带约束的Lévy过程风险控制理论及其应用
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摘要
在实际风险市场中,往往存在监管约束,如最低现金要求水平、投资约束等等.因此,本文讨论了金融保险中带监管约束的Levy过程随机控制问题及其应用.通过运用更新方法、逐段决定马氏过程(PDMP)方法、鞅论、拟变分不等式(QVI)以及数值渐进等工具,本文主要研究了监管约束下保险公司单位时间内的长期平均利润及相关的偿付能力问题(如期望折扣分红,分红总量的矩母函数,Gerber-Shiu期望贴现惩罚函数).本文研究内容的结构安排如下.
在第二至四章中,保险公司的风险模型带有监管约束,其中约束条件由监管部门决策.即,监管者通过执行最低现金要求水平及对保险公司的违规行为实行惩罚的方法来约束保险公司行为.该三章主要推广和深化了Tapiero等(1983)经典风险模型中联合监管部门-保险公司问题(监管成本函数最小条件下的保险公司利润最大化问题)的研究结果.第五章讨论了扩散风险模型中线性投资约束下的最优分红问题.最后,本文得到了随机保费收入风险模型在边际分红策略下的Gerber-Shiu期望贴现惩罚函数.
我们首先在第二章中研究了重尾情形下的经典风险模型,假设该模型在上述监管机制下动态变化.保险公司的问题是建立投资／分红／风险控制策略来最大化其单位时间内的长期平均利润,该控制策略为监管者所执行约束策略的函数.在正则重尾索赔分布下,我们得到了风险模型平稳分布的渐进解,并导出了保险公司问题下最优控制策略的基本渐进结果.最后本章以Pareto索赔分布为例,得到了在某类参数设置下保险公司渐进最优控制策略的闭解及其数值结果.
其次,第三章将监管约束下的经典风险模型推广至Levy风险模型.在给定的监管约束下,保险公司欲通过选择其投资／(非便宜)再保险／分红策略来最大化其单位时间内的长期平均利润.此外,当短期投资被转化为现金时,假设存在比例交易费用.我们导出了保险公司在单位时间内的长期平均利润函数及监管部门成本关于风险过程平稳分布的函数表达式.同时研究了无比例交易费用下的联合监管部门-保险公司问题,该联合问题的策略即为主从策略(Stakelberg strategies).最后,通过对平稳分布满足的Volterra积分方程的变量变形,得到了无比例交易费用下的联合监管部门-保险公司最优控制策略的渐进数值解.
第四章将PDMP方法和鞅应用到相同监管约束下Tapiero等(1983)经典风险模型中的相关偿付能力研究.不同于前两章中平均期望费用结构下单位时间内长期平均利润函数的研究,本章侧重讨论于折扣期望费用结构的值函数.此时,该风险模型有三个特性,即,借贷利率,短期和长期投资,边际分红.在绝对破产条件下,我们导出了期望折扣分红及其矩母函数,及Gerber-Shiu期望贴现惩罚函数所满足的积分-微分方程.并以指数索赔分布为例,我们得到了对应研究值函数的具体表达式及其数值结果.
第五章研究了线性扩散模型中带线性投资约束和分红交易费用的最优分红问题.其中公司作为小型投资者可将其盈余投资到经典Black-Scholes市场中,假设该投资行为不产生交易费用.本章的主要特点是在不允许卖空和无借款的情形下,对于投资行为存在一般线性约束条件,由此导致了正则-脉冲随机控制问题.在特征化值函数(期望净折扣分红)后,我们证明了值函数为对应拟变分不等式(QVI)的一阶连续粘性解.当正的市场风险价格下常数折扣存在时,称投资不能满足其资产损失的情形为非奇异情形.本章具体分析了非奇异情形中对应于QVI的三种可能情形,由此导出了值函数的具体构造形式及其最优投资／分红策略.此外,我们也给出了关于奇异情形的简单结论,并将本章结论应用到了具体的数值实例中.
最后一章研究了保费随机到达和红利边界下的破产问题,推广了Albrecher和Kainhofer (2002)和Bao(2006)中的结论.首先本章考虑了索赔到达间隔服从普通离散概率分布和非线性红利边界下的期望贴现惩罚函数,并得到无红利边界时的极限解；再将红利边界固定为某常数,考虑了平稳更新过程和PH更新过程中的结果.最后本章将结论具体应用于破产概率、破产前盈余的概率分布及破产前盈余到达红利边界的概率等.
