几类风险模型的风险理论及相关问题
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摘要
Levy过程是一具有独立平稳增量的随机过程，具有如马尔可夫性，无穷可分性等许多良好的性质，在金融数学中一直扮演着重要的角色．另一方面，风险理论中的许多风险模型，如经典风险模型(复合泊松过程)，带干扰的经典风险模型，布朗运动等，均是一些特殊的Levy过程．这时，不禁要问若风险模型中描述保险风险的基本盈余过程为一般的Levy过程，其破产问题会是什么样子的?近几年来，许多学者对基本盈余过程为谱负的Levy过程的风险模型进行了研究．在本文第二至六章中，利用逼近的的方法对具有正、负跳跃的Levy风险模型及具有投资收益的Levy风险模型进行了初步的研究并且讨论了两类由某特殊Levy过程决定的并且具有某分红策略的Ornstein-Uhlenbeck型风险过程的分红问题．
渐近估计方法是研究破产问题的一种常用方法．在第七、八章中，首先给出了随机游动阶梯高度及极大值的局部估计和尾估计以及一瑕疵更新方程解的渐近估计，然后将结果应用到风险理论中，得到了一些新结论．
迄今，大多数风险模型都假定索赔时间间隔与索赔量是独立的．若两者关，会对破产概率及其他相关的量产生怎样的影响．在本文的最后一章，我们研究了一类索赔时间间隔与索赔量相关且带干扰的风险模型．
根据内容本文分为以下九章：
第一章：我们介绍了Levy过程及一些轻尾，重尾分布族的定义以及Levy过程的一些基本定理．对于论文中用到的一些符号也给出了规定．
第二章：我们推广了Garrido and Morales[47]的Levy风险过程，引入了既有正跳跃又有负跳跃的Levy风险过程，其具有下面的形式U(t)=u+ct-S(t)，t≥0，过程{S(t)，t≥0}为一具有正、负跳跃的Levy过程，它包含了非连续收取的保费和索赔总量．在本章中，利用逼近的方法，得到了此风险过程的罚金折现函数Φ满足的更新方程及其Φ的一个级数表达式．在第四节中，我们还讨论了上风险过程在初始盈余值趋于无穷大时，其破产概率的一些渐近形式．
第三章：设U_t为一个具有风险投资回报的风险过程在t时刻的盈余值，具有以下形式U_t=e~(Mt)(u+∫_0~te~(-Ms)dR_s),t≥0，U_0=u．其中，保险公司的基本盈余过程为一Levy风险过程：R_t=ct-J_t+σB_t．索赔总量过程{J_t，t≥0)为一无漂移和干扰项的Levy过程．过程{e~(Mt)=e~(δt+rWt)，t≥0}是一几何布朗运动，δ>0，r是两个固定的参数．{B_t，t≥0}和{W_t，t≥0}为两个相互独立的一维标准布朗运动．在本章中，利用逼近的方法得到了破产概率满足的积分-微分方程．
第四章：假设某保险公司采用这样的财政策略：当公司的资产大于某一水平△(>0)时，超过△的部分进行投资，取得利息率为r(≥0)的收益；当公司的资产为负值但不低于某一特定的值(-c／δ)时，通过贷款来继续维持公司的运作，贷款利率为δ(>0)．设U(t)为采用上述策略后某保险公司在t时刻的盈余，满足下面的随机微分方程：索赔总量过程Z=：{Z(t)，t≥0}是一个无漂移部分的从属过程(Subordinator)．
通过研究绝对破产时刻，绝对破产前的瞬间盈余及绝对破产时的赤字三者的联合分布来研究上模型的绝对破产问题．首先，得到了当索赔总量过程为复合泊松过程时，联合分布满足的积分-微分方程，并给出了方程的一般解．利用上述结果我们得到了当索赔总量过程为从属过程时，联合分布的一个表达式．
第五章：设具有某一分红策略的保险公司在t时刻的盈余值过程R={R_t，t≥0)为一Ornstein-Uhlenbeck型风险过程，满足下面的随机微分方程dR_t=(μ+ρR_t-l(t))dt+σdW_t，其中，{W_t，t≥0}为一标准的布朗运动，μ，ρ，σ>0为三个固定的常数，l(t)表示在t时刻的分红率函数．本章，讨论了当l(t)为有界函数时，分红策略的最优分红问题．
第六章：设带有固定分红策略的α-平稳Ornstein-Uhlenbeck(1<α≤2)型风险过程X：={X_t，t≥0}满足下面的随机微分方程：dX_t=-λX_tdt+dZ_t-dD_t，t≥0，其中，λ>0为一固定的参数，{Z_t，t≥0)为谱负α-平稳过程．D：={D_t，t≥0)为到时刻t为止的分红总量．利用广义Wright’s超几何函数给出此模型分红总量折现值各阶矩的表达式．
第七章：在本章中，讨论了实数域上具有负均值的随机游动，得到了在指数估计不成立的条件下，此随机游动的阶梯高度和极大值的局部估计以及尾估计．随后，我们将这些结果应用到风险理论中的Sparre Andersen模型中．
第八章；本章，我们对一类瑕疵更新方程解进行了研究，得到了其解的非指数渐近表达式，并将结果应用到风险理论中．
第九章：在本章中，讨论了一个索赔相依且带干扰的风险模型，利用Laplace变换研究了破产时刻，破产前的瞬间盈余及破产时的赤字的联合分布，得到了此联合分布Laplace变换的一个表达式．当上风险模型具有部分分红策略时，研究了分红折现期望及矩母函数，得到了他们的一个表达式．
Levy processes are stochastic processes with independent and stationary increments, and play a fundamental role in Mathematical Finance. On the other hand, many important risk processes are also special cases of Levy processes, such as Brownian motion, the compound Poisson process, the compound Poisson process perturbed by Brownian motion. In Chapters 2-4 of this thesis, we study ruin problems for a Levy processes with positive and negative jumps and for some Levy processes with investment. In Chapters 5-6, we introduce two Ornstein-Uhlenbeck type risk processes driven by some Levy process. The optimal dividend problem and the moments of the cumulative dividend are studied, respectively.
The asymptotic problems of ruin problems are important topics in risk theory. In.Chapters7-8, when the conditions for the exponential estimate are not satisfied, a local asymptotic estimate and a tail asymptotic estimate for the distributions of ladder height and supremum for the random walk are derived and non-exponential asymptotic forms for solutions of defectiverenewal equations are obtained. All the results are applied to risk models. In last chapter, we consider a jump-diffusion model with a dependent setting, where the claim inter-occurrence times depend on the previous claim size.
