马氏环境或Copula相依下的精算模型
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摘要
本文利用马尔可夫过程,随机分析,更新测度,Copula连结函数等数学工具,研究了风险模型在随机环境以及Copula相依关系下的破产概率等问题.主要内容包括以下几个方面.
其一,我们给出经典风险模型的基本结构,对复合泊松模型和复合二项模型的研究情况进行了整理,同时也对经典模型的变化发展进行了梳理.
其二,考虑用一个马尔可夫环境来刻画随机环境,从而把复合二项模型放入到该环境中,利用上穿零点-更新测度的方法,依次给出了零点间隔的测度密度函数的表达式,破产时间和任意时刻的资产的联合分布,以及破产时间,破产前时刻资产和破产时赤字的联合分布的递归公式.并且在同样的思路下,考虑了破产前最大资产的分布.同时,给出了一个简单的例子,来说明上面得到的结果的可行性.
其三,考虑马尔可夫环境下二维复合二项模型,定义了三个不同的破产时间,包括最早出现破产时刻,均出现破产时刻,同时发生破产时刻.从而分别对它们的有限时间破产概率或者有限时间生存概率给出了递归公式,同时还给出了破产时具体赤字,以及发生破产时总资产赤字的相关结果.最后,利用上述部分结果,应用到了初始资产分配之一实际问题,给出了寻找最佳分配的方法.
其四,把Copula连结函数用到二维的风险模型中,从而考虑两个模型索赔额之间的相依关系.我们选择FGM Copula连结函数,首先对二维复合泊松模型给出了最早破产时刻定义下的生存概率满足的偏微分方程,然后对二维的复合二项模型,分别在连续型索赔额分布和离散型索赔分布下给出了不同定义的生存概率和破产概率的递归公式.特别在离散型分布下,对于其Copula函数的不唯一性进行了说明.
This dissertation focuses on the ruin problems in the specific situation of stochastic environment and Copula dependence by applying the mathematical tools such as Markovian process, stochastic analysis, renewal measure, and Cop-ulas. The main contents include the following aspects.
Firstly, the author proposes the application of the basic structure of the classical risk model and makes a clarification for both the study of the compound Poisson and the compound binomial models. The possible developments for the classical risk model are also clarified in order to draw out the discussion in the later chapters.
Secondly, the author suggests to put the compound binomial model into the stochastic environment which is created by the Markovian process. With the employing the method of up-crossing zero points-renewal measure, the expression of the renewal mass function and the recursive formula of the joint distributions of the ruin time, the surplus before ruin and the deficit at ruin will be provided successively. Meanwhile the supremum distribution for the surplus is also put into consideration with the same strategy. Furthermore, a simple example is provided to explain the feasibility of the conclusion which is drawn from the above part.
Thirdly, the author considers the two-dimensional compound binomial model in Markovian environment, and defines three types of ruin time, and the corre-sponding finite-time ruin probabilities. Then the recursive formula of the finite-time survival or ruin probabilities are derived separately for those three types. Meanwhile, some results about the specific deficits are also taken into considera-tion. Then, the allocation for the initial capital is considered using the finite-time survival probabilities.
Fourthly, during the process of applying the Copulas in the two-dimensional risk models, with the consideration of the dependence relation between the claim amounts, the author chooses the FGM Copula to be put into application. For the compound Poisson model, a partial integro-differential equation satisfied by the survival probability is derived. The recursive formula of the finite-time survival or ruin probabilities are then derived separately in the situation of continuous and discrete distribution of claim amount, for the compound binomial model. Furthermore, the uncertainty for the Copulas under the discrete case is also illustrated.
其一,我们给出经典风险模型的基本结构,对复合泊松模型和复合二项模型的研究情况进行了整理,同时也对经典模型的变化发展进行了梳理.
其二,考虑用一个马尔可夫环境来刻画随机环境,从而把复合二项模型放入到该环境中,利用上穿零点-更新测度的方法,依次给出了零点间隔的测度密度函数的表达式,破产时间和任意时刻的资产的联合分布,以及破产时间,破产前时刻资产和破产时赤字的联合分布的递归公式.并且在同样的思路下,考虑了破产前最大资产的分布.同时,给出了一个简单的例子,来说明上面得到的结果的可行性.
