Το κλασικό μοντέλο της θεωρίας κινδύνου με απαιτήσεις που εμφανίζουν χρονική υστέρηση
The classical model with of risk theory delayed claims
KeywordsΔιαχείριση κινδύνου ; Συναρτήσεις ; Εξισώσεις ; Στοχαστικές διαδικασίες ; Στατιστική ανάλυση
This thesis deals with the study of an extention to the classical compound Poisson risk model, in which two kinds of dependend claims are incorporated. Namely, main claims and by-claims are defined, where every by-claim is induced by the main claim and may be delayed for one time period with certain probability. The first chapter is an introductory section. A general description of the classical risk model and its surplus process is given. Also, the expected discounted penalty function of Gerber-Shiu, with and without dividend strategies, and several of its cases are also provided. In chapert two an integro-differential equation system for the Gerber-Shiu expected discounted penalty functions is derived and solved, using Laplace transforms, by proving that the Gerner-Shiu function satisfies some defective renewal equation. An exact representation for the solution of this equation is derived through an associated compound geometric distribution, and an analytic expression for this quantity is given when both the main claim and the by-claim amounts are exponentially distributed. Results for various ruin measures are provided, and a numerical example is given to clarify the proposed methodology. In chapter three, we consider an extention to the risk process studied in chapter two, perturbed by diffusion. An analytic expression for the solution of the involved integro-differential equation system for the Gerber-Shiu functions is given, when both the main claim and the by-claim amounts belong to the rational family of distributions. Numerical results are also provided. In chapter four, the same risk model of the previous chapter is considered in the presence of a multi-layer dividend strategy. A system of integro-differential equations for the expected discounted penalty function depending on the current surplus level, with certain initial and boundary conditions is obtained. To solve this, we derive a general solution to a certain second order integro-differential equation system. This solution is obtained by transforming this system to a Voltera-type system of integral equation of second kind, which is solved by using Laplace transforms provided an explicit expression for the Gerber-Shiu functions depending on the current surplus level. A numerical example is given to illustrate the applicability of the results. Finally, the appendix provides the definition and relevant properties of Laplace transform for a given function and a probability distribution function. In addition, lists several properties of Dickson-Hipp operator used in this study.