In this paper, a particular class of matrix-exponential distributions is described, also with respect to its use in risk theory, namely phase-type distributions. Phase-type distributions have the important advantage of being suitable for approximating most of other distributions as well as being mathematically tractable.
After a review on phase-type distributions and their properties, it is illustrated a possible use in risk theory. Modelling both interarrival claim times and individual claim sizes with this class of distributions an explicit formula for the probability of ultimate ruin is given
