高等破产论与风险排序

Ruin theory,ordering of risks and experience rating are three most active fields in actuarial mathematics with potential applications in insurance practice. In this project, some outstanding researches related to those three fields are carried out. The concrete contents are briefly described.respectively as follows: Ruin theory originated just a hundred years ago. In classical risk model, the following.assumption is made: the aggregate claim process is a compound Poisson process. In this process, the time parameter is continuous and the state space could be arbitrary. For simple identification,.we call this model as continuous classic risk model. With differences to this model, the classical risk model in fully discrete setting and the discrete time risk model with interest are treated in thisproject. In those models mentioned above,all risky indices related to ruin time （for example, the.ultimate ruin probability, the surplus immediately before ruin and the deficit at ruin etc.）are explored. Particularly, the asymptotic solutions and exponential bounds of the robabilitical laws of such indices are discussed with emphasis.Thus, the contents of classical risk model are greately extended. In addition, the crossed researches between ruin theory and mathematical finance are also carried out, and the applications of discrete stochastic order to classical risk model in fully discrete setting are explored too. The former is a new topic causing most attentions from.researchers in actuarial mathematics currently. The latter connects ruin theory with another active field in actuarial mathematics---Ordering of risks （i.e., the stochastic order）. This is an idea with innovation thinking. In contexts of actuarial mathematics, only those risks are treated which are caused by the.random losses and could be measured by the monetary units. Thus, the non_negative random variables are usually used to characterize the risks which equal just the absolute values of the random losses. So, the ordering of risks could be understood as the ordering of random variables, i.e., the stochastic order usually named in the literatures in applied probabilities. In this project, the extensions of （one dimensional）dominance order and some new properties of stop-loss order.are explored with the emphasis. Also, their some applications to the equilibrium analysis of no claim discount system and the compoud Poisson approximation to the individual risk model are explained. Those could be regarded as another group of ideals with innovation thinking. In addition, as we mentioned above, the discrete stochastic order is also applied to the classical risk model in fully discrete setting. In that case, the power order is first introduced by us for the.purpose of comparing the adjustment coefficients in different models..Experience rating is a general term to describe the techniques of determing the primums paidExperience rating is a general term to describe the techniques of determing the primums paid by policy holders. Such techniques are necessary because the policies in the portfolio of the nonlife insurance are heterogeneous and total number of policies is small. Thus, how to determine the primums properly is a key issue with challenges in theoretic aspect and potential applicable.background in insurance practice. In this project, the credibility model and no claim discount system are explored, which are two main approaches of experience rating. In fact, the extended researches of (one dimensional) dominance order is just introduced in order to discuss the.equilibrium analysis of no claim discount system

破产论与风险排序是当代精算数学理论研究中两个非常活跃的领域。现代概率论先进数学工具的引入、经济学与数学金融中最新研究成果的驱动，使得这两个领域中涌现出许多全新的研究方向，并呈现出精算数学与数学金融相互渗透、互动发展的新态势。本项目拟在此两领域的研究前沿开展高水平的理论研究工作，从而为推动现代精算数学的发展作出一点贡献。