Multivariate risk model of phase type

This paper is concerned with several types of ruin probabilities for a multivariate compound Poisson risk model, where the claim size vector follows a multivariate phase type distribution. First, an explicit representation for the convolution of a multivariate phase type distribution is derived, and then an explicit formula for the ruin probability that the total claim surplus exceeds the total initial reserve in infinite horizon is obtained. Furthermore, the effect of the dependence among various types of claims on this type of ruin probability is considered under the convex and supermodular orders. In addition, the bounds for other types of ruin probabilities are developed by utilizing the association of multivariate phase type distributions. Finally, some examples are presented to illustrate the results.