几类风险模型中的破产问题及最优控制问题研究
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摘要
本论文主要利用更新理论,随机控制理论,马氏过程及鞅论等数学工具,研究了几类风险模型的破产问题和最优控制问题.我们研究的风险模型大致可以分为两类,一类是具有随机收入的离散时间的风险模型,另一类是跳跃扩散风险模型(或称带扰动的复合Poisson风险模型).本论文研究内容的结构安排如下.
1.在第二章中,将经典的离散时间的复合二项风险模型中的固定保费收入的情况推广为一个二项过程,并考虑了保险公司的索赔会产生延迟索赔的现象,即一个具有延迟索赔和随机收入的复合二项风险模型.利用风险过程的平稳独立增量性和母函数的方法,得到了该模型在门槛分红策略下的破产之前的期望折现分红总量的一个显示表达式.另外,还通过两个具体的例子,分析了保险公司的初始资产和次索赔被延迟赔付的概率对破产之前的期望折现分红总量的影响.
2.在第三章中,以第二章的模型为基础,将经典的具有延迟索赔的风险模型中的一个主索赔一定会产生一个次索赔的情况推广为一个主索赔以某个概率产生一个次索赔的情况,即一个具有相依索赔和随机收入的复合二项风险模型.研究了该模型在门槛分红策略下的破产之前的期望折现分红总量,得到的结论能够包含第二章中所得的结果,最后还通过两个具体的例子分析了模型中各参数对破产之前的期望折现分红总量的影响.
3.在第四章中,对第三章的模型再进行深入的研究,用一个状态有限的时齐马氏链去刻画每个单位时间段的折现因子(或利率),这样就推广了以往的常利率的情况,即一个具有相依索赔、随机收入和随机利率的复合二项风险模型.通过对该模型在门槛分红策略下的破产之前的期望折现分红总量的研究,得到了它的一个一般表达式,最后在两个具体的例子中给出了它的解析表达式.
4.第五章中,在经典的复合Poisson风险模型的基础上,用一个标准的Brownian运动去刻画影响风险盈余过程的一些随机因素的干扰,然后讨论了保险公司对其股东和投保人均按阈值分红策略进行分红的情况,分别研究了该模型下的破产之前的期望折现分红总量和Gerber-Shin期望折罚函数,得到了它们所满足的积分-微分方程.先把它们所满足的积分-微分方程转化为与之等同的更新方程,再证明相应的更新方程的解的存在唯一性,最后利用更新迭代的办法,分别获得了它们的一种显示表达式.
5.第六章中,探讨了一个跳跃扩散风险模型的最优投资组合与比例再保险问题.采用不变方差弹性模型去刻画风险资产的价格过程,再保险公司按方差保费原理收取保费.针对现有文献中对跳跃扩散风险模型中的扩散项存在的两种不同的解释,同时讨论了以最终财富期望指数效用最大为目标的最优控制问题,分别得到了两种不同解释下的最优策略及其值函数的精确表达式.
6.第七章中,考虑用一个跳跃扩散风险模型去刻画保险公司的盈余过程,金融市场中的风险资产的价格过程由一个几何Levy过程来驱动.另外,对保险公司向再保险公司购买的比例再保险的自留水平加上了一个合理的限制,应用随机控制理论的方法,不仅得到了最优策略及其值函数的精确表达式,还通过具体的数值例子分析了模型中的不同的参数分别对最优策略的影响.
7.第八章中,假设保险公司的盈余过程服从跳跃扩散风险模型,该保险公司除了可以将其资产投资在Black-Scholes金融市场中,还可以通过购买再保险(或接受新业务)来转移一部分风险.研究了该保险公司和市场之间的双人零和博弈问题,应用随机微分博弈的方法,得到了最优最优策略及其值函数的精确表达式.另外,对跳跃扩散风险模型的扩散逼近情况,也求出了最优策略及其值函数的精确表达式.