Organization and outline of this thesis
Chapter 1: We introduce the definition of and some important theorems related to Levy processes, and the definitions of some light-tailed and heavy-tailed distributions are introduced also. Some notations which will be used throughout the thesis are set down in this chapter.
Chapter 2: Garrido and Morales [47] introduced a Levy risk process only with negative jumps, in which they studied the Gerber-Shiu function. The aim of this chapter is to extend their work to a general Levy risk process with positive and negative jumps, that is, the Levy risk process has the formU(t)=u + ct-S(t),t≥0,{S(t)} is a jump Levy process with positive and negative jumps. We prove that the GerberShiu functionΦsatisfies a renewal equation and a infinite series expression ofΦis obtained. Some asymptotic behaviors of the ruin probability as u→∞are discussed.
Chapter 3: In this chapter, we assume the surplus of the insurer at time t under some investment assumption is denoted by Ut:U_t=e~(Mt)(u+∫_0~te~(-Ms)dR_s),t≥0,U_0=u.where R_t=ct-J_t+σB_t is a Levy risk process. The aggregate claims process {Jt,t≥0}is a jump Levy process without drift and diffusion coefficient, started at 0. The process{e~(Mt)=e~(δt+rWt),t≥0}is a geometric Brownian motion,δ> 0 and r are constants, and{W_t,t≥0} is an one-dimensional Brownian motion independent of {Rt, t≥0}. In thischapter, it is shown that the ruin probabilities (by a jump or by oscillation) of the resulting surplus process satisfy certain integro-differential equations.
Chapter 4: Let U(t) be the surplus of insurer at time t with the liquid reserve level,. the credit interest force r≥0 and debit interest forceδ> 0. Then the surplus process {U(t), t≥0} satisfies the following stochastic differential equation:where u≥0 is the initial value and c > 0 is a constant premium rate, defined as c = (1 +θ)EZ(1), whereθ≥0 is the security loading factor. {Z(t), t≥0} is a subordinator with zero drift.
We study the absolute ruin questions by defining the joint distributed function of the absolute ruin time, the surplus immediately before absolute ruin and the deficit at absolute ruin. Using an approximation scheme, a general expression for the joint distributed function of the risk process driven by a subordinator is obtained.
Chapter 5: In this chapter, a controlled Ornstein-Uhlenbeck type model is studied whose reserve R_t is assumed to be governed by the stochastic differential equationdR_t=(μ+ρR_t-l(t))dt+σdW_t,where {W_t,t≥0} is a standard Brownian motion,μ,ρ,σ>0 are constants and l(t) is the rate of dividend payment at time t. It is shown how the optimal return function and the optimal dividend-payment strategy can be calculated when the rate function l(t) is restricted so that the function l(t) varies in [0, M] for some M <∞.
Chapter 6: In this chapter, we present a spectrally negative a-stable Ornstein-Uhlenbeck (1 <α≤2) type risk process X := {X_t,t≥0} with dividend barrier, which is the solution to the linear stochastic differential equationdX_t=-λX_tdt+dZ_t-dD_t,with X_0 = x, and the parameterλ> 0, Z := {Z_t,t≥0} be a spectrally negativeα-stableprocess, withα∈(1,2]. D := {D_t, t≥0} be the cumulative dividends process. The momentsof the present value of dividend payments until ruin are provided in terms of the Wright's generalized hypergeometric function 2ψ1.
Chapter 7: For a random walk on the real line with negative mean, we obtain a local asymptotic estimate and a tail asymptotic estimate for the distributions of ladder height and supremum for the random walk when the conditions for the exponential estimate are not satisfied. The results are applied to the Sparre Andersen model, some new results on the probability of ruin are presented.
Chapter 8: In this chapter, we derive non-exponential asymptotic forms for solutions of defective renewal equations. Applications of this result is given to the Gerber-Shiu discounted penalty function in the classical risk model.
Chapter 9: In this chapter, we consider a jump-diffusion model with a dependent setting, where the claim inter-occurrence times depend on the previous claim size. We study the joint distribution of the time of ruin, the surplus prior to ruin and the deficit at ruin, and an exact analytical expression for the Laplace transform of the the joint distribution is derived. We also study the dividend problem for the above model with a dividend strategy. A system of integro-differential equations with certain boundary conditions satisfied by the expected discounted dividend payments prior to ruin is derived and an infinite series expression of it is obtained.
