A correlated overflow model with a view towards applications in credit risk

In this paper we present and explicitly solve a specific multi-dimensional, correlated overflow problem. Due to the ordering of the underlying components in our model, explicit results are obtained for the probabilities under consideration; importantly, the results are not in terms of Laplace transforms. In our setting, each component behaves as a compound Poisson process with unit-sized upward jumps, decreased by a linear drift. The approach relies on a Beneš (1963) type argumentation, that is, the idea of partitioning the overflow event with respect to the last 'exceedance epoch'. This type of problems arises naturally in various branches of applied probability, and therefore has several applications. In this paper we point out two specific application areas: one in mathematical finance, and one in queueing. In the former, several 'obligors' (whose 'distances-to-default' are correlated, as is the case in practice) are considered, and it is quantified how likely it is that there are defaults before a given time T. In the latter, one is often interested in the workload or storage process; the results of this paper provide expressions for overflow probabilities in the context of specific coupled queuing systems.