This paper investigate the ruin problems and the optimal control prob-lems for several classes risk models by the renewal argument, stochastic control theory, Markov process and martingale theory. The risk model that we study can roughly be divided into two kinds:one is the discrete time risk model with random income, the other is the jump diffusion risk model (or the compound Poisson risk model perturbed by diffusion). The main ideas and contributions of this thesis is organized as follows.
1. In chapter2, we extend the deterministic premiums income in the clas-sical discrete time compound binomial risk model to the binomial process, and suppose that the claims maybe delayed. That is to say, we consider the com-pound binomial risk model with delayed claims and random income. Making use of the character of the stationary and independent increments of the risk process and the approach of generating function, under the constant dividend barrier, we obtain the explicit formula for the expected present value of total dividend payments prior to ruin. In addition, the impacts of the initial capi-tal and the delay of by-claims on the expected present value of total dividend payments prior to ruin are discussed by two examples.
2. In chapter3, based on the model in the chapter2, we extend the as-sumption that each main claim induces a by-claim with certainly in the most of the previous literature to the case that each main claim causes a by-claim with a certain probability. That is to say, we consider a compound binomial risk model with correlated claims and random income, and study the expected present value of total dividend payments prior to ruin in this model. Fortu-nately, the conclusion of this chapter can include the result as in chapter2. Finally, the influence of the parameters in this model on the expected present value of total dividend payments prior to ruin are discussed by two examples, respectively.
3. In chapter4, depth study on the model in the chapter3, almost all risk models described in the most of the previous literature relied on the assumption that the force of interest or the discount factor per period is a constant, in this chapter, we use a time-homogeneous Markov chain with a finite state space to model the one-period interest rates. That is to say, we consider a compound binomial risk model with correlated claims-, random income and stochastic interest, and study the expected present value of total dividend payments prior to ruin in this model. A general expression for the expected present value of total dividend payments prior to ruin are derived. Explicit results are obtained in two examples.
4. In chapter5, based on the classic compound Poisson risk model, and describing the disturbance of the stochastic factors by a stand Brownian mo-tion, double-threshold dividend strategy to shareholders and policy-holders is addressed here. A system of integro-differential equations with certain bound-ary conditions for the expected present value of total dividend payments prior to ruin and Gerber-Shiu expected discounted penalty function is derived and solved. Firstly, we translate the integro-differential equations into the renewal equations that identical to the integro-differential equations, then show that the solutions of the renewal equations are unique. Based on these, by iteration, their closed-form solutions are obtained.
5. In chapter6, we consider an optimal investment and proportional rein-surance problem of an insurer whose surplus process follows a jump diffusion model. In our model the insurer transfers part of the risk due to insurance claims via a proportional reinsurance and invests the surplus in a "simplified" financial market consisting of a risk-free asset and a risky asset. The dynamics of the risky asset are governed by a constant elasticity of variance model to incorporate conditional heteroscedasticity. The reinsurance premium is calcu-lated according to the variance principle. The diffusion term can explain the uncertainty associated with the surplus of the insurance company or the ad- ditional small claims. Our objective is to maximize the expected exponential utility of terminal wealth. This optimization problem is studied in two cases depending on the diffusion term's explanation. In all cases, by using techniques of stochastic control theory, closed-form expressions for the value function and optimal proportional reinsurance and investment policies are obtained.
6. In chapter7, we study the optimal investment and proportional reinsur-ance strategy for an insurance company with jump diffusion risk model. The dynamics of the risky asset are governed by a geometric Levy process. Un-der the criterion of maximizing the expected exponential utility from terminal wealth, with a constraint on the proportional reinsurance strategy, closed-form expressions for the value function and optimal strategy are obtained. Numerical examples are presented to show the impact of model parameters on the optimal strategies.
7. In chapter8, we discuss an optimal portfolio and reinsurance problem of an insurance company facing model uncertainty via a game theoretic ap-proach. The insurance company invests in a security market described by the Black-Scholes model. The risk process of the company is governed by either a jump-diffusion process or its diffusion approximation. The company can also transfer a certain proportion of the insurance risk to a reinsurance company by purchasing reinsurance. The optimal portfolio and reinsurance problem is formulated as two-player, zero-sum, stochastic differential games between the insurance and the market. We obtain closed-form solutions to the game prob-lems in both the jump-diffusion risk process and its diffusion approximation.
