Some Problems Of Ruin Probability Under A Two-Dimensional Risk Model

In the insurance mathematics category, the risk theory is one of its important branches, it mainly handles the problems of the ruin probability of the stochastic risk model in insurance business. The research of ruin theory begins from Filip Lundberg, an actuary of Sweden, who proposed the classical risk model, his work is considered as the basis of ruin theory. On this basis, Harald Cramer and his school set up a relationship between general stochastic process and risk theory. Afterwards, renewal theory and martingale method, introduced by Feller and Gerber, respectively, have deepen the research of ruin theory. However, as the complexity of the reality background, it is not very appropriate to characterize the risk process of the insurance company with classical risk model, so in recent years, the genelization of the classical risk model has attracted many mathematicians and actuaries, also some good results have been got, the risk model has been improved gradually.Compared with one-dimensional classical risk model, it is no doubt that multi-dimenional risk model will reflect the operating of insurance company more accurately and objectively, so it has more actual significance. But, the ruin theory under multi-dimenional risk model is very complex, even in the two-dimensional case. Wai-Sum Chan and his partner has studied the problems of ruin probability of a two-dimensional risk model, and got some simple bounds of ruin probabilities. Although the all precise expressions haven't been obtained, their exciting results is regarded as the basis of multi-dimenional risk problem, this area still needs more scholars to research.This paper is divided to three sections, In Chapter 1, we review the results of ruin theory over the past hundred years. In Chapter 2, we mainly introduce the results on ruin probabilities in a two-dimensional risk model which is studied by Wai-Sum Chan and his partner, on the basis of their exciting results, we state a new ruin time and ruin probability, point out its significance and get its explicit expression. In Chapter 3, we consider a two-dimensional risk model under random interest force, also the dependency of claims are assumed, we disscuse the ruin probability in this condition.In Chapter 1, firstly, we introduce the assumptions of Lundberg-Cramer classical risk model, then give the specific expression as Where U ( t ) is the surplus of an insurance company at time t≥0, u is the initial surplus,c is the rate at which the premiums are received, X i is the size of the i th claim, N ( t ) is the number of claims between time 0 and t .By researching, Lundberg and Cramer got these important results as follows:Theorem 1.1Specially, if the initial surplus .Theorem 1.2 (Lundberg inequality)Theorem 1.3 (Lundberg-Cramer approximation) There exsists a positive constantc , such thatThe second part of chapter 1, we simply introduce two important modern approaches in risk theory: Feller's renewal theory and Gerber's martingale method, their skills is regards as the brief tools to study the classical ruin theory. At last we summarize the genelization of the classical risk model: (1) The genelization of { N ( t ), t≥0}; (2) The interest force has been introduced in classical risk model; (3) The risk model which is perturbed by diffusion; (4) The premium received is not a fixed ratio as the time grows linearly, it depends on the insurance business; (5) The genelization from one-dimentional risk model to two-dimentional or multi-dimenti onal risk model; (6) The complete discrete classical model; (7) The ruin probability of other risk model.In Chapter 2, we mainly introduce the two-dimentional risk model which is set up by Wai-Sum Chan and his partner in 2003:In Section 2.1, firstly we introduce the meaning to set up this model, then define three different types of time of ruin and the corresponding probabilities; In Section 2.2, by changing the risk model from two-dimentional form to one-dimentional form, we discuss the simple bounds of ruin probabilities. In Section 2.3, we give the basic concept of Phase-type distribution at first, then get the explicit expression of the third type of ruin probabilities by using the property of Phase-type distribution. Besides,In Section 2.4,we define a new type of time of ruin and its corresponding probability, point out its practical significance and give its explicit expression.On the basis of the Chapter 2, in Chapter 3 we add the stochastic interest to the two-dimentional risk model. Section 3.1 introduce the conclusion of one-dimensional classical risk model with stochastic interest; In Section 3.2 ,we set up a two-dimentional risk model with stochastic interest as follows:Based on this new model, three types of ruin probabilities are defined. Considering the dependency of the two different claims, by using the method in Chapter 2 and the conclusion of one-dimensional classical risk model with stochastic interest we get the simple bounds of the ruin probabilities.