Advances of matrix–analytic methods in risk modelling

This work is concerned with the study of matrix–analytic methods with novel applications to the area of risk theory.
First, we review some topics of Applied Probability such as phase–type distributions, matrix–exponential distributions, Markovian arrival processes, Rational arrival processes, fluid flow processes and risk models. With these tools in hand, we propose a method to approximate the probability of ruin of any Cramér–Lundberg process using the theory of phase–type distributions, providing an error bound for such an approximation. With the goal of studying risk models with dependencies, we construct a class of bivariate distributions with given phase–type–distributed marginals and given Pearson’s correlation coefficient, which are later used to define different kinds of dependent Sparre–Andersen processes. Later on, we give an explicit formula for the probability of Parisian and cumulative Parisian ruin for a class of risk processes which are based on the theory of fluid flow processes. Next, we study some excursion properties of spectrally negative Lévy processes whenever they are inspected at an independent matrix–exponential time. Finally, inspired by the generalisation of the Markovian arrival process to the Rational arrival process, we construct a novel generalisation of the fluid flow process without Brownian components and study its first passage probabilities.