Ruin Problems Of Multivariate Risk Models

The multivariate risk model comes from the extension of the dimension of the classical risk model. It enriches the ruin theory as it provides more information on risk evaluation for financial organizations such as the insurance company. So it has both theoretical and application value to investigate on multivariate risk models.The paper is organized as follows:In chapter 1, the development history of ruin theory is reviewed, and general research methods of risk models, especially of multivariate risk models are summarized. Then two concepts that are often used in risk theory are introduced, which are Phase Type distribution and stochastic order, both univariate and multivariate. Definitions of usual stochastic order, increasing convex, convex order, increasing supermodular, supermodular order and related results are given. The construction of the paper is given at the end of this chapter.In chapter 2, the development history of multivariate risk models is reviewed, and two notably multivariate risk models i.e. the two-dimensional risk model(2003) and multivariate risk model of MPH(2004) are introduced. Then the extended multivariate risk model under constant interest force is established, simple bounds ofΨmin andΨmax, and the explicit expression ofΨsum are given. Also the effect of dependence among different types of claims is discussed in the end.In the first part of chapter 3, the multivariate risk model with bivariate claim process is established. Before approximating its first type of ruin probabilityΨmin by the improved Dickson & Waters method, the multivariate discrete compound binomial risk model is discussed. The recursive formula ofΨmin is proved to be true when we take 8=0. In the second part of the chapter, the premium process is also extended to the bivariate compound Poisson process, and the recursive formula of the correspondingΨmax is obtained.