渐近估计方法是研究破产问题的一种常用方法．在第七、八章中，首先给出了随机游动阶梯高度及极大值的局部估计和尾估计以及一瑕疵更新方程解的渐近估计，然后将结果应用到风险理论中，得到了一些新结论．
迄今，大多数风险模型都假定索赔时间间隔与索赔量是独立的．若两者关，会对破产概率及其他相关的量产生怎样的影响．在本文的最后一章，我们研究了一类索赔时间间隔与索赔量相关且带干扰的风险模型．
根据内容本文分为以下九章：
第一章：我们介绍了Levy过程及一些轻尾，重尾分布族的定义以及Levy过程的一些基本定理．对于论文中用到的一些符号也给出了规定．
第二章：我们推广了Garrido and Morales[47]的Levy风险过程，引入了既有正跳跃又有负跳跃的Levy风险过程，其具有下面的形式U(t)=u+ct-S(t)，t≥0，过程{S(t)，t≥0}为一具有正、负跳跃的Levy过程，它包含了非连续收取的保费和索赔总量．在本章中，利用逼近的方法，得到了此风险过程的罚金折现函数Φ满足的更新方程及其Φ的一个级数表达式．在第四节中，我们还讨论了上风险过程在初始盈余值趋于无穷大时，其破产概率的一些渐近形式．
第三章：设U_t为一个具有风险投资回报的风险过程在t时刻的盈余值，具有以下形式U_t=e~(Mt)(u+∫_0~te~(-Ms)dR_s),t≥0，U_0=u．其中，保险公司的基本盈余过程为一Levy风险过程：R_t=ct-J_t+σB_t．索赔总量过程{J_t，t≥0)为一无漂移和干扰项的Levy过程．过程{e~(Mt)=e~(δt+rWt)，t≥0}是一几何布朗运动，δ>0，r是两个固定的参数．{B_t，t≥0}和{W_t，t≥0}为两个相互独立的一维标准布朗运动．在本章中，利用逼近的方法得到了破产概率满足的积分-微分方程．
第四章：假设某保险公司采用这样的财政策略：当公司的资产大于某一水平△(>0)时，超过△的部分进行投资，取得利息率为r(≥0)的收益；当公司的资产为负值但不低于某一特定的值(-c／δ)时，通过贷款来继续维持公司的运作，贷款利率为δ(>0)．设U(t)为采用上述策略后某保险公司在t时刻的盈余，满足下面的随机微分方程：索赔总量过程Z=：{Z(t)，t≥0}是一个无漂移部分的从属过程(Subordinator)．
通过研究绝对破产时刻，绝对破产前的瞬间盈余及绝对破产时的赤字三者的联合分布来研究上模型的绝对破产问题．首先，得到了当索赔总量过程为复合泊松过程时，联合分布满足的积分-微分方程，并给出了方程的一般解．利用上述结果我们得到了当索赔总量过程为从属过程时，联合分布的一个表达式．
第五章：设具有某一分红策略的保险公司在t时刻的盈余值过程R={R_t，t≥0)为一Ornstein-Uhlenbeck型风险过程，满足下面的随机微分方程dR_t=(μ+ρR_t-l(t))dt+σdW_t，其中，{W_t，t≥0}为一标准的布朗运动，μ，ρ，σ>0为三个固定的常数，l(t)表示在t时刻的分红率函数．本章，讨论了当l(t)为有界函数时，分红策略的最优分红问题．
第六章：设带有固定分红策略的α-平稳Ornstein-Uhlenbeck(1<α≤2)型风险过程X：={X_t，t≥0}满足下面的随机微分方程：dX_t=-λX_tdt+dZ_t-dD_t，t≥0，其中，λ>0为一固定的参数，{Z_t，t≥0)为谱负α-平稳过程．D：={D_t，t≥0)为到时刻t为止的分红总量．利用广义Wright’s超几何函数给出此模型分红总量折现值各阶矩的表达式．
第七章：在本章中，讨论了实数域上具有负均值的随机游动，得到了在指数估计不成立的条件下，此随机游动的阶梯高度和极大值的局部估计以及尾估计．随后，我们将这些结果应用到风险理论中的Sparre Andersen模型中．
第八章；本章，我们对一类瑕疵更新方程解进行了研究，得到了其解的非指数渐近表达式，并将结果应用到风险理论中．
第九章：在本章中，讨论了一个索赔相依且带干扰的风险模型，利用Laplace变换研究了破产时刻，破产前的瞬间盈余及破产时的赤字的联合分布，得到了此联合分布Laplace变换的一个表达式．当上风险模型具有部分分红策略时，研究了分红折现期望及矩母函数，得到了他们的一个表达式．
Levy processes are stochastic processes with independent and stationary increments, and play a fundamental role in Mathematical Finance. On the other hand, many important risk processes are also special cases of Levy processes, such as Brownian motion, the compound Poisson process, the compound Poisson process perturbed by Brownian motion. In Chapters 2-4 of this thesis, we study ruin problems for a Levy processes with positive and negative jumps and for some Levy processes with investment. In Chapters 5-6, we introduce two Ornstein-Uhlenbeck type risk processes driven by some Levy process. The optimal dividend problem and the moments of the cumulative dividend are studied, respectively.
The asymptotic problems of ruin problems are important topics in risk theory. In.Chapters7-8, when the conditions for the exponential estimate are not satisfied, a local asymptotic estimate and a tail asymptotic estimate for the distributions of ladder height and supremum for the random walk are derived and non-exponential asymptotic forms for solutions of defectiverenewal equations are obtained. All the results are applied to risk models. In last chapter, we consider a jump-diffusion model with a dependent setting, where the claim inter-occurrence times depend on the previous claim size.
Organization and outline of this thesis
Chapter 1: We introduce the definition of and some important theorems related to Levy processes, and the definitions of some light-tailed and heavy-tailed distributions are introduced also. Some notations which will be used throughout the thesis are set down in this chapter.
Chapter 2: Garrido and Morales [47] introduced a Levy risk process only with negative jumps, in which they studied the Gerber-Shiu function. The aim of this chapter is to extend their work to a general Levy risk process with positive and negative jumps, that is, the Levy risk process has the formU(t)=u + ct-S(t),t≥0,{S(t)} is a jump Levy process with positive and negative jumps. We prove that the GerberShiu functionΦsatisfies a renewal equation and a infinite series expression ofΦis obtained. Some asymptotic behaviors of the ruin probability as u→∞are discussed.
Chapter 3: In this chapter, we assume the surplus of the insurer at time t under some investment assumption is denoted by Ut:U_t=e~(Mt)(u+∫_0~te~(-Ms)dR_s),t≥0,U_0=u.where R_t=ct-J_t+σB_t is a Levy risk process. The aggregate claims process {Jt,t≥0}is a jump Levy process without drift and diffusion coefficient, started at 0. The process{e~(Mt)=e~(δt+rWt),t≥0}is a geometric Brownian motion,δ> 0 and r are constants, and{W_t,t≥0} is an one-dimensional Brownian motion independent of {Rt, t≥0}. In thischapter, it is shown that the ruin probabilities (by a jump or by oscillation) of the resulting surplus process satisfy certain integro-differential equations.
Chapter 4: Let U(t) be the surplus of insurer at time t with the liquid reserve level,. the credit interest force r≥0 and debit interest forceδ> 0. Then the surplus process {U(t), t≥0} satisfies the following stochastic differential equation:where u≥0 is the initial value and c > 0 is a constant premium rate, defined as c = (1 +θ)EZ(1), whereθ≥0 is the security loading factor. {Z(t), t≥0} is a subordinator with zero drift.
We study the absolute ruin questions by defining the joint distributed function of the absolute ruin time, the surplus immediately before absolute ruin and the deficit at absolute ruin. Using an approximation scheme, a general expression for the joint distributed function of the risk process driven by a subordinator is obtained.
