【作者】 陈卓恒；

【导师】 刘次华；

【作者基本信息】 华中科技大学 ， 概率论与数理统计， 2006， 硕士

【摘要】 在传统风险理论中定义了三种风险模型：短期个体风险模型，短期聚合风险模型和长期聚合风险模型。本文介绍了这三种风险模型的定义和建立模型的背景，其中重点介绍后两种模型。在短期风险模型中，关于总索赔量分布的求法是一个中心又重要的话题。已知总索赔量分布的类型或是性质对于厘定费率，制定保费以及开发新险种等都有直接的指导和参考价值。关于短期风险模型中总索赔量分布的传统求法有很多种，精确求法有卷积法，矩母函数法，特征函数法等，近似求法则有正态分布近似法，平移Gamma分布近似法。但自从Panjer于1980年提出总索赔量分布的递推算法以来，很多的文献开始关注这方面的工作。递推算法在保单数目较多或者索赔数目较大的情况下具有良好的性质。由于开始提出的递推算法只适用于几类索赔次数分布，所以人们开始致力于对索赔次数分布类的推广。本文在前人工作基础上，对Panjer(a，b，k)类分布进行推广，得到一个新的分布类记为EX-Panjer(a，b，k)类，并得到此分布类的母函数所满足的微分方程。除了对索赔次数分布类进行推广以外，人们并不满足总索赔量分布只是一维情形。本文在索赔次数满足某类一维分布，索赔量为m维向量的情形下得到了总索赔量分布满足的递推公式，进一步推广了前人的结果，并对于实际应用时给出了更方便的表达式。递推算法在再保险实务中有极大的用处，本文在一个停止-损失在保险合同的净保费计算中，给出了数值例子。在长期聚合风险模型中，主要是对盈余水平的考察，这对于保险公司分配资会以及制定再保险计划等都有重要价值，本文运用随机游动的方法得到了破产前盈余通过给定水平概率的近似表达式，以及对它的精确估计。当个体索赔额服从指数分布时，给出了计算实例。然后在考虑破产以及不考虑破产两种情况下，运用鞅论的相关知识得到了盈余通过给定水平的的时刻的期望表达式，以及在破产前盈余通过给定值时刻的条件期望上界，所得结果较前人更加精确。文章最后进行了全文总结，并提出了进一步的研究方向和方法。更多还原

【Abstract】 There are three risk models in classical risk theory, namely short-time individualrisk model, short-time collective risk model and long-time collective risk model.The paperintroduced the definition and background of three risk models, emphatically on the last tworisk models.In short-time risk model,how to calculate the distribution of the total claimsamount is an important and centred topic.The type or property of the total distributionhas direct guidance and reference on how to rating,consulting premium and developingnew kinds of insurance.The classical calculation of the total claims amount distribution in short-time riskmodels have the method of convolution, moment generating function,character function,whichare exact methods.The approximate methods include normal distribution approxima-tion,Gamma distribution appproximation.But since Panjer introduced the recursive al-gorithm of the total claims amount distribution in 1980,many magazines began to do thiswork. The recursive algorithm has collective property to the too much policy or greatclaims amount.Because the recursive evaluation which we have refereed is suit for fewclaim number distribution,people begin to generalize the type of the claim number distri-bution.The thesis extends the Panjer(a,b,k)-class basically on previous work,then find anew distribution which named EX-Panjer(a,b,k) and the differential equation of the gen-erating moment.Besides of the type of distribution, we also have interested in multivariatecases.The thesis derive the recursive formula of the total claims distribution with one-dimension claim number and m-dimension claim severity which developed the previouspaper.and get convenient formula in practice.The recursive algorithm is useful in prac-tice.We also take a numberical example about calculation of net premium of a stop-lossinsurance contract.In long-time collective risk model,we mostly discuss the surplus process,which hasgreat value for insurance company to distribute fund and develop reinsurance plan.Thepaper discussed the surplus process in the classical model,using martingales and randomWalks we got the approximate formula about probability of that the insurer’s surplusreaches an upper barrier before the ruin and its precise estimate. And we also got theexact formula of expectation of hitting times, unconditionally on the event that ruin doesnot occur before the surplus exceeds.At the same time, upper barrier of the conditionalexpectation was given if we considered ruin in practice. At last,all the works in this thesis are summarized and some prospects are alsoproposed.更多还原