保险精算的信息熵方法
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摘要
生存分析和风险理论是精算研究中的两个核心问题。本论文以信息熵理论为基础,对生存分析中的死亡率修匀、死亡率预测和风险理论中的保费定价、破产概率等问题进行了研究,并给出了相应的计算模型和方法。本文的具体内容如下:
第一章为文献综述和选题背景。首先简要介绍了精算学研究的主要问题,并对生存分析、保费定价原理及破产概率的研究现状和主要方法进行了重点的介绍。然后,对信息熵方法在金融领域的应用进行了回顾,最后介绍了本文的研究工作。
第二章简要介绍了信息熵的概念和两种推断概率分布的准则,即最大熵原理和最小叉熵原理。利用凸规划的对偶理论,分别推导出最大熵和最小叉熵问题的对偶规划,从而可以极大地减少计算工作量和有助于两个熵优化原理的推广应用。并针对极大极小问题,给出了熵正则化和指数(乘子)罚函数两种不同的求解方法,并证明了两者之间的对偶等价关系。
第三章将熵优化方法作为一种统计推断的工具,应用到寿险精算中的生存分析问题,针对死亡率修匀和死亡率调整,建立了两个死亡率修匀模型(分别为最大熵修匀模型和多目标修匀模型)和一个死亡率预测模型(最小叉熵模型)。同现有的模型相比,这三个模型在保证计算精度的同时,在理论分析和计算效率上都有了很大的改进。
第四章将信息熵引入保费定价,依据熵作为概率不确定性度量的内涵,对以均值一方差为基础的几种保费计算原理进行了修正。在修正后的定价原理中,考虑了不同的概率选择可能带来的“系统”风险,为合理的保费定价提出了一个新的思想。
第五章提出了一个新的保费定价方法,即叉熵正则化方法,并导出了一个指数型的保费定价公式。该方法具有两个突出特点:(1)保费定价公式可以看作是对最大索赔额的光滑近似,其近似程度由一个光滑参数(相当于风险厌恶系数)控制;(2)在其推导过程里包含了一个由先验(预估)概率求后验(真实)概率的过程,并且给出了这一概率变换的显式表达式。
第六章将大偏差理论应用到风险理论中,对破产概率问题进行了研究,给
Survival analysis and risk theory are two central problems in actuarial science. Based on the informational entropy theory, this dissertation focus on several important problems, such as mortality graduation, mortality prediction, insurance pricing and ruin probability, and proposes corresponding models and computational algorithms. Main contents of this dissertation are as follows:In Chapter 1, the background and motivation of this dissertation are introduced together with a review of some existing methods and models for survival analysis and risk theory. Finally, main research work of this dissertation is discussed.Chapter 2 is devoted to introducing the entropy concept and entropy optimization principles, which are the tools for predicting the probability distribution. Based upon the duality theory of convex programming, we derive the corresponding dual programs of two entropy optimization problems. Finally, we introduce two methods (the entropy regularization and the exponential penalty) for solving the min-max problem, and explore the duality relationship between these two approaches.In Chapter 3, the graduation and prediction models for mortality are studied. Based on two entropy optimization principles, we present two graduation models and one mortality predicting model that include the maximum entropy graduation model, the multi-objective graduation model and the minimum cross entropy prediction model, respectively. They improve the existing methods in computational precision and efficiency.In chapter 4, the concept of informational entropy is fully explored in the context of insurance premium principles. An additional term of premium is added to the mean-variance based pricing models, in order to cover the systematic risk arising in the selection of probably distributions.In chapter 5, we propose a new premium approach, the entropy regularization method, and derive an exponential-type premium formula. The main feature of this new approach lies in the fact that a probability transformation process, from an
estimated one into a "real" one, is embedded in the premium derivation and transformation formula is explicitly given. This does not only make the present approach easily understood, but also establishes a firm link of it with some famous probability transformation and premium principles.In chapter 6, the large deviation theory is employed in risk theory. The ruin probability for the surplus progress is derived based upon the Cramer theorem. Two approaches to obtaining the rate function are discussed, of which the one is relied on Chernoff formula and the other is through the cross entropy. In particular, the latter method can be used to obtain the rate function under the complex probabilities.The last chapter gives a summary of the dissertation and some possible extensions of the present work.