Chapter 5: In this chapter, a controlled Ornstein-Uhlenbeck type model is studied whose reserve R_t is assumed to be governed by the stochastic differential equationdR_t=(μ+ρR_t-l(t))dt+σdW_t,where {W_t,t≥0} is a standard Brownian motion,μ,ρ,σ>0 are constants and l(t) is the rate of dividend payment at time t. It is shown how the optimal return function and the optimal dividend-payment strategy can be calculated when the rate function l(t) is restricted so that the function l(t) varies in [0, M] for some M <∞.
Chapter 6: In this chapter, we present a spectrally negative a-stable Ornstein-Uhlenbeck (1 <α≤2) type risk process X := {X_t,t≥0} with dividend barrier, which is the solution to the linear stochastic differential equationdX_t=-λX_tdt+dZ_t-dD_t,with X_0 = x, and the parameterλ> 0, Z := {Z_t,t≥0} be a spectrally negativeα-stableprocess, withα∈(1,2]. D := {D_t, t≥0} be the cumulative dividends process. The momentsof the present value of dividend payments until ruin are provided in terms of the Wright's generalized hypergeometric function 2ψ1.
Chapter 7: For a random walk on the real line with negative mean, we obtain a local asymptotic estimate and a tail asymptotic estimate for the distributions of ladder height and supremum for the random walk when the conditions for the exponential estimate are not satisfied. The results are applied to the Sparre Andersen model, some new results on the probability of ruin are presented.
Chapter 8: In this chapter, we derive non-exponential asymptotic forms for solutions of defective renewal equations. Applications of this result is given to the Gerber-Shiu discounted penalty function in the classical risk model.
Chapter 9: In this chapter, we consider a jump-diffusion model with a dependent setting, where the claim inter-occurrence times depend on the previous claim size. We study the joint distribution of the time of ruin, the surplus prior to ruin and the deficit at ruin, and an exact analytical expression for the Laplace transform of the the joint distribution is derived. We also study the dividend problem for the above model with a dividend strategy. A system of integro-differential equations with certain boundary conditions satisfied by the expected discounted dividend payments prior to ruin is derived and an infinite series expression of it is obtained.
引文
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[11] Asmussen S., Foss S. and Korshunov D. Asymptotics for sums of random variables with localsubexponential behaviour [J]. Journal of Theoretical Probability, 2003,16: 489-518.
[12] Bulmann H. Mathematical Methods in Risk Theory [M]. Springer-Verlag, Berlin, 1970.
[13] Bertoin J. Levy Processes [M]. Cambridge Tracts in Mathematics: Cambridge University Press,1996.
[14] Cai J. and Garrido J. Asymptotic forms and tails of convolutions of compound geometric distributions, with applications [M]. In Recent Advances in Statistical Methods, ed. Y. P. Chaubed, London: Imperial College Press, 1999, pp. 114-131.
[15] Cai J. and Dickson D. C. M. On the expected discounted penalty function at the ruin of a surplusprocess with interest [J]. Insurance: Mathematics and Economics, 2002, 30: 389-404.
[16] Cai J. Ruin probabilities and penalty functions with stochastic rates of interest [J]. StochasticProcesses and their Applications, 2004, 12: 53-78.
[17] Cai J. and Yang H. L. Ruin in the prturbed compound Poisson risk process under interest force[J]. Advances in Applied Probability, 2005, 37: 819-835.
[18] Cai J. and Xu C. M. On the decomposition of the ruin probability for a jump-diffusion surplus process compounded by a geometric Brownian motion [J]. North American Actuarial Journal, 2006, 10(2): 120-132.
[19] Cai J., Gerber H. U. and Yang H. L. Optimal dividends in an Ornstein-Uhlenbeck type model??with credit and debit interest [J]. North American Actuarial Journal, 2006, 10(2): 94-119.
[20] Cai J., Feng R. H. and Willmot G. E. The compound Poisson surplus model with interest andliquid reserves: analysis of the Gerber-Shiu discounted penalty function [J]. Methodology andComputing in Applied Probability, 2007, to appear: DOI 10.1007/sll009-007-9050-6.
[21] Cai J. On the time value of absolute ruin with debit interest [J]. Advances in Applied Probability,2007, 39: 343-359.
[22] Cheng K. and Wang W. Y. Approximations of a two-unit cold standby system's reliability behaviour [J]. Microelectronics and Reliability, 1996, 36(5): 627-630.
[23] Cheng Y., Tang Q. and Yang H. Approximations for moments of deficit at ruin with exponentialand subexponential claims [J]. Statistics & Probability Letters, 2002, 59: 367-378.
[24] Cheng Y. and Tang Q. Moments of the surplus before ruin and the deficit at ruin in the Erlang(2)risk process [J]. North American Actuarial Journal, 2003, 7: 1-12.
[25] Chiu S. N. and Yin C. C. The time of ruin, the surplus prior to ruin and the deficit at ruin for theclassical risk process perturbed by diffusion [J]. Insurance: Mathematics and Economics, 2003,33: 59-66.
[26] Chistykov V. P. A theorem on sums of independent positive random variables and its applicationsto branching random processes [J]. Theory of Probability and its Applications, 1964, 9: 640-648.
[27] Chover J. Ney P. and Wainger S. Degeneracy properties of subcritical branching process [J].Annals of Probability, 1973, 1: 663-673.
[28] Chover J., Ney P. and Wainger S. Function of probability measures [J]. Journal d'AnalyseMathematique, 1973, 26: 255-302.
[29] Cline D. B. H. Convolutions of distributions with exponential and subexponential tails [J]. Journalof the Australian Mathematical Society (Series A), 1987, 43:. 347-365.
[30] De Finetti Bruno, Su un'impostazione alternativa dell teoria collettiva del rischio [J]. Transactionsof the XVth International Congress of Actuaries, 1957, 2: 433-443.
[31] Denisov D., Foss S. and Korshunov D. Tail asymptotics for the surpremum of a random walkwhen the mean is not finite [J]. Queueing Systems, 2004, 46: 15-33.
[32] Dickson D. C. M. and Waters H. Some optimal dividends problems [J]. ASTTN Bulletin, 2004,34: 49-74.
[33] Doney R. A. and Kyprianou A. E. Overshoots and undershoots of Levy processes [J]. Annals ofApplied Probability, 2006, 16(1): 91-106.
[34] Doob J. L. The Brownian movement and stochastic equations [J]. Ann. Math., 1942, 43(2):351-369.
[35] Dufresne F., Gerber H. U. and Shiu E. S. W. Risk theory with the gamma process [I]. ASTINBulletin, 1991, 21(2): 177-192.
