7. Multivariate Ruin

In this chapter we focus on a multivariate variant of the conventional Cramér-Lundberg model. Imposing an ordering condition on the individual net cumulative claim processes, it turns out that the distribution of the joint running maximum can be derived, which can be used to evaluate ruin probabilities in a multivariate context. We start by analyzing the bivariate case, to then extend the reasoning to the higher-dimensional setting. The method relies upon the Kolmogorov forward equations underlying the associated queueing process. The solution reveals a so-called quasi-product form structure. We also point out how the results from this section can be translated into corresponding results for tandem queueing networks. We conclude the chapter by deriving the corresponding multivariate Gerber-Shiu metrics (including ruin times, undershoots, and overshoots).