Μελέτη ενός ημι-Μαρκοβιανού μοντέλου για τη διαδικασία πλεονάσματος ενός ασφαλιστικού χαρτοφυλακίου
Λώλου, Νικολέττα - Μαρία Η.
SubjectΜαρκοβιανές ανελίξεις ; Διαχείριση κινδύνου -- Οικονομετρικά μοντέλα ; Διαχείριση κινδύνου -- Στατιστικές μέθοδοι ; Διαχείριση χαρτοφυλακίου
This thesis deals with the study of a semi - Markov model for the surplus process of a portfolio under a unified methodology. Some relevant random variables associated with the ruin probability are the time of ruin, that is, the time that the surplus will first get a negative value, the surplus just before ruin and the deficit at ruin. The joint study of these three characteristics gives more information about the behavior of the surplus process than studying each one separately. Gerber and Shiu analyzed simultaneously the behavior of the above¬mentioned measures through a particular by now classical discounted penalty function at ruin. The thesis has the following structure. The first chapter is an introductory section regarding the basic notions of the thesis. First, a general description of the surplus process is given. Second, the general semi-Markov model and its involved parameters and variables are defined. Finally, the expected discounted penalty function of the Gerber-Shiu and several of its cases are also provided. The second chapter displays that the expected discounted penalty function of Gerber-Shiu satisfies an integro-differential equation which can be approached by means of Laplace transforms. Results for various ruin measures are provided and also the asymptotic behavior of the penalty function for light-tailed claim size is discussed. The following chapters discuss several cases included in the general model. In chapter three, the results obtained by the analysis performed in chapter two are specified for the classical compound Poisson risk model. In the fourth chapter, the case of Sparre Andersen renewal risk model with generalized two-stage Erlang distributed interclaim time is analyzed. Chapter fifth considers another special case of Sparre Andersen renewal model, the one which intermediate risk occurrence times follow phase-type distribution with two phases. Chapter six focuses on a certain generalization of the classical run model, where the distribution of the time between two claim occurrences depends on the previous claim size. This dependence case is also contained in the studied model and is analyzed by the proposed unified framework. A numerical example is given to clarify the proposed methodology. Finally, the appendix provides the definition of Laplace transform for a given function and a distribution, and lists its relevant properties used in this study.