[36] Dufresne F. and Gerber H. U., Risk theory for the compound Poisson process that is perturbed by diffusion [J]. Insurance: Mathematics and Economics, 1991,10: 51-59.
[37] Embrechts P., Goldie C. M. and Veraverbeke N. Subexponentiality and infinite divisibility [J]. Probability Theory and Related Fields, 1979; 49: 335-347.
[38] Embrechts P. and Veraverbeke N. Estimates for the probability of ruin with special emphasis on the possibility of large claims [J]. Insurance: Mathematics and Economics, 1982, 1: 55-72.
[39] Embrechts P. and Goldie C. M. On convolution tails [J]. Stochastic Processes and their Applications, 1982,13: 263-278.
[40] Embrechts P. and Omey E. A property of longtailed distributions [J]. Journal of Applied Probability, 1984, 21: 80-87.
[41] Embrechts P. and Schmidli H. Ruin estimation for a general insurance risk model [J]. Advances in Applied Probability, 1994, 26: 404-422.
[42] Embrechts P. Kluppelberg C. and Mikosch T. Modelling Extremal Events [M]. Berlin: Springer, 1997.
[43] Erdelyi A., Magnus W., Oberhettinger F. and Tricomi F. G. Higher Transcendental Func-tions(volume 3) [M]. McGraw-Hill, New York-Toronto-London, 1955.
[44] Feller W. An Introduction to Probability Theory and its Applications (Vol. II, 2nd ed.) [M]. New York: Wiley & Sons, 1971.
[45] Fleming W. H. and Rishel R. W. Deterministic and stochastic optimal control [M]. Springer-Verlag, 1975.
[46] Frolova A., Yuri K. and Serguei P. In the insurance business risky investments are dangerous [J]. Finance and Stochastic, 2002, 6: 227-235.
[47] Garrido J. and Morales M. On the expected discounted penalty function for a Levy risk process [J]. North American Actuarial Journal, 2006,10: 196-218.
[48] Gerber H. U. An extension of the renewal equation and its application in the collective theory of risk [J]. Skandinavisk Aktuarietidskrift, 1970: 205-210.
[49] Gerber H.U. Games of economic survival with discrete and continuous income process [J]. Operation Research, 1972, 20: 37-45.
[50] Gerber H. An Introduction to Mathematical Risk Theory [M]. S. S. Huebner Foundation. University of Pennsylvania, Philadelphia, 1979.
[51] Gerber H. U. On the probability of ruin in the presence of a linear dividend barrier [J]. Scandinavian Actuarial Journal, 1981: 105-115.
[52] Gerber H. U. and Shiu E. S. W. On the time value of ruin [J]. North American Actuarial Journal, 1998, 2: 48-72.
[53] Gerber H. U. and Landry B. On the discounted penalty at ruin in a jump-diffusion and the perpetual put option [J]. Insurance: Mathematics and Economics, 1998, 22: 263-276.
[54] Gerber H. U. and Shiu E. S. W. Optimal Dividends: Analysis with Brownian Motion [J]. North American Actuarial Journal, 2004, 8(1): 1-20.
[55] Gerber H. U. and Shiu E. S. W. The time value of ruin in a Sparre Andersen model [J]. North??American Actuarial Journal, 2005, 9: 49-69.
[56] Gerber H. U. and Shiu E. S. W. On optimal dividend strategies in the compound .Poisson model[J]. North American Actuarial Journal, 2006, 10(2): 76-93.
[57] Goldie C. M. and Kluppelberg C. Subexponential distributions [M]. In "A Practical Guide toHeavy-Tails: Statistical Techniques and Applications". Eds, Adler R J, Feldman R E, Taqqu MS, Birkhauser, 1998.
[58] H(?)gaard B. and Taksar M. Controlling risk exposure and dividends pay-out schemes: Insurancecompany example [J]. Mathematical Finance, 1999, 9 (2): 153-182.
[59] Huzak M., Sikic H. and Vondracek Z. Ruin probabilities and decompositions for general perturbedrisk processes [J]. Annals of Applied Probability, 2004, 14(3): 1378-1397.
[60] Hu F., Yin C. C. and Zong Z. J. Tail equivalence relationships for ruin probabilities in severalrisk models [J]. Applied Stochastic Models in Business and Industry, 2005, 21: 499-508.
[61] Kluppelberg C. On subexponential distributions and integrated tails [J]. Journal of AppliedProbability, 1988, 5: 132-141.
[62] Kluppelberg C. Subexponential distributions and characterizations of related classes [J]. Probability Theory and their Related Fields, 1989, 82: 259-269.
[63] Kluppelberg C, Kyprianou A. E. and Mailer R. A. Ruin probabilities and overshoots for generalLevy insurance risk processes [J]. Annals of Applied Probability, 2004,14(4): 1766-1801.
[64] Korshunov D. On distribution tail of the maximum of a random walks [J]. Stochastic Processesand their Applications, 1997, 72: 97-103.
[65] Lin X. S., Willmot G. E. and Drekic S. The classical risk model with a constant dividend barrier[J]. Insurance: Mathematics and Economics, 2003, 33: 551-566.
[66] Li S. and Garrido J. On ruin for the Erlang(n) risk process [J]. Insurance: Mathematics andEconomics, 2004, 34: 391-408.
[67] Li S. The distribution of the dividend payments in the compound Poisson risk model perturbedby diffusion [J]. Scandinavian Actuarial Journal, 2006: 1-13.
[68] Linz et al. Analystical and numerical methods for Volterra equations [M]. Studies 7, SIAM Studiesin Applied Mathematocs: Philadelpha, 1985.
[69] Muller A. M. and Pflug G. Asymptotic ruin probabilities for risk processes with dependentincrements [J]. Insurance: Mathematics and Economics, 2001, 28(3): 381-392.
[70] Morales M. On the expected discounted penalty function for a perturbed risk process driven bya subordinator [J]. Insurance: Mathematics and Economics, 2007, 40: 293-301.
[71] Ng K. W., Tang Q. and Yang H. Maxima of sums of heavy-tailed random variables [J]. ASTINBulletin, 2002, 32: 43-55.
[72] Nyrhinen H. Rough descriptions of ruin for a general class of surplus processes [J], Advances inApplied Probability, 1998, 30: 1008-1026.