1.在第二章中,将经典的离散时间的复合二项风险模型中的固定保费收入的情况推广为一个二项过程,并考虑了保险公司的索赔会产生延迟索赔的现象,即一个具有延迟索赔和随机收入的复合二项风险模型.利用风险过程的平稳独立增量性和母函数的方法,得到了该模型在门槛分红策略下的破产之前的期望折现分红总量的一个显示表达式.另外,还通过两个具体的例子,分析了保险公司的初始资产和次索赔被延迟赔付的概率对破产之前的期望折现分红总量的影响.
2.在第三章中,以第二章的模型为基础,将经典的具有延迟索赔的风险模型中的一个主索赔一定会产生一个次索赔的情况推广为一个主索赔以某个概率产生一个次索赔的情况,即一个具有相依索赔和随机收入的复合二项风险模型.研究了该模型在门槛分红策略下的破产之前的期望折现分红总量,得到的结论能够包含第二章中所得的结果,最后还通过两个具体的例子分析了模型中各参数对破产之前的期望折现分红总量的影响.
3.在第四章中,对第三章的模型再进行深入的研究,用一个状态有限的时齐马氏链去刻画每个单位时间段的折现因子(或利率),这样就推广了以往的常利率的情况,即一个具有相依索赔、随机收入和随机利率的复合二项风险模型.通过对该模型在门槛分红策略下的破产之前的期望折现分红总量的研究,得到了它的一个一般表达式,最后在两个具体的例子中给出了它的解析表达式.
4.第五章中,在经典的复合Poisson风险模型的基础上,用一个标准的Brownian运动去刻画影响风险盈余过程的一些随机因素的干扰,然后讨论了保险公司对其股东和投保人均按阈值分红策略进行分红的情况,分别研究了该模型下的破产之前的期望折现分红总量和Gerber-Shin期望折罚函数,得到了它们所满足的积分-微分方程.先把它们所满足的积分-微分方程转化为与之等同的更新方程,再证明相应的更新方程的解的存在唯一性,最后利用更新迭代的办法,分别获得了它们的一种显示表达式.
5.第六章中,探讨了一个跳跃扩散风险模型的最优投资组合与比例再保险问题.采用不变方差弹性模型去刻画风险资产的价格过程,再保险公司按方差保费原理收取保费.针对现有文献中对跳跃扩散风险模型中的扩散项存在的两种不同的解释,同时讨论了以最终财富期望指数效用最大为目标的最优控制问题,分别得到了两种不同解释下的最优策略及其值函数的精确表达式.
6.第七章中,考虑用一个跳跃扩散风险模型去刻画保险公司的盈余过程,金融市场中的风险资产的价格过程由一个几何Levy过程来驱动.另外,对保险公司向再保险公司购买的比例再保险的自留水平加上了一个合理的限制,应用随机控制理论的方法,不仅得到了最优策略及其值函数的精确表达式,还通过具体的数值例子分析了模型中的不同的参数分别对最优策略的影响.
7.第八章中,假设保险公司的盈余过程服从跳跃扩散风险模型,该保险公司除了可以将其资产投资在Black-Scholes金融市场中,还可以通过购买再保险(或接受新业务)来转移一部分风险.研究了该保险公司和市场之间的双人零和博弈问题,应用随机微分博弈的方法,得到了最优最优策略及其值函数的精确表达式.另外,对跳跃扩散风险模型的扩散逼近情况,也求出了最优策略及其值函数的精确表达式.
This paper investigate the ruin problems and the optimal control prob-lems for several classes risk models by the renewal argument, stochastic control theory, Markov process and martingale theory. The risk model that we study can roughly be divided into two kinds:one is the discrete time risk model with random income, the other is the jump diffusion risk model (or the compound Poisson risk model perturbed by diffusion). The main ideas and contributions of this thesis is organized as follows.
