Research On Ruin Probabilities In Mult-Type Risk Model Based On Erlang(n) Process

In the classical risk model, the claim number process is a Poisson process. One of the most important properties for Poisson distribution is that mean e-quals variance, but sometimes in insurance practice claim numbers will not entirely follow the Poisson distribution,with the variance being larger than the mean.In such a case, we can use the compound Poisson-Geometric process to describe the claim number process. And in the Poisson process there is no more than one claim occuring at each time point, while Erlang(n) process has n claims at each time point, which is more in line with the actual situation. This thesis maily studies the correlated structure of multi-type risk model and three problems are solved as following:Firstly, we study a risk model of the occurrence of the claim as the gener-alized Poisson process. We derive the explicit expression for joint distribution function with three characteristics:the time of ruin, the surplus before ruin, the deficit at ruin. By using the explicit joint distribution function, we obtain the ruin probability. Then we study the joint distribution function where the premium is a kind of exponential distribution, and the claim occurs as the gen-eralized Poisson process. And we also discuss the joint distribution function and ruin probability of generalized Erlang(n) process.Secondly, we do a study on two classes of correlated claims risk model, where the correlated two classes of claims counting process are transformed through model into independent Poisson-Geometric and generalized Erlang(n) processes. We decompose the Gerber-Shiu discounted penalty function into two parts. We get the integro-differential equations of Gerber-Shiu discounted penalty function, and the Lundberg equation is obtained by using the martin-gale method. And we also obtain the expression of Gerber-Shiu discounted penalty function.Finally, we study the ruin probability problems for a perturbed risk model with two classes of correlated aggregate claims process. The correlated two claims in the counting process are transformed through model into indepen-dent Poisson-Geometric and generalized Erlang(n) processes. The Laplace transforms of the Gerber-Shiu discounted penalty functions are obtained. If the densities of the two classes of claim distributions have rational Laplace transforms, explicit expressions and the corresponding ruin probability graph are given for the Gerber-Shiu discounted penalty functions.