摘要
风险理论是当前金融数学界和精算学界的重要研究内容之一,它通过研究保险业中的随机风险模型来处理保险公司所关心的几个精算量,如破产概率、破产时刻、破产赤字、破产前瞬时盈余、Gerber-Shiu期望折现罚金函数、期望折现分红函数、调节系数等。有关保险风险模型的早期研究可以追溯到Lundberg(1903)的结果,正是由于他的工作,奠定了保险风险理论的坚实基础,直到今天,已有大量的相关论文和学术专著对Lundberg(1903)的工作给出了各种各样的推广和深入研究,如后来出现的扰动风险模型、更新风险模型、绝对破产风险模型、马氏转换风险模型、相依风险模型等。
另外,带分红策略的风险模型也受到了广泛关注,这与分红本身的现实意义是分不开的。分红是指保险公司依据自身经营状况将部分盈余分配给股东或初始准备金提供者,分红的多少在一定程度上也反映了一个公司的经济效益与竞争实力。该策略最早是De Finitti(1957)在第十五届精算大会上提出的,他指出公司应当寻求破产前所有分红期望折现值的最大化。目前常见的分红策略有障碍分红策略、阈红利策略、分段分红策略、线性分红策略等。
基于上述背景,我的博士毕业论文主要致力于以下几个方面的研究：首先是建立与实际更接近的保险风险模型和问题,其次是根据当前的风险模型和问题的特点,充分发挥随机过程理论理论方法的作用,努力寻找解决问题的途径。最后,为了使研究成果对实践起到一个很好的指导作用,将尽可能给出问题的明确表达式或者数值例子。下面介绍各个章节的研究内容。
第一章介绍了几类保险风险模型与合流超几何方程的基础知识。
第二章考虑了阈红利策略下带有投资利率的绝对破产风险模型,获得了绝对破产前红利现值的矩母函数和n一阶矩函数、Gerber-Shiu期望折现罚金函数、首达红利边界时刻的拉普拉斯变换所满足的积分—微分方程及边界条件。在指数索赔条件下,得到了绝对破产前红利现值的n—阶矩函数和绝对破产时刻拉普拉斯变换的显示表达式。特别地,当n=1时,给出了数值例子,分析了阂值b、折现利息力、投资利率和贷款利率对期望折现分红函数的影响。
本章来自于Yu Wenguang, Huang Yujuan. On the time value of absolute ruin for a risk model with credit and debit interest under a threshold strategy. Science China Mathematics, under review.
第三章研究了阈红利策略下带有投资利率的扰动复合Poisson风险模型的绝对破产问题,导出了绝对破产前红利现值的矩母函数和n—阶矩函数、Gerber-Shiu期望折现罚金函数所满足的积分—微分方程及边界条件。当折现利息力α＝0时,在指数索赔条件下得到了绝对破产前红利现值的n—阶矩函数的显示表达式。特别地,当n=1和α>0时,给出了数值例子,分析了阈值b、折现利息力、投资利率和贷款利率对期望折现分红函数的影响。
本章来自于Yu Wenguang. Some results on absolute ruin in the perturbed insurance risk model with investment and debit interests. Economic Modelling,31(2013),625-634.
第四章研究了障碍分红策略下的马氏绝对破产风险模型,导出了绝对破产前红利现值的矩母函数和n—阶矩函数、Gerber-Shiu期望折现罚金函数所满足的积分—微分方程及边界条件,并给出了方程的矩阵表示。另外,进一步考虑了一类半马氏相依结构的绝对破产风险模型,在该框架下,对任一状态i时的即刻索赔,马尔可夫链的状态就会发生改变达到状态j,而理赔额的分布Fj(y)是依赖于新的状态j的。下一次索赔时间间隔服从参数为λj的指数分布。需要强调的是,在给定Zn-1和Zn的情况下,随机变量Wn和Xn是相互独立的,但在其连续索赔额的大小之间和连续索赔时间间隔之间存在自相关性,而在Wn和Xn之间存在交叉相关。
本章来自于Yu Wenguang, Huang Yujuan. Dividend payments and related prob-lems in a Markov-dependent insurance risk model under absolute ruin. American Journal of Industrial and Business Management,1(1)(2011),1-9.
Yu Wenguang, Huang Yujuan. The Markovian regime-switching risk model with constant dividend barrier under absolute ruin. Journal of Mathematical Finance,1(3)(2011),83-89.