[73] Paulsen J. Risk theory in a stochastic economic environment [J]. Stochastic Processes and their??Applications, 1993, 46: 327-361.
[74] Paulsen J. and Gjessing H. K. Ruin theory with stochastic return on investments [J]. Advancesin Applied Probability, 1997, 29: 965-985.
[75] Patie P. Two-sided exit problem for a spectrally negative α-stable Ornstein-Uhlenbeck processand the Wright's generalized hypergeometric function [J]. Electronic Communications in Probability, 2007,12: 146-160.
[76] Po-Fang Hsieh and Sibuya Y. Basic theory of ordinary differential equations [M], Springer, NewYork, 1999.
[77] Protter P. Stochastic Integration and Differential Equations[M]. New York: Springer, 1990.
[78] Ross S. Bounds on the delay distribution in GI/G/1 queues [J]. Journal of Applied Probability,1974,11: 417-421.
[79] Ross S. Stochastic Processes (2nd edition) [M]. Wiley, New York, 1996.
[80] Rolski T., Schmidli H. and Schmidt V. Stochastic Processes for Insurance and Finance [M]. NewYork: Wiley 1999.
[81] Sato K. Levy Processes and Infinitely Divisible Distributions [M]. Cambridge University Press,Cambridge, 1999.
[82] Sgibnev M. S. Banach algebras of measures of class S(v) [J]. Siberian Math J, 1988, 29: 647-655.
[83] Srivastava H.M. and Karlsson P.W. Multiple Gaussian Hypergeometric Series[M]. Ellis HorwoodSeries: Math. Appl. Ellis Horwood Ltd., Chichester, 1985.
[84] Stam A. J. Regular variation of the tail of a subordinated probability distribution [J]. Advancesin Applied Probability, 1973, 5(2): 308-327.
[85] Sundt B. and Teuels J. L. Ruin estimates under interest force [J]. Insurance: Mathematics andEconomics, 1995,15: 7-22.
[86] Tang Q. An asymptotic relationship for ruin probabilities under heavy-tailed claims [J]. Sciencein China (Series A). 2002, 45: 632-639.
[87] Tsai C. C. L. and Willmot G. E. A generalized defective renewal equation for the surplus processperturbed by diffusion [J]. Insurance: Mathematics and Economics, 2002, 30: 51-66.
[88] Veraverbeke N. Asymptotic behavior of Wiener-Hopf factors of a random walk [J]. StochasticProcesses and their Applications, 1977, 5: 27-37.
[89] Wang G. J. and Wu R. Some distributions for classical risk processes that is perturbed by diffusion[J], Insurance: Mathematics and Economics, 2000, 26: 15-24.
[90] Wang G. J. A decomposition of the ruin probability for the risk process perturbed by diffusion[J]. Insurance: Mathematics and Economics, 2001, 28: 49-59.
[91] Wang G. J. and Wu R. Distributions for the risk process with a stochastic return on investment[J]. Stochastic Processes and their Applications, 2001, 95: 329-341.
[92] Wan N. Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion [J], Insurance: Mathematics and Economics, 2007, 40(3): 509-523.
[93] Willmot G., Cai J. and Lin X. S. Lundberg inequalities for renewal equations [J], Advances in Applied Probability, 2001, 33: 674-689.
[94] Yang H. and Zhang L. Spectrally negative Levy processes with applications in risk theory [J]. Advances in Applied Probability, 2001, 33(1): 281-291.
[95] Yin C. C. A local theorem for the probability of ruin [J]. Science in China (Series A). 2004, 47: 711-721.
[96] Yin C. C. and Zhao J. S. Non-exponential asymptotics for the solutions of renewal equations with applications [J], Journal of Applied Probability, 2006, 43(3): 815-824.
[97] Zachary S. A note on Veraverbeke's theorem [J]. Queueing Systems, 2004, 46: 9-14.
[98] Zhou X.W. On a classical risk model with a constant dividend barrier [J], North American Actuarial Journal, 2005, 9: 95-108.
[2] Albrecher H. and Kantor J. Simulation of ruin probabilities for risk processes of Markovian type[J]. Methodology and Computing in Applied Probability, 2002, 8: 111-127.
[3] Albrecher H., Kainhofer R. and Tichy R. F. Simulation methods in ruin models with non-lineardividend barriers [J]. Mathematics and Computers in Simulation, 2003, 62: 277-287.
[4] Albrecher H. and Boxma O. J. A ruin model with dependence between claim sizes and claimintervals [J]. Insurance: Mathematics and Economics, 2004, 35: 245-254.
[5] Albrecher H. and Boxma O. J. On the discounted penalty function in a Markov-dependent riskmodel [J]. Insurance: Mathematics and Economics, 2005, 37: 650-672.
[6] Albrecher H., Hartinger J. and Tichy R. F. On the distribution of dividend payments and the discounted function in a risk model with linear dividend barrier [J]. Scandinavian Actuarial Journal, 2005, 2: 103-126.
[7] Asmussen S. and Taskar M. Controlled diffusion models for optimal dividend pay-out [J]. Insurance: Mathematics and Economics, 1997, 20: 1-15.
[8] Asmussen S. A probabilistic look at the Wiener-Hopf equation [J]. SLAM Review, 1998, 40:189-201.
[9] Asmussen S: Ruin Probabilities [M]. World Scientific, Singapore, 2000.
[10] Asmussen S., Kalashnikov V., Konstantinides D., Kluppelbergd C. and Tsitsiashvili G. A local limit theorem for random walk maxima with heavy tails [J]. Statistics & Probability Letters, 2002, 56: 399-404.
[11] Asmussen S., Foss S. and Korshunov D. Asymptotics for sums of random variables with localsubexponential behaviour [J]. Journal of Theoretical Probability, 2003,16: 489-518.
[12] Bulmann H. Mathematical Methods in Risk Theory [M]. Springer-Verlag, Berlin, 1970.
[13] Bertoin J. Levy Processes [M]. Cambridge Tracts in Mathematics: Cambridge University Press,1996.
[14] Cai J. and Garrido J. Asymptotic forms and tails of convolutions of compound geometric distributions, with applications [M]. In Recent Advances in Statistical Methods, ed. Y. P. Chaubed, London: Imperial College Press, 1999, pp. 114-131.
[15] Cai J. and Dickson D. C. M. On the expected discounted penalty function at the ruin of a surplusprocess with interest [J]. Insurance: Mathematics and Economics, 2002, 30: 389-404.
