Research On Several Kinds Of Insurance And Risk Models

Insurance and risk problem is one of the important research contents of financial mathematics. There are many basic tools and important methods in dealing with these problems. In this thesis we will start from the basic classic risk model, by using the knowledge and the theory of stochastic process, we consider the ruin probability for a risk model with random income and other problems. These problems play an important role in in actuarial research.Because the classical risk process has excellent properties and after the development of more than one hundred years, the various actuarial results in the classical risk model are almost perfect. In the classical risk model, the only income of one company is assumed to be the premium it received. But in real practice, the insurance company may earn money from different ways, for example, they may invest in the stock market and other financial markets to make money. Besides, with the development of modern insurance, the insurance companies can start their own enterprise, for instance, bank and so on. This dissertation is devoted to introduce the random income to the risk model, that is, except the premium income (received at a constant rate, c, for instant), the insurer has random income. This is also the research direction of a lot of experts and scholars. Our research is carried out in two cases:the continuous case and the discrete case.In chapter2we induct the series expression of the joint distribution of time to ruins suplus immediately before ruin and the deficit at ruin for one class of risk model with random income.In chapter3we get the expression of the Gerber-Shiu penalty function of the risk model.We deduce one upper bound and one lower bound for the ruin probability when the distributions of the jumps are all common ones.In chapter4we consider the dividend problem with constant dividend barrier for a risk model with random income. In the research of the dividend problem, we mainly discuss the expectation of the total dividend payment. For the first time we obtain the Laplace transform of the first-exit time (first exit time from a bounded area). With the help of the result of the first-exit time we obtain the Laplace transform of the time to ruin and the expected dividend payment in the risk model.In chapter5, we consider the case of discrete time risk model. Concretly speaking, we first consider the Gerber-Shiu penalty function for the discrete time risk model, then we putforward the expection of the time to ruin and the finite time ruin probability. Besides we consider the dividend problem with constant dividend barrier, we obtain the recursion formula for the dividend payment and functions satisfied by the total utility of the dividend payment.In chapter6, we consider a class of more extensive process, the double-sided jumps renewal model with interest. By the application of the Martingale approach we obtain one upper bound for the ultimate ruin probability. As an aplication, we consider the upper bound of the ruin probability for the double-sided jumps classical risk model with interest. This is an extension of the classical risk model.In chapter7, we consider the dividend problem for the share-holding corporation. By the introduction of the controlled feedback process, we consider the new capital process (influenced by the dividend barrier) and some new resluts about the dividend problem.In chapter8, we consider the merge of the share-holding corporations. Through the comparison of the indexs:the ablity of resist risk(measured by the probablitity that the capital of the corporation below zero)> the earnings of the corporation(measured by the dividends given to the share holders) and others, we can judge whether the merger is good or not.In the last chapter, we consider the calculation of the capacity of the logistic center. By the introduction of the barrier strategy we get the optimum forcast of the capacity.