第五章研究了一类具有随机分红和随机保费收入的离散风险模型,其中保费收入过程和索赔过程均服从复合二项过程。当公司盈余达到或超过界限b时,红利以概率q0进行支付1单位。我们导出了期望折现罚金函数满足的递推公式,作为应用,给出了破产概率、破产赤字分布函数、破产赤字矩母函数的递推公式。最后给出数值例子,分析了相关参数对破产概率的影响。
本章来自于Yu Wenguang. Randomized dividends in a discrete insurance risk model with stochastic premium income. Mathematical Problems in Engineering,2013(2013),1-9.
第六章研究了一类具有相依结构的风险模型,即两次理赔间隔决定了下次理赔额的分布,当理赔额服从指数分布时,得到了Gerber-Shiu期望折现罚金函数所满足的积分—微分方程及拉普拉斯变换,作为应用给出了破产时刻,破产赤字及破产前瞬时盈余的拉普拉斯变换。最后,在具有障碍分红策略下的同一风险模型中,分析了Gerber-Shiu期望折现罚金函数和期望折现分红函数所满足的积分—微分方程。
本章来自于Yu Wenguang, Huang Yujuan. Some results on a risk model with dependence between claim sizes and claim intervals.数学杂志,33(5)(2013),781-787.
第七章研究了一类带有随机保费收入的马氏转换风险模型(也叫马氏调制风险模型),其中,保费收入过程、索赔过程和折现利息力过程均受马氏过程控制,本章的目的是研究期望折现罚金函数所满足的积分方程。作为该积分方程的应用,当状态个数仅为1个时,且索赔额服从指数分布时,给出了破产时刻、破产前瞬时盈余和破产赤字的拉普拉斯变换的明确表达式。最后,给出了数值例子,讨论了相关参数对上述精算量的影响。
本章来自于Yu Wenguang. On the expected discounted penalty function for a Markov regime-switching risk model with stochastic premium income. Discrete Dynam-ics in Nature and Society,2013(2013),1-9.
Risk theory plays an important role in financial mathematics and actuary, it through the study of stochastic risk model in the insurance industry to deal with several actuarial variables, such as the ruin probability, ruin time, deficit at ruin, surplus immediately prior to ruin, Gerber-Shiu expected discounted penalty function, expected discounted dividend function, adjustment coefficient, etc. Early research on insurance risk model can be traced back to the results of Lundberg(1903). It was because of his work, which lay a solid foundation for the insurance risk theory. Until today there are a large number of related papers and monographs, which generalize the work of Lundberg(1903) and in-depth study, such as the perturbed risk model to come, renewal risk model, compound binomial risk model, absolute ruin risk model, Markov regime-switching risk model and dependent risk model, etc.
In addition, the dividend strategy risk model are also received widespread attention, which is inseparable with the realistic significance of dividends. Dividends mean the in-surance company pays certain surplus to the shareholders or the initial reserve provider. The dividend amount also reflects a company's economic efficiency and strength. The dividend strategy was first discussed by De Finitti(1957) at the15th International Congress of Actuaries in1957. He pointed out that the company should maximize the expected discounted dividends before ruin. The current common dividend strategies are barrier dividend strategy, threshold dividend strategy, band dividend strategy, linear dividend strategy, etc.
On the basis of these background, my doctoral dissertation will be devoted to doing some researches in the following aspects:Firstly, I will make the insurance risk model and problem more practical. Secondly, according to the characteristics of the current risk model and problem, giving full play to the role of the theory of stochastic process, I will try to find the way to solve the problem. Finally, in order to make the research results have a very good guide to practice, I will try my best to give the explicit expressions or numerical examples. In the following, I will introduce the content of every Chapter.
Chapter1. We introduce several insurance risk models and confluent hypergeo-metric equation, etc.
Chapter2. We consider the absolute ruin risk model with credit interest under the threshold dividend strategy, and obtain the integro-differential equations with boundary conditions satisfied by the moment-generating function and n-th moment of the present value of all dividends until absolute ruin, Gerber-Shiu expected discounted penalty func-tion, the Laplace transform of the first time to reach the dividend barrier. When the claim sizes have a exponential distribution, we get the explicit expressions for the n-th moment of the present value of all dividends until absolute ruin and the Laplace trans-form of absolute ruin time. In particular, in the case of n=1we provide the numerical examples and illustrate the impacts of threshold b, discount interest force, credit interest and debit interest on the expected discounted dividend function.
