金融保险中的几类风险模型
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摘要
本论文利用更新理论、马氏过程、随机控制及鞅论等数学工具，主要研究了金融保险中几种风险过程的破产问题。对破产概率的上界，破产前瞬时盈余和破产时赤字的分布，破产时罚金折现期望函数的性质及新风险业务的最优比例进行了分析。具体表现在以下几个方面：
1．将经典风险模型中确定的保费收入推广为复合Poisson过程，并采用Wiener过程来刻画随机因素的干扰，即考虑了带扩散扰动项的双复合Poisson风险模型。利用风险过程的平稳独立增量性，得到了破产概率的一般表达式和Lundberg不等式，且通过数值例子，分析了初始盈余、保费收入、索赔支付对破产概率及调节系数的影响；利用风险过程的齐次强马氏性，给出了破产前最大盈余分布及破产时赤字分布的积分方程。
2．讨论了Erlang(2)风险过程的罚金折现期望函数。将常利率引入Erlang(2)风险模型中，利用更新理论，导出了罚金期望值的微积分方程，以及破产前瞬时盈余和破产时赤字联合分布的递推公式，并给出了无利率的特殊情形下罚金函数的瑕疵更新方程与级数表达式。然后，建立了一类新红利策略下的Erlang(2)风险过程，通过一定的数学技巧，获得并求解了罚金折现函数所满足的微积分方程，得到了有红利界限时与无红利界限时罚金折现函数之间的关系。
3．探讨了马氏环境下的Cox风险模型，即其强度过程是马氏跳过程。首先，考察常利率因素影响下的Cox风险过程，通过后向差分法，获得了条件罚金期望值与平稳情形时罚金期望值的积分方程。其次，考虑了带干扰的且保费收入依赖索赔强度的Cox风险模型下的罚金函数，在一定条件下，给出了罚金期望值的瑕疵更新方程和渐近性质。最后，建立了具有双险种风险业务的Cox风险过程，且两类索赔的到达过程通过含有一个共同的计数过程而相关，利用鞅技巧，导出了破产概率的上界估计。
4．考虑了一类具有时间相依索赔的风险模型下的罚金折现函数，其中，主索赔可能会引起副索赔，且该副索赔以概率θ与主索赔同时发生，以概率1-θ延迟到下一个时间段发生。我们通过Rouché’s定理和Laplace变换，求解了关于罚金折现函数的微积分方程，且给出了数值结果。
5．将利率因素引入到一类保险风险的控制问题中，研究了常利率下保险公司为最小化破产概率，其新风险业务最优比例的选取问题。采用带漂移的Brownian运动刻画风险业务，通过随机控制的方法，推导并求解了相应的Hamilton-Jacobi-Bellman方程，得到了破产概率的最小值及最优比例的显示表达式。最后，通过数值例子，分析了利率及初始资本对破产概率和最优比例的影响。
The paper investigate the ruin problem of several risk processes in finance and insurance by the renewal argument, Markov process, stochastic control and martingale theory. The upper bound of ruin probability, the distributions of the surplus immediately before ruin and the deficit at ruin, the property of the expected discounted penalty function at ruin and the optimal proportion of new risk business are discussed. The main ideas and contributions of this thesis are as follows:
1. Extending the deterministic premiums income in the classical risk model to the compound Poisson process, and describing the disturbance of the stochastic factors by the Wiener process, we consider the double compound Poisson risk model perturbed by diffusion. Making use of the character of the stationary and independent increments of the risk process, we get the general formula and the Lundberg inequality of the ruin probability, and analyze the effect of the initial surplus, the income of premium and the claim payment on the ruin probability and the adjustment coefficient by a numerical example. The integral equations of the distributions of the supremum surplus before ruin and the deficit at ruin are given by the homogeneous strong Markov property of the risk process.
2. The expected discounted penalty function in Erlang(2) risk process is discussed. Introducing the constant interest rate into the Erlang(2) risk model, we derive the integro-differential equaton for the expected value of the penalty, and the recursive formula for the joint distribution of the surplus immediately before ruin and the deficit at ruin by the renewal argument. And the defective renewal equation, the series representation of the penalty function in the special interest-free case are given. Next, the Erlang(2) risk process with a new threshold dividend strategy are developed. Using certain mathematical technique, we get and solve the integro-differential equation satisfied by the discounted penalty function. The relation between the discounted penalty functions with or without the threshold dividend is obtained.
