Séminaires DSA - Ruin probabilities in risk models with dependent and phase-type distributed claims and inter-arrivals

In this talk we consider risk-reserve processes where we allow for dependency between claims and inter--arrivals in several ways. Between claims we may either have a (deterministic) linear increase in the reserve or a stochastic development governed by a Brownian motion with a drift. Assuming claims and inter--arrivals being phase--type distributed, we develop methods for calculating explicit or exact ruin probabilities of different kind (classical infinite horizon, Parisian and finite--time Parisian) by representing the original risk--reserve process in terms of an equivalent fluid flow model with an optional Brownian component. We shall pay special attention to the construction and control of the (Pearson) correlation between claim sizes and inter-arrival times using a copula method based on order statistics for the construction of bivariate phase—type distributions. We provide a numerical study regarding the effect of the correlation and different scenarios.