Research On The Theory Of Ruin In The Risk Models With Two Types Of Delayed Claims

Risk is a measure of uncertainty. Risks are everywhere for most people, whether they are going into the natural environment, or living in a social environment. Risk theory is the general theory used for the insurance companies to guide their business and for decision makers to quantify the risk quantification and prediction. As an important part of modern applied mathematics, investment and insurance industries have been applied among a very wide range. The main method of risk theory is using probability theory and stochastic process theory to construct the corresponding mathematical model to quantify the described risks process. In order to enhance scientific model selection, eventually formed a relatively complete theory-the ruin theory.Application of ruin theory can provide effective tools and methodologies for the risk managers, insurance companies can also measure the degree of operational soundness. It can be used by insurance companies to design appropriate financial warning system, also can be used by the insurance regulatory departments to build management indicators.First, this paper researches a compound Poission risk model with two types of by-claims. Assume that the occurrence of the main claim may not induce any by-claims, or induces one of the two by-claims, or induces both of the two by-claims. And the two types of by-claims may be delayed to the occurrence time of the next main claim with certain probability, or may occur simultaneously with certain probability. An integro-differntial equation system is obtained by applying the law of total probability. Furthermore, the Gerber-Shiu expected discounted penalty function expression can be obtained by using Laplace transform, Laplace Final-vlaue theorem, diagnonal dominance theorem. Then the paper focuses on the survival probability of this model, An integro-differntial equation system and expression of the survival probability are also obtained. And the conclusion can be degenerate to the existing conclusions under other conditions. Finally, numerical results are provided to illustrate the influence of the parameters on the survival probability.