  In this paper, we establish a large deviations principle for a multivariate
compound process induced by a multivariate Hawkes process with random marks.
Our proof hinges on showing essential smoothness of the limiting cumulant of
the multivariate compound process, resolving the inherent complication that
this cumulant is implicitly characterized through a fixed-point representation.
We employ the large deviations principle to derive logarithmic asymptotic
results on the marginal ruin probabilities of the associated multivariate risk
process. We also show how to conduct rare event simulation in this multivariate
setting using importance sampling and prove the asymptotic efficiency of our
importance sampling based estimator. The paper is concluded with a systematic
assessment of the performance of our rare event simulation procedure.
