保險風(fēng)險模型的破產(chǎn)理論與分紅策略研究
發(fā)布時間：2018-09-19 06:12
【摘要】：風(fēng)險理論是當(dāng)前金融數(shù)學(xué)界和精算學(xué)界的重要研究內(nèi)容之一,它通過研究保險業(yè)中的隨機風(fēng)險模型來處理保險公司所關(guān)心的幾個精算量,如破產(chǎn)概率、破產(chǎn)時刻、破產(chǎn)赤字、破產(chǎn)前瞬時盈余、Gerber-Shiu期望折現(xiàn)罰金函數(shù)、期望折現(xiàn)分紅函數(shù)、調(diào)節(jié)系數(shù)等。有關(guān)保險風(fēng)險模型的早期研究可以追溯到Lundberg(1903)的結(jié)果,正是由于他的工作,奠定了保險風(fēng)險理論的堅實基礎(chǔ),直到今天,已有大量的相關(guān)論文和學(xué)術(shù)專著對Lundberg(1903)的工作給出了各種各樣的推廣和深入研究,如后來出現(xiàn)的擾動風(fēng)險模型、更新風(fēng)險模型、絕對破產(chǎn)風(fēng)險模型、馬氏轉(zhuǎn)換風(fēng)險模型、相依風(fēng)險模型等。 另外,帶分紅策略的風(fēng)險模型也受到了廣泛關(guān)注,這與分紅本身的現(xiàn)實意義是分不開的。分紅是指保險公司依據(jù)自身經(jīng)營狀況將部分盈余分配給股東或初始準(zhǔn)備金提供者,分紅的多少在一定程度上也反映了一個公司的經(jīng)濟效益與競爭實力。該策略最早是De Finitti(1957)在第十五屆精算大會上提出的,他指出公司應(yīng)當(dāng)尋求破產(chǎn)前所有分紅期望折現(xiàn)值的最大化。目前常見的分紅策略有障礙分紅策略、閾紅利策略、分段分紅策略、線性分紅策略等。 基于上述背景,我的博士畢業(yè)論文主要致力于以下幾個方面的研究：首先是建立與實際更接近的保險風(fēng)險模型和問題,其次是根據(jù)當(dāng)前的風(fēng)險模型和問題的特點,充分發(fā)揮隨機過程理論理論方法的作用,努力尋找解決問題的途徑。最后,為了使研究成果對實踐起到一個很好的指導(dǎo)作用,將盡可能給出問題的明確表達(dá)式或者數(shù)值例子。下面介紹各個章節(jié)的研究內(nèi)容。 第一章介紹了幾類保險風(fēng)險模型與合流超幾何方程的基礎(chǔ)知識。 第二章考慮了閾紅利策略下帶有投資利率的絕對破產(chǎn)風(fēng)險模型,獲得了絕對破產(chǎn)前紅利現(xiàn)值的矩母函數(shù)和n一階矩函數(shù)、Gerber-Shiu期望折現(xiàn)罰金函數(shù)、首達(dá)紅利邊界時刻的拉普拉斯變換所滿足的積分—微分方程及邊界條件。在指數(shù)索賠條件下,得到了絕對破產(chǎn)前紅利現(xiàn)值的n—階矩函數(shù)和絕對破產(chǎn)時刻拉普拉斯變換的顯示表達(dá)式。特別地,當(dāng)n=1時,給出了數(shù)值例子,分析了閡值b、折現(xiàn)利息力、投資利率和貸款利率對期望折現(xiàn)分紅函數(shù)的影響。 本章來自于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. 第三章研究了閾紅利策略下帶有投資利率的擾動復(fù)合Poisson風(fēng)險模型的絕對破產(chǎn)問題,導(dǎo)出了絕對破產(chǎn)前紅利現(xiàn)值的矩母函數(shù)和n—階矩函數(shù)、Gerber-Shiu期望折現(xiàn)罰金函數(shù)所滿足的積分—微分方程及邊界條件。當(dāng)折現(xiàn)利息力α＝0時,在指數(shù)索賠條件下得到了絕對破產(chǎn)前紅利現(xiàn)值的n—階矩函數(shù)的顯示表達(dá)式。特別地,當(dāng)n=1和α0時,給出了數(shù)值例子,分析了閾值b、折現(xiàn)利息力、投資利率和貸款利率對期望折現(xiàn)分紅函數(shù)的影響。 本章來自于Yu Wenguang. Some results on absolute ruin in the perturbed insurance risk model with investment and debit interests. Economic Modelling,31(2013),625-634. 