A Class Of Multivariate Risk Model

In the risk theory, the most model which is discussed is the one-dimensional risk model, the research of the univariate risk process. But because of the multiplication of main body in the insurance market, we have to use multivariate risk model to describe this kind of situation; In addition, in the market each insurance company deals with some important services, such as charging the insurance premium, payment of claims and so on, usually in the discrete time. In view of two above kinds of situation, this article applies the discrete time multivariate risks model describing the actual market management condition.Firstly, the author discusses a discrete time one-dimensional model. This model describes such a single risk process, whose claim process is a sequence of random variables which follow a Poisson distribution with parameterλ. Using the knowledge of martingale, the author obtains its ruin probability. And then introducing an integral operator, the expression of the finite time ruin deficit distribution is obtained, and the limit of the finite time ruin deficit distribution is derived. Using the relationship of the ruin probability and the finite time ruin deficit distribution, the finite time ruin probability is obtained.Secondly, the article proposes a simplest discrete time multivariate risk model—the competitive two-dimensional model, and defines two different types of ruin probability. In term of some results of other's references and the properties of the conditional expectation, the article obtains the finite time ruin probability and ruin probability for the insurance market, and deduces the finite time ruin probability and the ruin probability of the individual insurance company.Finally, based on the competitive two-dimensional risk model, the author discusses the competitive n-dimension model. By introducing the matrix of market assignment describing the market proportion of each company in the insurance market, the author establishes the competitive n-dimensional model. Similarly, two different types of ruin are defined, and the expressions of them are obtained as well as the responding ruin probability of each insurance company are obtained.