1. In chapter2, we extend the deterministic premiums income in the clas-sical discrete time compound binomial risk model to the binomial process, and suppose that the claims maybe delayed. That is to say, we consider the com-pound binomial risk model with delayed claims and random income. Making use of the character of the stationary and independent increments of the risk process and the approach of generating function, under the constant dividend barrier, we obtain the explicit formula for the expected present value of total dividend payments prior to ruin. In addition, the impacts of the initial capi-tal and the delay of by-claims on the expected present value of total dividend payments prior to ruin are discussed by two examples.
2. In chapter3, based on the model in the chapter2, we extend the as-sumption that each main claim induces a by-claim with certainly in the most of the previous literature to the case that each main claim causes a by-claim with a certain probability. That is to say, we consider a compound binomial risk model with correlated claims and random income, and study the expected present value of total dividend payments prior to ruin in this model. Fortu-nately, the conclusion of this chapter can include the result as in chapter2. Finally, the influence of the parameters in this model on the expected present value of total dividend payments prior to ruin are discussed by two examples, respectively.
3. In chapter4, depth study on the model in the chapter3, almost all risk models described in the most of the previous literature relied on the assumption that the force of interest or the discount factor per period is a constant, in this chapter, we use a time-homogeneous Markov chain with a finite state space to model the one-period interest rates. That is to say, we consider a compound binomial risk model with correlated claims-, random income and stochastic interest, and study the expected present value of total dividend payments prior to ruin in this model. A general expression for the expected present value of total dividend payments prior to ruin are derived. Explicit results are obtained in two examples.
4. In chapter5, based on the classic compound Poisson risk model, and describing the disturbance of the stochastic factors by a stand Brownian mo-tion, double-threshold dividend strategy to shareholders and policy-holders is addressed here. A system of integro-differential equations with certain bound-ary conditions for the expected present value of total dividend payments prior to ruin and Gerber-Shiu expected discounted penalty function is derived and solved. Firstly, we translate the integro-differential equations into the renewal equations that identical to the integro-differential equations, then show that the solutions of the renewal equations are unique. Based on these, by iteration, their closed-form solutions are obtained.
5. In chapter6, we consider an optimal investment and proportional rein-surance problem of an insurer whose surplus process follows a jump diffusion model. In our model the insurer transfers part of the risk due to insurance claims via a proportional reinsurance and invests the surplus in a "simplified" financial market consisting of a risk-free asset and a risky asset. The dynamics of the risky asset are governed by a constant elasticity of variance model to incorporate conditional heteroscedasticity. The reinsurance premium is calcu-lated according to the variance principle. The diffusion term can explain the uncertainty associated with the surplus of the insurance company or the ad- ditional small claims. Our objective is to maximize the expected exponential utility of terminal wealth. This optimization problem is studied in two cases depending on the diffusion term's explanation. In all cases, by using techniques of stochastic control theory, closed-form expressions for the value function and optimal proportional reinsurance and investment policies are obtained.
6. In chapter7, we study the optimal investment and proportional reinsur-ance strategy for an insurance company with jump diffusion risk model. The dynamics of the risky asset are governed by a geometric Levy process. Un-der the criterion of maximizing the expected exponential utility from terminal wealth, with a constraint on the proportional reinsurance strategy, closed-form expressions for the value function and optimal strategy are obtained. Numerical examples are presented to show the impact of model parameters on the optimal strategies.
7. In chapter8, we discuss an optimal portfolio and reinsurance problem of an insurance company facing model uncertainty via a game theoretic ap-proach. The insurance company invests in a security market described by the Black-Scholes model. The risk process of the company is governed by either a jump-diffusion process or its diffusion approximation. The company can also transfer a certain proportion of the insurance risk to a reinsurance company by purchasing reinsurance. The optimal portfolio and reinsurance problem is formulated as two-player, zero-sum, stochastic differential games between the insurance and the market. We obtain closed-form solutions to the game prob-lems in both the jump-diffusion risk process and its diffusion approximation.
引文
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