Ανάλυση δισδιάστατων στοχαστικών διαδικασιών πλεονάσματος στη θεωρία χρεοκοπίας
This diploma aims to study proportional reinsurance, dynamic proportional reinsurance, and dividend barrier strategies in the light of a two-dimensional ruin model. As an insurance company in one of its portfolios may have more than one risk class, we will expand the classic ruin model to more dimensions, with each dimension describing a risk class. Also in many cases, the risk classes (sub portfolios) can be linked together, so we will add another stochastic process to describe their interaction, in particular. In Chapter 1 we will recapitulate the classic model of risk theory by showing some basic results for Poisson's stochastic processes, the rejuvenation equation, the Gerber-Shiu function as well as some dams, and the Laplace transformation for the generalized Erlang process. In Chapter 2 we will begin by extending the classical model to m-dimensions by giving the corresponding function of Gerber-Shiu. In addition, we will specialize it in the two-dimensional, which is the main purpose of this diploma thesis, with some results for the phase-type distributions for some applications. In Chapter 3 we will give a lightweight different model that applies to dynamic proportional reinsurance and proportional reinsurance as well as some key risks for the probability of ruin and Laplace transformation of joint ruin time. Finally, in chapter 4, a two-dimensional risk process is considered in which each individual class applies a dividend barrier strategy. The insurance portfolios of the insurer are related as they are subject to the common shocks that cause the claims. In order to analyse the expected discounted dividends up to the common time of ruin of the two-dimensional process, it is proposed with a distinct temporal counterpart of the model and a two-dimensional extension of the Dickson-Waters discretization (Dickson and Waters (1991)) is applied using a two- Panjer (Walhin and Paris (2000)). Detailed numerical examples under different dependencies through common shocks, probability theories (Copulas) and proportional reinsurance are taken into account, and applications are given to optimal problems in reinsurance, main distribution and dividends. It is also clarified that the optimal pair of dividend bars maximizing the dividend function depends on the original surplus level. Finally, we propose the type of dividend strategy.