This chapter is mainly based on the paper:
Yu Wenguang, Huang Yujuan. On the time value of absolute ruin for a risk model with credit and debit interest under a threshold strategy. Science China Mathematics, under review.
Chapter3. We study the absolute ruin problems for the perturbed compound poisson risk model with credit interest under the threshold dividend strategy, and get the integro-differential equations with boundary conditions satisfied by the moment-generating function and n-th moment of the present value of all dividends until absolute ruin, Gerber-Shiu expected discounted penalty function. When the claim sizes follow exponential distribution, we derive the explicit expressions for the n-th moment of the present value of all dividends until absolute ruin when discount interest force α=0. Specially, when n=1and α>0we provide the numerical examples and explain the impacts of threshold b, discount interest force, credit interest and debit interest on the expected discounted dividend function.
This chapter is mainly based on the paper:
Yu Wenguang. Some results on absolute ruin in the perturbed insurance risk model with investment and debit interests. Economic Modelling,31(2013),625-634.
Chapter4. We study the absolute ruin Markov risk model under the barrier dividend strategy, and derive the integro-differential equations with boundary conditions satisfied by the moment-generating function and n-th moment of the present value of all dividends until absolute ruin and Gerber-Shiu expected discounted penalty function. In addition, we further consider a class of semi-Markovian dependent absolute ruin risk model. Under the framework, at each instant of a claim, the Markov chain jumps to a state j, and the distribution Fj(y) of the claim depends on the new state j. Then the next interarrival time is exponentially distributed with parameter A^. Note that given the states Zn-1and Zn, the quantities Wn and Xn are independent, but there is autocorrelation among consecutive claim sizes and among consecutive interclaim times as well as cross-correlation between Wn and Xn.
This chapter is mainly based on the paper:
Yu Wenguang, Huang Yujuan. Dividend payments and related problems in a Markov-dependent insurance risk model under absolute ruin. American Journal of In-dustrial and Business Management,1(1)(2011),1-9.
Yu Wenguang, Huang Yujuan. The Markovian regime-switching risk model with constant dividend barrier under absolute ruin. Journal of Mathematical Finance,1(3)(20183-89.
Chapter5. We study a discrete risk model with randomized dividends and s-tochastic premium income, where the premium income process and claim process follow compound binomial process. The insurer pays a dividend of1with a probability go when the surplus is greater than or equal to a nonnegative integer b. We derive the recursion formulas for the expected discounted penalty function. As applications, we present the recursion formulas for the ruin probability, the distribution function of the deficit at ruin and the generating function of the deficit at ruin. Finally, numerical examples are also given to illustrate the effect of the related parameters on the ruin probability.
This chapter is mainly based on the paper:
Yu Wenguang. Randomized dividends in a discrete insurance risk model with s-tochastic premium income. Mathematical Problems in Engineering,2013(2013),1-9.
Chapter6. We consider the risk model with a dependent setting where the time between two claim occurrences determines the distribution of the next claim size. An integro-differential equation for some Gerber-Shiu expected discounted penalty function for the exponentially distributed claim sizes is derived. Applications of the integro-differential equation are given to the Laplace transform of the time of ruin, the deficit at ruin, the surplus immediately before ruin occurs. Finally, we analyze the Gerber-Shiu expected discounted penalty function and the expected discounted dividend function in the same risk model with a constant dividend barrier.
This chapter is mainly based on the paper:
Yu Wenguang, Huang Yujuan. Some results on a risk model with dependence between claim sizes and claim intervals.数学杂志,33(5)(2013),781-787.
Chapter7. We study a Markovian regime-switching risk model (also called Markov modulated risk model) with stochastic premium income, in which the premium income process, the claim process and discount interest force process are driven by Markovian regime-switching process. The purpose of this section is to study the integral equations satisfied by the expected discounted penalty function. Applications of the integral equa-tions are given to be explicit expression of Laplace transform of the time of ruin, the deficit at ruin and the surplus immediately before ruin occurs in the case of one state and exponential distribution. Finally, numerical example is also given to illustrate the effect of the related parameters on these quantities.
This chapter is mainly based on the paper:
Yu Wenguang. On the expected discounted penalty function for a Markov regime switching risk model with stochastic premium income. Discrete Dynamics in Nature and Society,2013(2013),1-9.
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