Renewal Risk Model, Bankruptcy Issues And Bonus Issues,

With the vigorous development of the insurance industry and its promising prospects, more and more experts and scholars are focusing their attention on the study of this field of which the ruin theory-the core content of the risk theory is a very important research direction. In addition, due to the increasingly competitive insurance industry as well as people's gradually higher understanding level of insurance products, insurance products with dividend have entered people's real life and begun to attract their attention. At present, the ruin theory, the constant dividend strategy and the threshold dividend strategy of the classical risk model and generalized Erlang(n) risk model have been studied. Under these circumstances, in order to make further research in the risk theory, using the famous expected discounted penalty function, this thesis firstly analyzes the ruin probability of a Sparre Andersen risk model in which the claim inter-arrival distribution is a mixture of an Erlang(n) distribution and an Erlang(m) distribution, then studies a risk model involving two independent classes of insurance risks with a threshold dividend strategy, we assumed that the two claim number processes are independent Poisson and generalized Erlang(2) processes, respectively.We consider the ruin probability of a Sparre Andersen risk model in which the claim inter-arrival distribution is a mixture of two Erlang distributions in Chapter 1. In Section 1.1, we give a brief introduction to this model and the definition of the ruin time, the ultimate ruin probability and the expected discounted penalty (Gerber-Shiu) functionφδ(u); Section 1.2 obtains a integro-differential equation aboutφδ(u); We give a generalized Lundberg's equation and prove this equation has exactly m roots with positive real parts in Section 1.3; In Section 1.4, we derive the Laplace transform and renewal equation of φδ(u), then we get the ultimate ruin probability in Section 1.5; finally, we consider an example.In Chapter 2, we consider a risk model involving two independent classes of insurance risks with a threshold dividend strategy, we assume that the two claim number processes are independent Poisson and generalized Erlang(2) processes, respectively. In Section 2.1, we give a brief introduction to this model; Two integro-diflFerential equations systems for the Gerber-Shiu discounted penalty functions are shown in Section 2.2; Section 2.3 derives two generalized renewal equations; Finally, explicit results are derived when the claims are exponentially distributed.