Κατανομές τύπου φάσεων και οι εφαρμογές τους στην αναλογιστική επιστήμη
Phase type distributions and their applications in actuarial science
KeywordsΚατανομές τύπου φάσεων ; Μαρκοβιανές αλυσίδες ; Μοντέλο συλλογικού κινδύνου ; Phase type renewal theory
In recent years the interest in Phase Type distributions has been greatly increased. The simplest examples of Phase Type distributions are convolutions of exponential distributions and models of finite mixtures. Given that discrete Phase Type distributions are dense in the class of distributions on ℕ0, it is then clear that continuous Phase Type distributions are dense in the class of distributions on ℝ+. This property makes them a very versatile modeling tool. The purpose of this thesis is to present Phase Type distributions and some of their applications in actuarial science. More specifically, in the first chapter, a brief presentation of Markov chains in both discrete and continuous time is given. In the sequel we introduce and study Phase Type distributions both in the discrete and continuous case, presenting a variety of examples and applications. In the second chapter we study the collective risk model in relation to Phase Type distributions. Using theoretical results, examples, and applications, we show that Phase Type distributions can be applied to this field of actuarial science. Finally, in the third chapter we deal with ruin theory models. The application of Phase Type distributions to ruin theory can be an additional tool for actuaries. Through applications and using appropriate theoretical results, we are able to find the probability of ruin using Phase Type distributions in some cases.