保险风险模型的破产理论与分红策略研究
发布时间：2018-09-19 06:12
【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α0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.
[Abstract]:Risk theory is one of the important research contents in financial mathematics and Actuarial science. It deals with several actuarial quantities that insurance companies care about by studying stochastic risk models in insurance industry, such as ruin probability, ruin time, ruin deficit, instantaneous surplus before ruin, Gerber-Shiu expected discount penalty function and expected discount dividend letter. The early research on the insurance risk model can be traced back to Lundberg (1903). It is precisely because of his work that he laid a solid foundation for the insurance risk theory. Up to now, a large number of related papers and academic monographs have given a variety of extensions and in-depth research on Lundberg (1903) work, such as later. Disturbance risk model, update risk model, absolute ruin risk model, Markov transformation risk model, dependent risk model and so on.
In addition, the risk model with dividend strategy has been widely concerned, which is inseparable from the practical significance of dividend itself. Dividend refers to the insurance company distributes part of the surplus to shareholders or providers of initial reserve according to its own operating conditions. The amount of dividend also reflects the economic benefits and competition of a company to a certain extent. Strength. The strategy was first proposed by De Finitti (1957) at the 15th Actuarial Conference. He pointed out that companies should seek to maximize the expected discount value of all dividends before bankruptcy.
Based on the above background, my doctoral dissertation is mainly devoted to the following aspects: firstly, establishing insurance risk models and problems closer to reality; secondly, according to the characteristics of current risk models and problems, giving full play to the role of stochastic process theory and methods, and striving to find a solution to the problem. In order to make the research results play a good guiding role in practice, the explicit expressions or numerical examples of the problem will be given as far as possible. The following sections will introduce the research contents.
The first chapter introduces the basic knowledge of several kinds of insurance risk models and confluent hypergeometric equations.
In Chapter 2, we consider the absolute ruin risk model with investment interest rate under threshold dividend policy. We obtain the moment generating function of the present value of the absolute pre-ruin dividend and the first-order moment function, the Gerber-Shiu expected discounted penalty function, the integral-differential equation and the boundary conditions satisfied by the Laplace transform at the time of the first dividend boundary. Under the condition of absolute bankruptcy, the n-moment function of dividend present value before absolute bankruptcy and the Laplace transform of absolute bankruptcy time are obtained. Especially, when n=1, numerical examples are given to analyze the effects of threshold b, discounted interest force, investment interest rate and loan interest rate on the expected discounted dividend function.
This chapter is from 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.
In Chapter 3, the absolute ruin problem of the perturbed compound Poisson risk model with investment interest rate under threshold dividend strategy is studied. The moment generating function and n-order moment function of the present value of the dividend before absolute ruin are derived. The integral-differential equation and boundary conditions satisfied by Gerber-Shiu's expected discounted penalty function are derived. In particular, when n = 1 and alpha 0, numerical examples are given to analyze the effects of threshold b, discounted interest force, investment interest rate and loan interest rate on the expected discounted dividend function.
This chapter is from Yu Wenguang. Some results on absolute ruin in the perturbed insurance risk model with investment and debit interests. Economic Modelling, 31 (2013), 625-634.
In Chapter 4, we study Markov's absolute ruin risk model under barrier dividend policy, derive the moment generating function and n-order moment function of the present dividend value before absolute ruin, the integral-differential equation and boundary conditions satisfied by Gerber-Shiu expected discounted penalty function, and give the matrix representation of the equation. Under this framework, the state of Markov chain will change to state j for any instant claim in state i, and the distribution of claim amount Fj (y) depends on the new state J. The next claim interval obeys the exponential distribution of parameter lambda J. It should be emphasized that the state of the Markov chain will change to state j for any instant claim in state I. In the case of Zn and Wn, the random variables Wn and Xn are independent of each other, but there is an autocorrelation between the amount of continuous claims and the time interval of continuous claims, while there is a cross correlation between Wn and Xn.
This chapter is from Yu Wenguang, Huang Yujuan. Dividend payments and related prob-lems in a Markov-dependent insurance risk model under absolute ruin. American Journal of Industry 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.
In Chapter 5, we study a discrete risk model with random dividend and premium income, in which both premium income process and claim process obey compound binomial process. When the earnings of a company reach or exceed the limit b, the dividend pays one unit with probability q0. We derive a recursive formula for the expected discounted penalty function as follows. The recursive formulas of ruin probability, ruin deficit distribution function and ruin deficit moments generating function are given. Finally, numerical examples are given to analyze the influence of relevant parameters on ruin probability.
This chapter is from Yu Wenguang. Randomized dividends in a discrete insurance risk model with stochastic premium income. Mathematical Problems in Engineering, 2013 (2013), 1-9.
In Chapter 6, we study a kind of risk model with dependent structure, that is, the distribution of the next claim is determined by the interval between two claims. When the claim is exponential distribution, we obtain the integral-differential equation and Laplace transformation satisfied by Gerber-Shiu's expected discounted penalty function. As an application, we give the ruin time, ruin deficit and break. Lastly, the integral-differential equations of Gerber-Shiu expected discount penalty function and expected discount dividend function are analyzed in the same risk model with barrier dividend strategy.
This chapter is from Yu Wenguang, Huang Yujuan. Some results on a risk model with dependence between claim sizes and claim intervals. Mathematical Journal, 33 (5) (2013), 781-787.
In Chapter 7, we study a Markov transformation risk model with stochastic premium income. The premium income process, claim process and discounted interest force process are all controlled by Markov process. The purpose of this chapter is to study the integral equation satisfied by the expected discounted penalty function. When the number of States is only one and the claim amount obeys exponential distribution, the explicit expressions of Laplace transformation for ruin time, instantaneous surplus before ruin and ruin deficit are given.
