The Ruin Probabity On Compound Negative Binomial Risk Model

Risk theory is a hot study topic in the realm of modern actuarial science and mathematics, and ruin theory is the core of risk theory. The study of the risk theory started from the PhD paper of Flip Lundberg, who was a Swedish actuarial scientist, in 1903, he first brought up an important stochastic process, Poisson process, in his paper. Cramer developed Lundberg' work based on the mathematics, and formed the base theory of the classical risk theory. At present, the classical risk model has been deeply researched and many important theories have been educed, which have established a stable foundation for the further development of the risk theory. As the theory has many defects, many studiers have been improving it.This article has outlined the research and development of the risk theory, and summarized the negative binomial risk model in detail in the domestic and overseas present research situation as well as the constituent and the main results of the classics separate risk model. Based on this, in view of the gradually complex and specific situation on present actual insurance business, in this article I have proposed the mixed negative binomial risk model, and studied the bankrupt parameter of this model, so that the actual operation situation of insurance company can be reflected more truly and accurately, and it's helpful for the insurance company to make the overall plan arrangement. In all kinds of complicated issues of the insurance business, the insurer carries on the classification according to the risk certain characteristics, but the chit in the same kind still is inevitable to exist some kind of degree the non-homogeneity. Therefore, regarding the claim number of times model's description of the indentical kind of chit combination, firstly must determinate the claim number of times model of each chit in the definite policy combination, then in the basis identical kind of chit's non-homogeneity, determinate the parameter distribution rule in its model, finally describe the claim number of times model of this chit combination completely again.This article firstly introduced the survival probability,the final ruin probability of the compound negative binomial risk model, the survival probability in limited time; then made the insurance premium collection times and each time gathers in the negative binomial risk model be random variable, proposed the compound double negative binomial risk model, studied the digital characteristics of the earnings process in this model, the ruin probability in the limited time, final ruin probability and so on. In view of the gradual increases of the insurance types in the present insurance practice, I promoted the compound double negative binomial risk model further, proposed the compound mufti-negative binomial risk model of double insurance types, and studied the digital characteristics of earning process in this model, final ruin probability, Lundberg inequality and so on. In fact, in the process of the insurance company's operation, there are some random factors once for a while, which makes insurance company have some uncertain incomes or the disbursements, so I also studied the bankrupt problems of the double compound negative binomial risk model in double insurance types with disturbance items.Under the above each kind of model, all are obtained the corresponding earnings sequence nature, namely the earnings process is a steady independent increment process, which obtained the concrete expression form of ruin probability by recursive method and Martingale methodEspecially more importantly had found the upper boundary of ruin probability, that is the Lundberg inequalityIt is widely applied in the insurance system's risk analysis for its strong feasibility, which has the important theoritical and the practical significance!...