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

Insurance plays an important role which cannot be ignored in the development of today’s society. The research on the ruin probability and the related problem formed an important research field, ruin theory. Building risk model accord with the actual situation of the insurance company is significant to the insurance company. In reality, the claim process is not always the stationary and independent increments process. And the delayed claims can be consudered as the IBNR (Incurred But Not Reported) claims, it’s necessary for the insurance company to build up reserves for these claims. Most study about the risk model with delayed claims assumes that there is only one kind of by-claims that induced by the occurrence of the main claims. However, the occurrence of the main claim X may induce by-claim Y or its occurrence may induce by-claim Z due to various random factor. In other words, the occurrence of the main claims may induce different kinds of by-claims. Several risk models were studied based on the above consideration.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. An integro-differntial equation system is obtained by applying the law of total probability. Furthermore, the survival probability expression can be obtained by using Laplace transform, Laplace Final-vlaue theorem and Rouche theorem. Based on the assumption that all the claims amounts are same exponentially distributed, we get the explicit expression of the survival probability. Finally, numerical results are provided to illustrate the influence of the parameters on the survival probability.Then the paper focus on the Gerber-Shiu discounted penalty function with random premium. Under the condition that there are two types of by-claims, we assume the occurrence of the main claim will induce one by-claim and whether the by-claim delay or not depend on the comparison between the main claim size and the threshold. The process of solving the Gerber-Shiu discounted penalty function is given and the defective renewal equation the Gerber-Shiu discounted penalty function satisfied is discussed. We also show the numerical example when the Gerber-Shiu discounted penalty function degenerate to the ruin probability function.Finally, two kinds of risk model with n types of by-claims are studied. Assuming each main claim will induce a by-claim. One risk model assumes that counting process of the main claims obey Poisson distribution and whether the by-claims delay or not depend on the comparison between the main claim size and the threshold; the other one assumes the arrival time of the main claims obey Erlang(2) distribution and the by-claims will be delayed with certain probability. The expression of survival can be obtained and numerical examples are given.