**2009, Vol. 19, Issue 1****, an ****item ****5**

**HEILPERN S., Probability of ruin for a dependent, two-dimensional Poisson
process**

A two-dimensional, dependent Poisson risk process is investigated in the paper. Claims are divided into two classes. Within each class claims have the same distribution, but claims belonging to different classes can have different distributions and the corresponding counting processes can be dependent. This dependence is induced by a common factor. Three models of ruin based on different conditions for the time of ruin are investigated: the first time of ruin, the time of ruin in both classes and the time of ruin for the sum of the processes. The probabilities of ruin and the influence of the degree of dependence on the probability of ruin are studied for each model. Examples where the claims have exponential distributions are investigated. The method of simulation is used for the two first models. A formula based on phase-type distributions is used for the last model.

Keywords: Poisson process, claim, model of ruin, simulation, phase-type distribution