The Bankruptcy Of The Types Of Risk Models

Risk theory as the most important part of Actuarial Analysis mainly consider risks of insurance companies. In 1903, Filip Lundberg first used the probability theory and stochastic process in research of risk theory and built the classic risk model. The Cramer e.t. generalized the work of Filip Lundberg and built a series of risk model. They abroad use the poisson process , markov process, martingale and renewal process in the research of risk theory. This dissertation is devoted to the development of ruin theory in three kinds of risk model. Concretely, three aspects of works are considered:In the first part, we use generalized Poisson process extending the classical Poisson risk model and build a new model- generalized Poisson model. The new model is built which the occurrence of both the claims and company income is described by generalized Poisson process. Then the "Lundberg inequality" and the formula for the ruin probability in the new model through martingale theory are concluded. We consider the generalized Poisson mode perturbed by diffusion subsequently. The martingale property this kind of surplus process are discussed. By a martingale method its ruin probability and "Lundberg inequality" are obtained.In the second part, renewal risk model and stationary renewal model are considered which claims occur as an Erlang(2) process. The integral-differential equations of ultimate ruin probability are obtained. In particular, we get the explicit solution of ultimate ruin probability in the case where the individual claim amount distribution is exponential. Finally, we calculate the ruin probability of a concrete example and analyze the variety of ruin probability affected by initial reserve and gross risk premium rate.In the last part, we put the generalized markov chain into research of risk theory. We consider the generalized poisson-renewal risk model and the discrete time insurance risk model with interest. The surplus at claims occurrence times is homogeneous Markov chain. By using the transition probability, we obtain several important ruin probabilities and distributions in the risk theory: the ruin probability in finite time, the ultimate ruin probability, the distribution of the ruin time ,the distribution of surplus immediately before ruin and the deficit at ruin.