第一章为文献综述和选题背景。首先简要介绍了精算学研究的主要问题,并对生存分析、保费定价原理及破产概率的研究现状和主要方法进行了重点的介绍。然后,对信息熵方法在金融领域的应用进行了回顾,最后介绍了本文的研究工作。
第二章简要介绍了信息熵的概念和两种推断概率分布的准则,即最大熵原理和最小叉熵原理。利用凸规划的对偶理论,分别推导出最大熵和最小叉熵问题的对偶规划,从而可以极大地减少计算工作量和有助于两个熵优化原理的推广应用。并针对极大极小问题,给出了熵正则化和指数(乘子)罚函数两种不同的求解方法,并证明了两者之间的对偶等价关系。
第三章将熵优化方法作为一种统计推断的工具,应用到寿险精算中的生存分析问题,针对死亡率修匀和死亡率调整,建立了两个死亡率修匀模型(分别为最大熵修匀模型和多目标修匀模型)和一个死亡率预测模型(最小叉熵模型)。同现有的模型相比,这三个模型在保证计算精度的同时,在理论分析和计算效率上都有了很大的改进。
第四章将信息熵引入保费定价,依据熵作为概率不确定性度量的内涵,对以均值一方差为基础的几种保费计算原理进行了修正。在修正后的定价原理中,考虑了不同的概率选择可能带来的“系统”风险,为合理的保费定价提出了一个新的思想。
第五章提出了一个新的保费定价方法,即叉熵正则化方法,并导出了一个指数型的保费定价公式。该方法具有两个突出特点:(1)保费定价公式可以看作是对最大索赔额的光滑近似,其近似程度由一个光滑参数(相当于风险厌恶系数)控制;(2)在其推导过程里包含了一个由先验(预估)概率求后验(真实)概率的过程,并且给出了这一概率变换的显式表达式。
第六章将大偏差理论应用到风险理论中,对破产概率问题进行了研究,给
Survival analysis and risk theory are two central problems in actuarial science. Based on the informational entropy theory, this dissertation focus on several important problems, such as mortality graduation, mortality prediction, insurance pricing and ruin probability, and proposes corresponding models and computational algorithms. Main contents of this dissertation are as follows:In Chapter 1, the background and motivation of this dissertation are introduced together with a review of some existing methods and models for survival analysis and risk theory. Finally, main research work of this dissertation is discussed.Chapter 2 is devoted to introducing the entropy concept and entropy optimization principles, which are the tools for predicting the probability distribution. Based upon the duality theory of convex programming, we derive the corresponding dual programs of two entropy optimization problems. Finally, we introduce two methods (the entropy regularization and the exponential penalty) for solving the min-max problem, and explore the duality relationship between these two approaches.In Chapter 3, the graduation and prediction models for mortality are studied. Based on two entropy optimization principles, we present two graduation models and one mortality predicting model that include the maximum entropy graduation model, the multi-objective graduation model and the minimum cross entropy prediction model, respectively. They improve the existing methods in computational precision and efficiency.In chapter 4, the concept of informational entropy is fully explored in the context of insurance premium principles. An additional term of premium is added to the mean-variance based pricing models, in order to cover the systematic risk arising in the selection of probably distributions.In chapter 5, we propose a new premium approach, the entropy regularization method, and derive an exponential-type premium formula. The main feature of this new approach lies in the fact that a probability transformation process, from an
estimated one into a "real" one, is embedded in the premium derivation and transformation formula is explicitly given. This does not only make the present approach easily understood, but also establishes a firm link of it with some famous probability transformation and premium principles.In chapter 6, the large deviation theory is employed in risk theory. The ruin probability for the surplus progress is derived based upon the Cramer theorem. Two approaches to obtaining the rate function are discussed, of which the one is relied on Chernoff formula and the other is through the cross entropy. In particular, the latter method can be used to obtain the rate function under the complex probabilities.The last chapter gives a summary of the dissertation and some possible extensions of the present work.
引文
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