Μοντέλα κινδύνου με στοχαστικά ασφάλιστρα, εξαρτήσεις και στρατηγικές μερίσματος

Abstract

This thesis generalizes the classical risk model in which the total premium income paid by customers follows a linear function of time with a constant positive premium rate. A more realistic model taking into account the uncertainty of customer arrivals is the one with stochastic premiums. The current work studies various models for the surplus process with the amounts of individual premiums being random, as well as some dependence structures between individual claim size amounts, individual premium size amounts, inter-claim times and inter-premium times. Various risk measures are considered, such as the ruin probability or the ruin time, through the analysis of the expected discounted penalty function, also known as Gerber-Shiu function. For some of the models under consideration, the distribution of dividend payments to shareholders is additionally studied, considering for the respective stochastic surplus process of the portfolio various dividend payment strategies. Chapter 1 is an introductory section which gives basic concepts from ruin theory, a brief description of the classical compound Poisson model, known as the Cramer-Lundberg model, the expected discounted penalty function, and gives the solution of the defective renewal equation for the Gerber-Shiu function. In Chapter 2 two risk models with stochastic premiums without dependencies are examined. In the first model the aggregate premium amount is equal to the number of insured persons which is described by a Poisson process, while in the second model premiums and claims occur in the time according to compound Poisson processes. Chapter 3 considers two risk models with stochastic premiums and dependencies. The case in which the size of a claim controls the distributions of the time until the arrival of the next claim and individual premium sizes is first examined, while in the second case under consideration the time between successive claims determines the distributions of the next claim size and individual premium size. Chapter 4 examines two models with stochastic premiums, dependencies and dividend strategies. Namely, in the first model, under a constant dividend barrier, there is dependence between the claim sizes and their interarrival times, while in the second model, under a threshold dividend strategy, dependencies between premiums and inter-premium times as well as dependencies between claim sizes and their inter-claim times are considered. The Appendix provides the relevant mathematical tools (Laplace transform, Dickson-Hipp operator, etc.) used in this study. Mathematica was used to perform symbolic calculations, arithmetic operations, and graphs.