[16] Cai J. Ruin probabilities and penalty functions with stochastic rates of interest [J]. StochasticProcesses and their Applications, 2004, 12: 53-78.
[17] Cai J. and Yang H. L. Ruin in the prturbed compound Poisson risk process under interest force[J]. Advances in Applied Probability, 2005, 37: 819-835.
[18] Cai J. and Xu C. M. On the decomposition of the ruin probability for a jump-diffusion surplus process compounded by a geometric Brownian motion [J]. North American Actuarial Journal, 2006, 10(2): 120-132.
[19] Cai J., Gerber H. U. and Yang H. L. Optimal dividends in an Ornstein-Uhlenbeck type model??with credit and debit interest [J]. North American Actuarial Journal, 2006, 10(2): 94-119.
[20] Cai J., Feng R. H. and Willmot G. E. The compound Poisson surplus model with interest andliquid reserves: analysis of the Gerber-Shiu discounted penalty function [J]. Methodology andComputing in Applied Probability, 2007, to appear: DOI 10.1007/sll009-007-9050-6.
[21] Cai J. On the time value of absolute ruin with debit interest [J]. Advances in Applied Probability,2007, 39: 343-359.
[22] Cheng K. and Wang W. Y. Approximations of a two-unit cold standby system's reliability behaviour [J]. Microelectronics and Reliability, 1996, 36(5): 627-630.
[23] Cheng Y., Tang Q. and Yang H. Approximations for moments of deficit at ruin with exponentialand subexponential claims [J]. Statistics & Probability Letters, 2002, 59: 367-378.
[24] Cheng Y. and Tang Q. Moments of the surplus before ruin and the deficit at ruin in the Erlang(2)risk process [J]. North American Actuarial Journal, 2003, 7: 1-12.
[25] Chiu S. N. and Yin C. C. The time of ruin, the surplus prior to ruin and the deficit at ruin for theclassical risk process perturbed by diffusion [J]. Insurance: Mathematics and Economics, 2003,33: 59-66.
[26] Chistykov V. P. A theorem on sums of independent positive random variables and its applicationsto branching random processes [J]. Theory of Probability and its Applications, 1964, 9: 640-648.
[27] Chover J. Ney P. and Wainger S. Degeneracy properties of subcritical branching process [J].Annals of Probability, 1973, 1: 663-673.
[28] Chover J., Ney P. and Wainger S. Function of probability measures [J]. Journal d'AnalyseMathematique, 1973, 26: 255-302.
[29] Cline D. B. H. Convolutions of distributions with exponential and subexponential tails [J]. Journalof the Australian Mathematical Society (Series A), 1987, 43:. 347-365.
[30] De Finetti Bruno, Su un'impostazione alternativa dell teoria collettiva del rischio [J]. Transactionsof the XVth International Congress of Actuaries, 1957, 2: 433-443.
[31] Denisov D., Foss S. and Korshunov D. Tail asymptotics for the surpremum of a random walkwhen the mean is not finite [J]. Queueing Systems, 2004, 46: 15-33.
[32] Dickson D. C. M. and Waters H. Some optimal dividends problems [J]. ASTTN Bulletin, 2004,34: 49-74.
[33] Doney R. A. and Kyprianou A. E. Overshoots and undershoots of Levy processes [J]. Annals ofApplied Probability, 2006, 16(1): 91-106.
[34] Doob J. L. The Brownian movement and stochastic equations [J]. Ann. Math., 1942, 43(2):351-369.
[35] Dufresne F., Gerber H. U. and Shiu E. S. W. Risk theory with the gamma process [I]. ASTINBulletin, 1991, 21(2): 177-192.
[36] Dufresne F. and Gerber H. U., Risk theory for the compound Poisson process that is perturbed by diffusion [J]. Insurance: Mathematics and Economics, 1991,10: 51-59.
[37] Embrechts P., Goldie C. M. and Veraverbeke N. Subexponentiality and infinite divisibility [J]. Probability Theory and Related Fields, 1979; 49: 335-347.
[38] Embrechts P. and Veraverbeke N. Estimates for the probability of ruin with special emphasis on the possibility of large claims [J]. Insurance: Mathematics and Economics, 1982, 1: 55-72.
[39] Embrechts P. and Goldie C. M. On convolution tails [J]. Stochastic Processes and their Applications, 1982,13: 263-278.
[40] Embrechts P. and Omey E. A property of longtailed distributions [J]. Journal of Applied Probability, 1984, 21: 80-87.
[41] Embrechts P. and Schmidli H. Ruin estimation for a general insurance risk model [J]. Advances in Applied Probability, 1994, 26: 404-422.
[42] Embrechts P. Kluppelberg C. and Mikosch T. Modelling Extremal Events [M]. Berlin: Springer, 1997.
[43] Erdelyi A., Magnus W., Oberhettinger F. and Tricomi F. G. Higher Transcendental Func-tions(volume 3) [M]. McGraw-Hill, New York-Toronto-London, 1955.
[44] Feller W. An Introduction to Probability Theory and its Applications (Vol. II, 2nd ed.) [M]. New York: Wiley & Sons, 1971.
[45] Fleming W. H. and Rishel R. W. Deterministic and stochastic optimal control [M]. Springer-Verlag, 1975.
[46] Frolova A., Yuri K. and Serguei P. In the insurance business risky investments are dangerous [J]. Finance and Stochastic, 2002, 6: 227-235.
[47] Garrido J. and Morales M. On the expected discounted penalty function for a Levy risk process [J]. North American Actuarial Journal, 2006,10: 196-218.
[48] Gerber H. U. An extension of the renewal equation and its application in the collective theory of risk [J]. Skandinavisk Aktuarietidskrift, 1970: 205-210.
[49] Gerber H.U. Games of economic survival with discrete and continuous income process [J]. Operation Research, 1972, 20: 37-45.
[50] Gerber H. An Introduction to Mathematical Risk Theory [M]. S. S. Huebner Foundation. University of Pennsylvania, Philadelphia, 1979.
[51] Gerber H. U. On the probability of ruin in the presence of a linear dividend barrier [J]. Scandinavian Actuarial Journal, 1981: 105-115.
[52] Gerber H. U. and Shiu E. S. W. On the time value of ruin [J]. North American Actuarial Journal, 1998, 2: 48-72.