3. The Cox risk models in a markovian environment are investigated, namely, the intensity process of the Cox process is markovian jump process. Firstly, in the Cox risk model where the constant interest force is included, we obtain the integral equation for the conditional expected value of the penalty and the expected value of the penalty which is in the stationary case by the backward differential argument. Secondly, we consider the penalty function of the Cox risk model with the premium rate which varies with the claim intensity and disturbed by diffusion. The defective renewal equation and asymptotic property for the expected value of the penalty are given in certain circumstances. Finally, we establish the Cox risk process with two kinds of risk business. Further, there is a certain correlation between the claim arrival processes of the different risk business. The estimation of sharp upper bound of the ruin probability are deduced by the martingale technique.
4. We consider the discounted penalty function in a kind of the risk model with time-correlated claims. In this model, the main claim can induce the by claim which may happen with the main claim simultaneously with probabilityθ, and may delay to next time epoch with probability 1—θ. The integro-differential equation of the discounted penalty function is solved by the Rouché's theorem and Laplace transform, and the numerical result is given.
5. The interest force is introduced into a kind of control problem of the insurance risk. The optimal proportion of new risk business under constant interest rate is studied. Optimal is meant in the sense of minimizing the ruin probability of the insurance company. The risk business are assumed to follow a Brownian motion with drift. We derive and solve the Hamilton-Jacobi-Bellman equation by the stochastic control method. The close-form of the minimum ruin probability and the optimal proportion are found. Finally, the influence of the interest rate and the initial capital on the probability of ruin and the optimal proportion is discussed by the numerical example.
1．将经典风险模型中确定的保费收入推广为复合Poisson过程，并采用Wiener过程来刻画随机因素的干扰，即考虑了带扩散扰动项的双复合Poisson风险模型。利用风险过程的平稳独立增量性，得到了破产概率的一般表达式和Lundberg不等式，且通过数值例子，分析了初始盈余、保费收入、索赔支付对破产概率及调节系数的影响；利用风险过程的齐次强马氏性，给出了破产前最大盈余分布及破产时赤字分布的积分方程。
2．讨论了Erlang(2)风险过程的罚金折现期望函数。将常利率引入Erlang(2)风险模型中，利用更新理论，导出了罚金期望值的微积分方程，以及破产前瞬时盈余和破产时赤字联合分布的递推公式，并给出了无利率的特殊情形下罚金函数的瑕疵更新方程与级数表达式。然后，建立了一类新红利策略下的Erlang(2)风险过程，通过一定的数学技巧，获得并求解了罚金折现函数所满足的微积分方程，得到了有红利界限时与无红利界限时罚金折现函数之间的关系。
3．探讨了马氏环境下的Cox风险模型，即其强度过程是马氏跳过程。首先，考察常利率因素影响下的Cox风险过程，通过后向差分法，获得了条件罚金期望值与平稳情形时罚金期望值的积分方程。其次，考虑了带干扰的且保费收入依赖索赔强度的Cox风险模型下的罚金函数，在一定条件下，给出了罚金期望值的瑕疵更新方程和渐近性质。最后，建立了具有双险种风险业务的Cox风险过程，且两类索赔的到达过程通过含有一个共同的计数过程而相关，利用鞅技巧，导出了破产概率的上界估计。
4．考虑了一类具有时间相依索赔的风险模型下的罚金折现函数，其中，主索赔可能会引起副索赔，且该副索赔以概率θ与主索赔同时发生，以概率1-θ延迟到下一个时间段发生。我们通过Rouché’s定理和Laplace变换，求解了关于罚金折现函数的微积分方程，且给出了数值结果。
5．将利率因素引入到一类保险风险的控制问题中，研究了常利率下保险公司为最小化破产概率，其新风险业务最优比例的选取问题。采用带漂移的Brownian运动刻画风险业务，通过随机控制的方法，推导并求解了相应的Hamilton-Jacobi-Bellman方程，得到了破产概率的最小值及最优比例的显示表达式。最后，通过数值例子，分析了利率及初始资本对破产概率和最优比例的影响。
The paper investigate the ruin problem of several risk processes in finance and insurance by the renewal argument, Markov process, stochastic control and martingale theory. The upper bound of ruin probability, the distributions of the surplus immediately before ruin and the deficit at ruin, the property of the expected discounted penalty function at ruin and the optimal proportion of new risk business are discussed. The main ideas and contributions of this thesis are as follows:
1. Extending the deterministic premiums income in the classical risk model to the compound Poisson process, and describing the disturbance of the stochastic factors by the Wiener process, we consider the double compound Poisson risk model perturbed by diffusion. Making use of the character of the stationary and independent increments of the risk process, we get the general formula and the Lundberg inequality of the ruin probability, and analyze the effect of the initial surplus, the income of premium and the claim payment on the ruin probability and the adjustment coefficient by a numerical example. The integral equations of the distributions of the supremum surplus before ruin and the deficit at ruin are given by the homogeneous strong Markov property of the risk process.
