Θεωρία χρεοκοπίας για μοντέλα κινδύνων με δύο κλάσεις στοχαστικών απαιτήσεων
Ruin theory for risk models with two classes of stochastic claims
Initially, Mathematical risk theory is obtained essentially at the beginning of the 20th century, with mathematicians Filip Lundberg and Harald Cramer incorporating the theory of contemplative requirements into actuarial science. Recently, Gerber and Shiu, in their work 'On the time value of ruin', gave unprecedented dimensions to mathematical risk theory. In particular, they managed to incorporate all the risk measures of an insurance institution into a single function, the expected discounted penalty function. The main hypothesis in the above models is the independence between the time of occurrence of the risks and the amount of damage resulting from their occurrence. In fact, the dependency between the time of occurrence of risks and the magnitude of damage is not fully reflected. In this thesis, we will deal with the study of Gerber-Shiu's expected discounted penalty function, where the contemplative process of total compensation is produced by two classes of risk. In the first chapter, we will first make an introduction, introducing risk measures, the contemplative surplus procedure, and also giving the definition of the expected penalty discount function. We will then give the description of the same model, this time by designing it for two classes of stochastic claims. In the second chapter, we deal with the assumption that claim number procedures are independent Poisson procedures and generalized Erlang(𝑛), respectively. We will then give results for the Gerber-Shiu function, for two classes of stochastic requirements. In the third chapter, we consider both claim number procedures to be renewal procedures, with phase-type claim times. We will re-product whole-differential equations, as well as roots of the Lundberg equation, adapted to our hypothesis. In the fourth chapter, we will deal with the study of the maximum surplus, just before bankruptcy. Our study will also be done in two categories of risk model, where both procedures have claim times that follow phase-type distribution. Finally, the fifth chapter provides a study of stochastic premiums, for two classes of stochastic requirements, and gives relevant results for Laplace transformations of expected penalty discount functions.