其三,考虑马尔可夫环境下二维复合二项模型,定义了三个不同的破产时间,包括最早出现破产时刻,均出现破产时刻,同时发生破产时刻.从而分别对它们的有限时间破产概率或者有限时间生存概率给出了递归公式,同时还给出了破产时具体赤字,以及发生破产时总资产赤字的相关结果.最后,利用上述部分结果,应用到了初始资产分配之一实际问题,给出了寻找最佳分配的方法.
其四,把Copula连结函数用到二维的风险模型中,从而考虑两个模型索赔额之间的相依关系.我们选择FGM Copula连结函数,首先对二维复合泊松模型给出了最早破产时刻定义下的生存概率满足的偏微分方程,然后对二维的复合二项模型,分别在连续型索赔额分布和离散型索赔分布下给出了不同定义的生存概率和破产概率的递归公式.特别在离散型分布下,对于其Copula函数的不唯一性进行了说明.
This dissertation focuses on the ruin problems in the specific situation of stochastic environment and Copula dependence by applying the mathematical tools such as Markovian process, stochastic analysis, renewal measure, and Cop-ulas. The main contents include the following aspects.
Firstly, the author proposes the application of the basic structure of the classical risk model and makes a clarification for both the study of the compound Poisson and the compound binomial models. The possible developments for the classical risk model are also clarified in order to draw out the discussion in the later chapters.
Secondly, the author suggests to put the compound binomial model into the stochastic environment which is created by the Markovian process. With the employing the method of up-crossing zero points-renewal measure, the expression of the renewal mass function and the recursive formula of the joint distributions of the ruin time, the surplus before ruin and the deficit at ruin will be provided successively. Meanwhile the supremum distribution for the surplus is also put into consideration with the same strategy. Furthermore, a simple example is provided to explain the feasibility of the conclusion which is drawn from the above part.
Thirdly, the author considers the two-dimensional compound binomial model in Markovian environment, and defines three types of ruin time, and the corre-sponding finite-time ruin probabilities. Then the recursive formula of the finite-time survival or ruin probabilities are derived separately for those three types. Meanwhile, some results about the specific deficits are also taken into considera-tion. Then, the allocation for the initial capital is considered using the finite-time survival probabilities.
Fourthly, during the process of applying the Copulas in the two-dimensional risk models, with the consideration of the dependence relation between the claim amounts, the author chooses the FGM Copula to be put into application. For the compound Poisson model, a partial integro-differential equation satisfied by the survival probability is derived. The recursive formula of the finite-time survival or ruin probabilities are then derived separately in the situation of continuous and discrete distribution of claim amount, for the compound binomial model. Furthermore, the uncertainty for the Copulas under the discrete case is also illustrated.
引文
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[3]Krolzjg, Marcellino, Mizon (2002). A Markov-Switching Vector Equilibrium Correction Model of the UK Labour Market. Empirical Economics 27,233-254.
[4]石柱鲜,刘俊生,吴泰岳(2007).利用多变量马尔科夫转移因子模型对我国经济周期的经验研究.数理统计与管理26,821-829.
[5]李金全,李庆华(2009).中国经济周期的阶段性划分和经济波动的非对称性-基于马尔可夫区制转移模型的研究.社会科学战线6,85-90.
[6]Baum, L.E., Petrie, T. (1966). Statistical Inference for Probabilistic Func-tions of Finite State Markov Chains. Annals of Mathematical Statistics 37, 1554-1563.
[7]Rabiner, L.R. (1989). A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. Proceedings of the IEEE 77(2),257-286.
[8]Choi, H., Baraniuk, R.G. (2001). Multiscale Image segmentation Using Wavelet-domain Hidden Markov Models. IEEE Transactions on Image Pro-ceedings 10(9),1309-1321.
[9]Do, M.N., Vetterli, M. (2002). Rotation Invariant Texture Characteriza-tion and Retrieval Using Steerable Wavelet-domain Hidden Markov Models. IEEE Transactions on Image Proceedings 4(4),517-527.
[10]金谷雷,汪旭升,朱军(2005).水稻14-3-3蛋白家族的生物信号学分析.遗传学报32(7),726-732.
[11]Alessandro R.. Giampiero, M.G. (2006). Volatility Estimation via Hidden Markov Models. J. Empirical Finance 13(2),203-230.
[12]杨洁(2006).马尔可夫链在管理决策中的初步应用.商场现代化467.
[13]曹晓丽(2006).马尔可夫模型在人力资源预测中的应用.统计与决策7,42-43.
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