In practical risk markets, there always exist regulations such as minimum cash requirement, investment constraints and so on. In the context of stochastic control in finance and insurance, this thesis investigates Levy processes under regulations and its applications. By use of renewal argument, PDMP method, martingale theory, QVI and numerical approximations, this thesis focuses on value functions, including the long run average profit function per unit time of an insurance firm and the related solvency studies (e.g. the expected discounted dividends, the moment generating function of total dividends, the Gerber-Shiu expected discounted penalty function) and so on. The thesis is organized as follows.
In chapters 2-4, the risk model of an insurance firm is investigated under regulation imposed by the regulatory authority. That is, the regulator exercises a minimum cash requirement level and penalties for violating it to regulate the insurance firm. These three chapters extend and deepen the studies in Tapiero et al. (1983), where a joint insurance corporation-regulatory authority problem was investigated in a classical risk model. In the fourth chapter, the dividend optimization problem is investigated for a diffusion model under linear investment constraints. At last, the Gerber-Shiu expected discounted penalty function is obtained with stochastic income and a barrier dividend strategy.
First, we consider a classical risk model with heavy tailed claims, included in a regulation mechanism of minimum cash requirement. The problem of the insurance firm is to establish an investment and risk exposure policy as well as a barrier dividend strategy, which maximizes the long run average profit per unit time. The strategy of the insurance firm is a function of the strategy used by the regulator. For regularly varying tailed claim size distributions, we find the asymptotics of the stationary distribution of the risk model and derive fundamental asymptotic results of the insurance firm's problem. In the special case of Pareto claim size distributions with special parameters setting, the asymptotic optimal control policy is found in closed form, as well as numerical results.
Then, chapter 3 investigates a Levy risk model with the same regulation as in Chap-ter 2. Under the given regulation, the insurance corporation maximizes its long run av-erage profit per unit time, by choosing its investment/(non-cheap) reinsurance/dividend policy. In addition, it is assumed that proportional transaction cost occurs, when short term investments is converted into cash. Explicit expressions of the long run average profit per unit time, of the regulatory authority's cost function are derived. For the case of non transaction cost, a joint insurance corporation-regulatory authority problem is also investigated, which is in the concept of Stackelberg strategies. Finally, by variable transformations in the numerical solution of Volterra integral equations for the station-ary distributions, the resulting values of the optimal control policy without traction cost are approximated numerically.
The PDMP method and martingales are used to solvency studies in Chapter 4 for the classical risk model under the same regulation as in Tapiero et al. (1983). Chap-ter 4 focuses on the discounted value functions, which are different from the long run average profit function in Chapter 1 and Chapter 2. The risk model includes three features, namely debit interest, short-term and long-term invested interest, barrier div-idend strategy. We derive integro-differential equations under absolute ruin for the ex-pected discounted dividends and its moment generating function, and the Gerber-Shiu expected discounted penalty function. In the case of exponential claim amounts, explicit expressions of the corresponding value functions are obtained, as well as their numerical illustrations.
Chapter 5 investigates the dividend optimization problem of a linear diffusion model with linear investment constraints and dividend transaction costs. Moreover a corpora-tion as a small investor can invest its reserve in a classical Black-Scholes market without paying transaction fees. The main feature of this chapter is that there exists general linear constraints on investments including the special case of short-sale and borrowing constraints. This results in a regular-impulse stochastic control problem. By character-izing the value function (the expected discounted dividends), then it is a once continuous viscosity solution of the corresponding quasi-variational inequalities (QVI). The nontriv-ial case is that the investment can't meet the loss of wealth due to discounting with positive market risk price. In this case, delicate analysis is carried out on QVI w.r.t three possible situations, leading to an explicit construction of the value functions to-gether with the optimal investment/dividend policies. We also give a brief conclusion of other trivial cases and apply the derived results into explicit examples numerically.