[53] Gerber H. U. and Landry B. On the discounted penalty at ruin in a jump-diffusion and the perpetual put option [J]. Insurance: Mathematics and Economics, 1998, 22: 263-276.
[54] Gerber H. U. and Shiu E. S. W. Optimal Dividends: Analysis with Brownian Motion [J]. North American Actuarial Journal, 2004, 8(1): 1-20.
[55] Gerber H. U. and Shiu E. S. W. The time value of ruin in a Sparre Andersen model [J]. North??American Actuarial Journal, 2005, 9: 49-69.
[56] Gerber H. U. and Shiu E. S. W. On optimal dividend strategies in the compound .Poisson model[J]. North American Actuarial Journal, 2006, 10(2): 76-93.
[57] Goldie C. M. and Kluppelberg C. Subexponential distributions [M]. In "A Practical Guide toHeavy-Tails: Statistical Techniques and Applications". Eds, Adler R J, Feldman R E, Taqqu MS, Birkhauser, 1998.
[58] H(?)gaard B. and Taksar M. Controlling risk exposure and dividends pay-out schemes: Insurancecompany example [J]. Mathematical Finance, 1999, 9 (2): 153-182.
[59] Huzak M., Sikic H. and Vondracek Z. Ruin probabilities and decompositions for general perturbedrisk processes [J]. Annals of Applied Probability, 2004, 14(3): 1378-1397.
[60] Hu F., Yin C. C. and Zong Z. J. Tail equivalence relationships for ruin probabilities in severalrisk models [J]. Applied Stochastic Models in Business and Industry, 2005, 21: 499-508.
[61] Kluppelberg C. On subexponential distributions and integrated tails [J]. Journal of AppliedProbability, 1988, 5: 132-141.
[62] Kluppelberg C. Subexponential distributions and characterizations of related classes [J]. Probability Theory and their Related Fields, 1989, 82: 259-269.
[63] Kluppelberg C, Kyprianou A. E. and Mailer R. A. Ruin probabilities and overshoots for generalLevy insurance risk processes [J]. Annals of Applied Probability, 2004,14(4): 1766-1801.
[64] Korshunov D. On distribution tail of the maximum of a random walks [J]. Stochastic Processesand their Applications, 1997, 72: 97-103.
[65] Lin X. S., Willmot G. E. and Drekic S. The classical risk model with a constant dividend barrier[J]. Insurance: Mathematics and Economics, 2003, 33: 551-566.
[66] Li S. and Garrido J. On ruin for the Erlang(n) risk process [J]. Insurance: Mathematics andEconomics, 2004, 34: 391-408.
[67] Li S. The distribution of the dividend payments in the compound Poisson risk model perturbedby diffusion [J]. Scandinavian Actuarial Journal, 2006: 1-13.
[68] Linz et al. Analystical and numerical methods for Volterra equations [M]. Studies 7, SIAM Studiesin Applied Mathematocs: Philadelpha, 1985.
[69] Muller A. M. and Pflug G. Asymptotic ruin probabilities for risk processes with dependentincrements [J]. Insurance: Mathematics and Economics, 2001, 28(3): 381-392.
[70] Morales M. On the expected discounted penalty function for a perturbed risk process driven bya subordinator [J]. Insurance: Mathematics and Economics, 2007, 40: 293-301.
[71] Ng K. W., Tang Q. and Yang H. Maxima of sums of heavy-tailed random variables [J]. ASTINBulletin, 2002, 32: 43-55.
[72] Nyrhinen H. Rough descriptions of ruin for a general class of surplus processes [J], Advances inApplied Probability, 1998, 30: 1008-1026.
[73] Paulsen J. Risk theory in a stochastic economic environment [J]. Stochastic Processes and their??Applications, 1993, 46: 327-361.
[74] Paulsen J. and Gjessing H. K. Ruin theory with stochastic return on investments [J]. Advancesin Applied Probability, 1997, 29: 965-985.
[75] Patie P. Two-sided exit problem for a spectrally negative α-stable Ornstein-Uhlenbeck processand the Wright's generalized hypergeometric function [J]. Electronic Communications in Probability, 2007,12: 146-160.
[76] Po-Fang Hsieh and Sibuya Y. Basic theory of ordinary differential equations [M], Springer, NewYork, 1999.
[77] Protter P. Stochastic Integration and Differential Equations[M]. New York: Springer, 1990.
[78] Ross S. Bounds on the delay distribution in GI/G/1 queues [J]. Journal of Applied Probability,1974,11: 417-421.
[79] Ross S. Stochastic Processes (2nd edition) [M]. Wiley, New York, 1996.
[80] Rolski T., Schmidli H. and Schmidt V. Stochastic Processes for Insurance and Finance [M]. NewYork: Wiley 1999.
[81] Sato K. Levy Processes and Infinitely Divisible Distributions [M]. Cambridge University Press,Cambridge, 1999.
[82] Sgibnev M. S. Banach algebras of measures of class S(v) [J]. Siberian Math J, 1988, 29: 647-655.
[83] Srivastava H.M. and Karlsson P.W. Multiple Gaussian Hypergeometric Series[M]. Ellis HorwoodSeries: Math. Appl. Ellis Horwood Ltd., Chichester, 1985.
[84] Stam A. J. Regular variation of the tail of a subordinated probability distribution [J]. Advancesin Applied Probability, 1973, 5(2): 308-327.
[85] Sundt B. and Teuels J. L. Ruin estimates under interest force [J]. Insurance: Mathematics andEconomics, 1995,15: 7-22.
[86] Tang Q. An asymptotic relationship for ruin probabilities under heavy-tailed claims [J]. Sciencein China (Series A). 2002, 45: 632-639.
[87] Tsai C. C. L. and Willmot G. E. A generalized defective renewal equation for the surplus processperturbed by diffusion [J]. Insurance: Mathematics and Economics, 2002, 30: 51-66.
[88] Veraverbeke N. Asymptotic behavior of Wiener-Hopf factors of a random walk [J]. StochasticProcesses and their Applications, 1977, 5: 27-37.
[89] Wang G. J. and Wu R. Some distributions for classical risk processes that is perturbed by diffusion[J], Insurance: Mathematics and Economics, 2000, 26: 15-24.
[90] Wang G. J. A decomposition of the ruin probability for the risk process perturbed by diffusion[J]. Insurance: Mathematics and Economics, 2001, 28: 49-59.
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