第四章研究了障礙分紅策略下的馬氏絕對破產(chǎn)風(fēng)險模型,導(dǎo)出了絕對破產(chǎn)前紅利現(xiàn)值的矩母函數(shù)和n—階矩函數(shù)、Gerber-Shiu期望折現(xiàn)罰金函數(shù)所滿足的積分—微分方程及邊界條件,并給出了方程的矩陣表示。另外,進(jìn)一步考慮了一類半馬氏相依結(jié)構(gòu)的絕對破產(chǎn)風(fēng)險模型,在該框架下,對任一狀態(tài)i時的即刻索賠,馬爾可夫鏈的狀態(tài)就會發(fā)生改變達(dá)到狀態(tài)j,而理賠額的分布Fj(y)是依賴于新的狀態(tài)j的。下一次索賠時間間隔服從參數(shù)為λj的指數(shù)分布。需要強調(diào)的是,在給定Zn-1和Zn的情況下,隨機變量Wn和Xn是相互獨立的,但在其連續(xù)索賠額的大小之間和連續(xù)索賠時間間隔之間存在自相關(guān)性,而在Wn和Xn之間存在交叉相關(guān)。 本章來自于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. 第五章研究了一類具有隨機分紅和隨機保費收入的離散風(fēng)險模型,其中保費收入過程和索賠過程均服從復(fù)合二項過程。當(dāng)公司盈余達(dá)到或超過界限b時,紅利以概率q0進(jìn)行支付1單位。我們導(dǎo)出了期望折現(xiàn)罰金函數(shù)滿足的遞推公式,作為應(yīng)用,給出了破產(chǎn)概率、破產(chǎn)赤字分布函數(shù)、破產(chǎn)赤字矩母函數(shù)的遞推公式。最后給出數(shù)值例子,分析了相關(guān)參數(shù)對破產(chǎn)概率的影響。 本章來自于Yu Wenguang. Randomized dividends in a discrete insurance risk model with stochastic premium income. Mathematical Problems in Engineering,2013(2013),1-9. 第六章研究了一類具有相依結(jié)構(gòu)的風(fēng)險模型,即兩次理賠間隔決定了下次理賠額的分布,當(dāng)理賠額服從指數(shù)分布時,得到了Gerber-Shiu期望折現(xiàn)罰金函數(shù)所滿足的積分—微分方程及拉普拉斯變換,作為應(yīng)用給出了破產(chǎn)時刻,破產(chǎn)赤字及破產(chǎn)前瞬時盈余的拉普拉斯變換。最后,在具有障礙分紅策略下的同一風(fēng)險模型中,分析了Gerber-Shiu期望折現(xiàn)罰金函數(shù)和期望折現(xiàn)分紅函數(shù)所滿足的積分—微分方程。 本章來自于Yu Wenguang, Huang Yujuan. Some results on a risk model with dependence between claim sizes and claim intervals.數(shù)學(xué)雜志,33(5)(2013),781-787. 第七章研究了一類帶有隨機保費收入的馬氏轉(zhuǎn)換風(fēng)險模型(也叫馬氏調(diào)制風(fēng)險模型),其中,保費收入過程、索賠過程和折現(xiàn)利息力過程均受馬氏過程控制,本章的目的是研究期望折現(xiàn)罰金函數(shù)所滿足的積分方程。作為該積分方程的應(yīng)用,當(dāng)狀態(tài)個數(shù)僅為1個時,且索賠額服從指數(shù)分布時,給出了破產(chǎn)時刻、破產(chǎn)前瞬時盈余和破產(chǎn)赤字的拉普拉斯變換的明確表達(dá)式。最后,給出了數(shù)值例子,討論了相關(guān)參數(shù)對上述精算量的影響。 本章來自于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.
【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2014
【分類號】：F840.4;F224
本文編號：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.
【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2014
【分類號】：F840.4;F224
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