At last, the ruin problem is investigated with stochastic income and barrier dividend strategy, which extends the results of Albrecher and Kainhofer (2002) and Bao (2006). Firstly, this chapter considers the expected discounted penalty with common distributed claim amounts and non-linear dividend barrier, and obtains the limit solution without barrier dividends; then the results of stationary renewal process and PH renewal process are derived for fixed constant dividend barrier. Finally, the conclusions are applied in ruin probabilities, probability distribution of surplus prior to ruin and the probability of surplus arriving at dividend barrier before ruin, as well as numerical examples.
在第二至四章中,保险公司的风险模型带有监管约束,其中约束条件由监管部门决策.即,监管者通过执行最低现金要求水平及对保险公司的违规行为实行惩罚的方法来约束保险公司行为.该三章主要推广和深化了Tapiero等(1983)经典风险模型中联合监管部门-保险公司问题(监管成本函数最小条件下的保险公司利润最大化问题)的研究结果.第五章讨论了扩散风险模型中线性投资约束下的最优分红问题.最后,本文得到了随机保费收入风险模型在边际分红策略下的Gerber-Shiu期望贴现惩罚函数.
我们首先在第二章中研究了重尾情形下的经典风险模型,假设该模型在上述监管机制下动态变化.保险公司的问题是建立投资／分红／风险控制策略来最大化其单位时间内的长期平均利润,该控制策略为监管者所执行约束策略的函数.在正则重尾索赔分布下,我们得到了风险模型平稳分布的渐进解,并导出了保险公司问题下最优控制策略的基本渐进结果.最后本章以Pareto索赔分布为例,得到了在某类参数设置下保险公司渐进最优控制策略的闭解及其数值结果.
其次,第三章将监管约束下的经典风险模型推广至Levy风险模型.在给定的监管约束下,保险公司欲通过选择其投资／(非便宜)再保险／分红策略来最大化其单位时间内的长期平均利润.此外,当短期投资被转化为现金时,假设存在比例交易费用.我们导出了保险公司在单位时间内的长期平均利润函数及监管部门成本关于风险过程平稳分布的函数表达式.同时研究了无比例交易费用下的联合监管部门-保险公司问题,该联合问题的策略即为主从策略(Stakelberg strategies).最后,通过对平稳分布满足的Volterra积分方程的变量变形,得到了无比例交易费用下的联合监管部门-保险公司最优控制策略的渐进数值解.
第四章将PDMP方法和鞅应用到相同监管约束下Tapiero等(1983)经典风险模型中的相关偿付能力研究.不同于前两章中平均期望费用结构下单位时间内长期平均利润函数的研究,本章侧重讨论于折扣期望费用结构的值函数.此时,该风险模型有三个特性,即,借贷利率,短期和长期投资,边际分红.在绝对破产条件下,我们导出了期望折扣分红及其矩母函数,及Gerber-Shiu期望贴现惩罚函数所满足的积分-微分方程.并以指数索赔分布为例,我们得到了对应研究值函数的具体表达式及其数值结果.
第五章研究了线性扩散模型中带线性投资约束和分红交易费用的最优分红问题.其中公司作为小型投资者可将其盈余投资到经典Black-Scholes市场中,假设该投资行为不产生交易费用.本章的主要特点是在不允许卖空和无借款的情形下,对于投资行为存在一般线性约束条件,由此导致了正则-脉冲随机控制问题.在特征化值函数(期望净折扣分红)后,我们证明了值函数为对应拟变分不等式(QVI)的一阶连续粘性解.当正的市场风险价格下常数折扣存在时,称投资不能满足其资产损失的情形为非奇异情形.本章具体分析了非奇异情形中对应于QVI的三种可能情形,由此导出了值函数的具体构造形式及其最优投资／分红策略.此外,我们也给出了关于奇异情形的简单结论,并将本章结论应用到了具体的数值实例中.
最后一章研究了保费随机到达和红利边界下的破产问题,推广了Albrecher和Kainhofer (2002)和Bao(2006)中的结论.首先本章考虑了索赔到达间隔服从普通离散概率分布和非线性红利边界下的期望贴现惩罚函数,并得到无红利边界时的极限解；再将红利边界固定为某常数,考虑了平稳更新过程和PH更新过程中的结果.最后本章将结论具体应用于破产概率、破产前盈余的概率分布及破产前盈余到达红利边界的概率等.