This chapter is from 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.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：F840.4;F224
本文编号：2249280
[Abstract]:Risk theory is one of the important research contents in financial mathematics and Actuarial science. It deals with several actuarial quantities that insurance companies care about by studying stochastic risk models in insurance industry, such as ruin probability, ruin time, ruin deficit, instantaneous surplus before ruin, Gerber-Shiu expected discount penalty function and expected discount dividend letter. The early research on the insurance risk model can be traced back to Lundberg (1903). It is precisely because of his work that he laid a solid foundation for the insurance risk theory. Up to now, a large number of related papers and academic monographs have given a variety of extensions and in-depth research on Lundberg (1903) work, such as later. Disturbance risk model, update risk model, absolute ruin risk model, Markov transformation risk model, dependent risk model and so on.
In addition, the risk model with dividend strategy has been widely concerned, which is inseparable from the practical significance of dividend itself. Dividend refers to the insurance company distributes part of the surplus to shareholders or providers of initial reserve according to its own operating conditions. The amount of dividend also reflects the economic benefits and competition of a company to a certain extent. Strength. The strategy was first proposed by De Finitti (1957) at the 15th Actuarial Conference. He pointed out that companies should seek to maximize the expected discount value of all dividends before bankruptcy.
Based on the above background, my doctoral dissertation is mainly devoted to the following aspects: firstly, establishing insurance risk models and problems closer to reality; secondly, according to the characteristics of current risk models and problems, giving full play to the role of stochastic process theory and methods, and striving to find a solution to the problem. In order to make the research results play a good guiding role in practice, the explicit expressions or numerical examples of the problem will be given as far as possible. The following sections will introduce the research contents.
The first chapter introduces the basic knowledge of several kinds of insurance risk models and confluent hypergeometric equations.
In Chapter 2, we consider the absolute ruin risk model with investment interest rate under threshold dividend policy. We obtain the moment generating function of the present value of the absolute pre-ruin dividend and the first-order moment function, the Gerber-Shiu expected discounted penalty function, the integral-differential equation and the boundary conditions satisfied by the Laplace transform at the time of the first dividend boundary. Under the condition of absolute bankruptcy, the n-moment function of dividend present value before absolute bankruptcy and the Laplace transform of absolute bankruptcy time are obtained. Especially, when n=1, numerical examples are given to analyze the effects of threshold b, discounted interest force, investment interest rate and loan interest rate on the expected discounted dividend function.
This chapter is from 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.
In Chapter 3, the absolute ruin problem of the perturbed compound Poisson risk model with investment interest rate under threshold dividend strategy is studied. The moment generating function and n-order moment function of the present value of the dividend before absolute ruin are derived. The integral-differential equation and boundary conditions satisfied by Gerber-Shiu's expected discounted penalty function are derived. In particular, when n = 1 and alpha 0, numerical examples are given to analyze the effects of threshold b, discounted interest force, investment interest rate and loan interest rate on the expected discounted dividend function.
This chapter is from Yu Wenguang. Some results on absolute ruin in the perturbed insurance risk model with investment and debit interests. Economic Modelling, 31 (2013), 625-634.
In Chapter 4, we study Markov's absolute ruin risk model under barrier dividend policy, derive the moment generating function and n-order moment function of the present dividend value before absolute ruin, the integral-differential equation and boundary conditions satisfied by Gerber-Shiu expected discounted penalty function, and give the matrix representation of the equation. Under this framework, the state of Markov chain will change to state j for any instant claim in state i, and the distribution of claim amount Fj (y) depends on the new state J. The next claim interval obeys the exponential distribution of parameter lambda J. It should be emphasized that the state of the Markov chain will change to state j for any instant claim in state I. In the case of Zn and Wn, the random variables Wn and Xn are independent of each other, but there is an autocorrelation between the amount of continuous claims and the time interval of continuous claims, while there is a cross correlation between Wn and Xn.
This chapter is from Yu Wenguang, Huang Yujuan. Dividend payments and related prob-lems in a Markov-dependent insurance risk model under absolute ruin. American Journal of Industry 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.
In Chapter 5, we study a discrete risk model with random dividend and premium income, in which both premium income process and claim process obey compound binomial process. When the earnings of a company reach or exceed the limit b, the dividend pays one unit with probability q0. We derive a recursive formula for the expected discounted penalty function as follows. The recursive formulas of ruin probability, ruin deficit distribution function and ruin deficit moments generating function are given. Finally, numerical examples are given to analyze the influence of relevant parameters on ruin probability.
This chapter is from Yu Wenguang. Randomized dividends in a discrete insurance risk model with stochastic premium income. Mathematical Problems in Engineering, 2013 (2013), 1-9.
In Chapter 6, we study a kind of risk model with dependent structure, that is, the distribution of the next claim is determined by the interval between two claims. When the claim is exponential distribution, we obtain the integral-differential equation and Laplace transformation satisfied by Gerber-Shiu's expected discounted penalty function. As an application, we give the ruin time, ruin deficit and break. Lastly, the integral-differential equations of Gerber-Shiu expected discount penalty function and expected discount dividend function are analyzed in the same risk model with barrier dividend strategy.
This chapter is from Yu Wenguang, Huang Yujuan. Some results on a risk model with dependence between claim sizes and claim intervals. Mathematical Journal, 33 (5) (2013), 781-787.
In Chapter 7, we study a Markov transformation risk model with stochastic premium income. The premium income process, claim process and discounted interest force process are all controlled by Markov process. The purpose of this chapter is to study the integral equation satisfied by the expected discounted penalty function. When the number of States is only one and the claim amount obeys exponential distribution, the explicit expressions of Laplace transformation for ruin time, instantaneous surplus before ruin and ruin deficit are given.
This chapter is from 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.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：F840.4;F224
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