Claims Sparse Two-dimensional Problem Of Ruin

With the rapid development of insurance industry, insurance companies’ scale of operation is becoming larger and larger and the types of insurances tend to be diverse and independent. Only considering a risk model that contains only one type of insurance can’t meet the risk management needs of insurance companies. Therefore, a risk model that contains more than one type of insurances is needed. To be closer to reality, the relationships of different insurances are also considered to the model. This type of models will play a major role to the running of insurance companies and the supervision of regulatory authorities.Therefore, consider a risk model that contains two types of insurances. And the numbers of claims are sparse dependent:Assume that N1(t)=N11(t)+n21(t),and N2(t)=N22(t)+N12(t),{N11(t),t>0} is a Poisson process with parameter λ11, and {N12(t), t>0} is a p1-sparse process of {N11(t), t>0}. Similarly,{N22(t), t>0} is a Poisson process with parameter λ22, and {N21(t),t>0} is a p2-sparse process of {N22(t),t>0}.{N22(t),t>0} and {N11(t),t>0} are independent. Firstly, the models are combined as one risk model with two types of insurances. In the model, the numbers of claims are sparse dependent. The ruin probability has been researched. In addition, the influence of sparse process to c and ruin probability has been discussed. Secondly, the models are seen as a risk model with two-dimension. In this model, a partial integro-differential equation for the survival probability is obtained, and a recursive formulas for calculating the survival probability in the case that F1and F2are both exponentially distributed is given.