2. The expected discounted penalty function in Erlang(2) risk process is discussed. Introducing the constant interest rate into the Erlang(2) risk model, we derive the integro-differential equaton for the expected value of the penalty, and the recursive formula for the joint distribution of the surplus immediately before ruin and the deficit at ruin by the renewal argument. And the defective renewal equation, the series representation of the penalty function in the special interest-free case are given. Next, the Erlang(2) risk process with a new threshold dividend strategy are developed. Using certain mathematical technique, we get and solve the integro-differential equation satisfied by the discounted penalty function. The relation between the discounted penalty functions with or without the threshold dividend is obtained.
3. The Cox risk models in a markovian environment are investigated, namely, the intensity process of the Cox process is markovian jump process. Firstly, in the Cox risk model where the constant interest force is included, we obtain the integral equation for the conditional expected value of the penalty and the expected value of the penalty which is in the stationary case by the backward differential argument. Secondly, we consider the penalty function of the Cox risk model with the premium rate which varies with the claim intensity and disturbed by diffusion. The defective renewal equation and asymptotic property for the expected value of the penalty are given in certain circumstances. Finally, we establish the Cox risk process with two kinds of risk business. Further, there is a certain correlation between the claim arrival processes of the different risk business. The estimation of sharp upper bound of the ruin probability are deduced by the martingale technique.
4. We consider the discounted penalty function in a kind of the risk model with time-correlated claims. In this model, the main claim can induce the by claim which may happen with the main claim simultaneously with probabilityθ, and may delay to next time epoch with probability 1—θ. The integro-differential equation of the discounted penalty function is solved by the Rouché's theorem and Laplace transform, and the numerical result is given.
5. The interest force is introduced into a kind of control problem of the insurance risk. The optimal proportion of new risk business under constant interest rate is studied. Optimal is meant in the sense of minimizing the ruin probability of the insurance company. The risk business are assumed to follow a Brownian motion with drift. We derive and solve the Hamilton-Jacobi-Bellman equation by the stochastic control method. The close-form of the minimum ruin probability and the optimal proportion are found. Finally, the influence of the interest rate and the initial capital on the probability of ruin and the optimal proportion is discussed by the numerical example.
引文
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[2] Lundberg F. I. Aterforsokering av Kollektivrisker. Uppsala: Almqvist and Wiksell, 1903.
[3] Lundberg F. I. Forsakringsteknisk Riskutjamning. Stockholm: F. Englunds boktryckeri A, 1926.
[4] Cramer H. On the Mathematical Theory of Risk. Stockholm: Skandia Jubilee Volume, 1930.
[5] Cramer H. On Some Questions Connected with Mathematical Risk. Statistics, 1954, 2: 99-123.
[6] Cramer H. Collective Risk Theory. Stockholm: Skandia Jubilee Volume, 1955.
[7] Gerber H. U. An Introduction to Mathematical Risk Theory. Philadelphia, S. S. Hsubner Foundation Monograph Series 8，1979．数学风险论导引(中译本，成世学，严颖译)北京：世界图书出版社发行公司，1997．
[8] Grandell J. Aspects of Risk Theory. New York: Springer-Verlag, 1991.
[9] Gerber H. U. Martingales in Risk Theory. Mitteilungen der Schweizerischen Vereinigung der Versicherungsmathematiker, 1973: 205-216.
[10] Harrison J. M. Ruin Problem with Compounding Assets. Stochastic Processes and their Application, 1977, 5: 67-79.
[11] Paulsen J. Risk Theory in a Stochastic Economic Environment. Stochastic Processes and their Application, 1993, 46: 327-361.
[12] Dassios A., Embrechts P. Martingale and Insurance Risk. Communications in Statistics-Stochastic Models, 1989, 5; 181-217.
[13] Gerber H. U. An Extension to the Renewal Equation and its Application in the Collective Theory of Risk. Scandinavisk Aktuarietidskrift, 1970: 205-210.
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