With Dividends Sparse Risk Model The Expectations Of The Discounted Penalty Function

Ruin theory for long-term and stable business of the insurance company has the extremely vital significance. At present, due to dividends insurance can provide a good opportunity to protect customers from risk and enable customers to maximize revenue. So dividend and ruin issue has become a hot topic which was studied by more and more scholars. In the Lundberg-Cramer classical risk model, the premium amounts and the claim amounts are always assumed to be indenpendent. In fact, this hypothes is inade-quate to depict the realistic circumstances. So many cases of dependency risk model are studied by more and more scholars.Based on the Lundberg-Cramer classical risk model, we put the barrier dividend into the model and generalize it to more general situation. Thus we get a dependent risk model. If it satisfied:the premium income is not constant, but a random variables. We consider the premium arriving numble process{N(t)} is a Poisson process with pa-rameterλ>0 while the claim numble process{N1(t)} is the p-thinning process of the premium arriving number process. Under such a model, we obtain the integro equation, the integro-differential equation and the recursive formula for the expected discounted penalty function. Based on this, we find some expressions for the Laplace transform of the integro-differential equation when the premium and the claim sizes are some special distributed. At last, using the integro-differential equation, we give the explicit solutions of the time of ruin, the deficit at ruin and the surplus before ruin when the premium and the claim sizes are exponentially distributed, so it is significant to study the dependent risk model.According to the contents, this paper is divided into four chapters:In the first chapter, mainly, we give the dependent risk modelThen we introduce the research status of expected discounted penalty function, div-idend, random premium income and some main achievements of scholars.In the second chapter, we obtain the integro equation for the expected discounted penalty function where and the recursive formula In the third chapter, we study some expressions for the Laplace transform of the integro-differential equation when the premium and the claim sizes are some special dis-tributed. In the forth chapter, we obtain the explicit solutions of the time of ruinψb(u), the deficit at ruinζb(u) and the surplus before ruinηb(u) when the premium and the claim sizes are exponentially distributed 曲阜师范大学硕士学位论文μ(u)=(p((ab2+b3-a)e(-a-b)b-a))/((a+b)2)[1-((ap-b)e(ap-b-a)b[(1-p)(ap-b)+b])/(ap(ap-b-a))-(a2-b2e(-a-b)b)/(a(a+b))]-1e(ap-b)u Keywords:Thinning process；Barrier dividend；Random premium income；Ex-pected discounted penalty function；Integro-differential equation.