extensively tested in a series of numerical experiments.
