Research On Ruin Problems With Two-type By-claims And Delay Peirod Based On Compound Binomial Risk Model

The classic compound binomial risk model is one of the basic discrete risk models. Inyears of study, the number of claim types is increasing. The pervious study is adding the timecorrelated by-claim to the main claim, and the by-claim may happen with a certain probabilityor when the main claim exceeds a certain threshold. The by-claim may happen at the periodwhen the main claim happens, or the by-claim maybe delayed to the next period. The model isextended in this paper, the main content is as follows:1. The extension of the compound binomial risk model with two-type by-claims.Because of the various types of insurance in insurance companies, we consider a compoundbinomial risk model. In the model, we assume that the main claim(such as vehicle insurance)induces not only one but two by-claims(maybe property insurance and life insurance), each ofthe two types of by-claims may happen in the same period in which the main claim happens,or may be delayed to the next period. Under this model, the insurance company can face anddeal with more kinds of situation, and adjust policy strategy in time, then reduce the ruinprobability.2. The submodels and the recursive formula of joint distribution. Through importingthree submodels, adding random variables in the initial model, and using the law of totalprobability, summation and simultaneous equations, we obtain the expression of the jointdistribution f (u, x, y)of the surplus just before ruin and deficit at ruin when initial surplus is0. Then we use the moment generating function method to obtain the recursive formula of thejoint distribution when initial surplus is u.3. Calculation of ruin probability. Due to that the surplus just before ruin and deficit atruin can be of any value, through the previous calculation of the model and the submodelswhen ruin occurs at time1, we sum the surplus just before ruin and deficit at ruin each from0to infinity, then obtain the ruin probability when initial surplus is0. The method to calculateruin probability when initial surplus is any value, that is, to do the same operation to f (0, x, y),we can obtain the recursive formula of ruin probability when initial surplus is u.