Ανάλυση του πλεονάσματος για ανανεωτικές στοχαστικές διαδικασίες ασφαλιστικών κινδύνων
Surplus analysis of renewal stochastic insurance risk processes
KeywordsΑναλογισμός ; Πλεόνασμα ; Συνάρτηση Gerber-Shiu ; Μοντέλο κινδύνου Poisson ; Χρόνος χρεοκοπίας ; Πιθανότητα χρεοκοπίας
The main study of this thesis is the surplus analysis of renewal stochastic insurance risk processes. We analyze important quantities such as the ruin probability, the time of ruin, the surplus prior to ruin, the deficit after ruin occurs. In Chapter 1 we proceed to a technical introduction of what follows. We present the Lagrange polynomial, the Dickson – Hipp operator, the defective renewal equation and we define the Erlang distribution and the Coxian distribution. In Chapter 2 the Gerber – Shiu function under the classical Poisson risk model is presented. Further results of this function combined with the defective renewal equation are given. An extension of the Gerber – Shiu function is given in Chapter 3, this time under the dependent Sparre – Andersen risk model In Chapter 4 we give further analysis to Erlang and Coxian distributions and we make use of their results to draw conclusions about the claim sizes and the interclaim times. In Chapter 5 the time of ruin distribution under the classical Poisson risk model is presented, while we also give the joint distribution of the time of ruin and the deficit after ruin occurs. In Chapter 6 we draw conclusions about estimating the aggregate claim costs and the number of claims until ruin.