In practical risk markets, there always exist regulations such as minimum cash requirement, investment constraints and so on. In the context of stochastic control in finance and insurance, this thesis investigates Levy processes under regulations and its applications. By use of renewal argument, PDMP method, martingale theory, QVI and numerical approximations, this thesis focuses on value functions, including the long run average profit function per unit time of an insurance firm and the related solvency studies (e.g. the expected discounted dividends, the moment generating function of total dividends, the Gerber-Shiu expected discounted penalty function) and so on. The thesis is organized as follows.
In chapters 2-4, the risk model of an insurance firm is investigated under regulation imposed by the regulatory authority. That is, the regulator exercises a minimum cash requirement level and penalties for violating it to regulate the insurance firm. These three chapters extend and deepen the studies in Tapiero et al. (1983), where a joint insurance corporation-regulatory authority problem was investigated in a classical risk model. In the fourth chapter, the dividend optimization problem is investigated for a diffusion model under linear investment constraints. At last, the Gerber-Shiu expected discounted penalty function is obtained with stochastic income and a barrier dividend strategy.
First, we consider a classical risk model with heavy tailed claims, included in a regulation mechanism of minimum cash requirement. The problem of the insurance firm is to establish an investment and risk exposure policy as well as a barrier dividend strategy, which maximizes the long run average profit per unit time. The strategy of the insurance firm is a function of the strategy used by the regulator. For regularly varying tailed claim size distributions, we find the asymptotics of the stationary distribution of the risk model and derive fundamental asymptotic results of the insurance firm's problem. In the special case of Pareto claim size distributions with special parameters setting, the asymptotic optimal control policy is found in closed form, as well as numerical results.
Then, chapter 3 investigates a Levy risk model with the same regulation as in Chap-ter 2. Under the given regulation, the insurance corporation maximizes its long run av-erage profit per unit time, by choosing its investment/(non-cheap) reinsurance/dividend policy. In addition, it is assumed that proportional transaction cost occurs, when short term investments is converted into cash. Explicit expressions of the long run average profit per unit time, of the regulatory authority's cost function are derived. For the case of non transaction cost, a joint insurance corporation-regulatory authority problem is also investigated, which is in the concept of Stackelberg strategies. Finally, by variable transformations in the numerical solution of Volterra integral equations for the station-ary distributions, the resulting values of the optimal control policy without traction cost are approximated numerically.
The PDMP method and martingales are used to solvency studies in Chapter 4 for the classical risk model under the same regulation as in Tapiero et al. (1983). Chap-ter 4 focuses on the discounted value functions, which are different from the long run average profit function in Chapter 1 and Chapter 2. The risk model includes three features, namely debit interest, short-term and long-term invested interest, barrier div-idend strategy. We derive integro-differential equations under absolute ruin for the ex-pected discounted dividends and its moment generating function, and the Gerber-Shiu expected discounted penalty function. In the case of exponential claim amounts, explicit expressions of the corresponding value functions are obtained, as well as their numerical illustrations.
Chapter 5 investigates the dividend optimization problem of a linear diffusion model with linear investment constraints and dividend transaction costs. Moreover a corpora-tion as a small investor can invest its reserve in a classical Black-Scholes market without paying transaction fees. The main feature of this chapter is that there exists general linear constraints on investments including the special case of short-sale and borrowing constraints. This results in a regular-impulse stochastic control problem. By character-izing the value function (the expected discounted dividends), then it is a once continuous viscosity solution of the corresponding quasi-variational inequalities (QVI). The nontriv-ial case is that the investment can't meet the loss of wealth due to discounting with positive market risk price. In this case, delicate analysis is carried out on QVI w.r.t three possible situations, leading to an explicit construction of the value functions to-gether with the optimal investment/dividend policies. We also give a brief conclusion of other trivial cases and apply the derived results into explicit examples numerically.
At last, the ruin problem is investigated with stochastic income and barrier dividend strategy, which extends the results of Albrecher and Kainhofer (2002) and Bao (2006). Firstly, this chapter considers the expected discounted penalty with common distributed claim amounts and non-linear dividend barrier, and obtains the limit solution without barrier dividends; then the results of stationary renewal process and PH renewal process are derived for fixed constant dividend barrier. Finally, the conclusions are applied in ruin probabilities, probability distribution of surplus prior to ruin and the probability of surplus arriving at dividend barrier before ruin, as well as numerical examples.
引文
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