基于时间不一致性和约束的保险公司最优决策研究
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摘要
随机最优控制或动态规划方法是解决经济金融中许多动态优化问题的有力工具.随着经济金融理论的不断丰富和发展,出现了许多的时间不一致性随机控制问题,所谓的时间不一致性是指不满足Bellman最优性原理,以至于动态规划方法也不能应用.因此,研究时间不一致性随机控制问题,特别是研究它的时间一致性控制或策略就显得十分必要.常见的时间不一致性随机控制问题有著名的动态均值-方差模型,双曲折现的最优投资消费问题等.另外,许多的经济金融模型都需要考虑一些现实的限制,因此,带约束的随机优化问题也是一个非常重要和具有挑战性的研究领域.本文主要考虑了两个时间不一致随机控制问题和一个约束随机优化问题.首先研究了时间一致性投资再保险策略.其次讨论了时间一致性分红策略,最后讨论了带偿付能力约束的分红优化问题.
第一章首先介绍了时间不一致性随机控制问题及约束随机优化问题产生的背景；其次介绍了本文的主要工作；最后,列出了一些解决时间不一致随机控制问题的预备知识.
第二章考虑了风险厌恶依赖于状态时的时间一致性投资再保险策略.假设盈余过程为扩散过程,保险公司可以购买比例再保险并且在金融市场投资.金融市场由一个无风险资产和多种风险资产组成,其中风险资产的价格服从几何布朗运动.在此情形下,我们分别考虑两个优化问题,其中一个是投资-再保险问题,另外一个是只有投资的情形.特别地,当考虑风险厌恶动态地依赖于当前财富时,模型更符合现实.我们使用由Bjork和Murgoci(2010)所发展的方法,通过相应的扩展HJB方程导出了这两个问题的时间一致性策略.结果表明我们的时间一致性策略也依赖于当前财富,这比当风险厌恶为常数的情形更合理.
第三章研究了对偶模型下具有非指数折现函数的时间一致性分红策略.一个公司要分红给股东,折现函数是非指数的,并且公司的财富过程用一个对偶模型来描述.我们的目的是寻找一个分红策略来最大化到公司破产为止支付给股东的红利的期望折现值.非指数折现函数会导致这个问题是时间不一致的.但是我们要寻找时间一致性策略,把我们的问题看做一个非合作博弈,导出的均衡策略就是时间一致的.我们给出一个扩展的Hamilton-Jacobi-Bellman方程系统和验证定理来导出均衡策略和均衡值函数.对一种伪指数函数的情形,我们给出了均衡策略和均衡值函数的解析表达式.另外,给出了数值结果来例证我们的结果并且分析参数对结果的影响.
第四章研究了跳扩散模型下带偿付能力约束的分红优化问题.假设保险公司的盈余遵从跳扩散模型,考虑公司在偿付能力约束或破产概率约束下的最优分红问题.由已有的文献知道,当不考虑破产概率约束时,跳扩散模型下的最优分红策略为障碍分红策略.附加了破产概率约束后,分红优化问题变得比较复杂,甚至很难求解.我们利用随机分析和偏微分积分方程的方法和技巧来考虑一个约束分红优化问题,通过分析破产概率的一些性质,给出了分红优化问题的最优策略及最优值函数.
Stochastic optimal control or dynamic programming method is a powerful tool to solve many dynamic optimization problems in the economics and finance. With the economic and financial theory are unceasingly rich and the development, a lot of time-inconsistent stochastic control problems appears. The time-inconsistency means that the Bellman Optimality Principle does not hold, as a consequence, dynamic programming cannot be applied. Therefore, it is very necessary to s-tudy the time-inconsistent stochastic control problems, especially, to study their time-consistent control or strategy. The celebrated dynamic mean-variance model and optimal investment and consumption under hyperbolic discounting are both time-inconsistent stochastic control problems. In addition, many of the economic and financial models need to consider some realistic restrictions. As a result, the stochastic optimization problems with constraints are also very important and are challenging research fields. This thesis considers two time-inconsistent stochastic control problems and a stochastic optimization problem with the constraint. First, we study the time-consistent investment-reinsurance strategy. Second, we consider the time-consistent dividend-payment strategy. Finally, we discuss the dividend optimization problem with solvency constraint.
In Chapter1, we firstly introduce the background of the time-inconsistent stochastic control problems and stochastic optimization problems with constraints. Then state the main results of this thesis. Finally, we list some preliminaries to solve the time-inconsistent stochastic control problems.
Chapter2is devoted to studying the time-consistent investment and reinsur-ance strategies with state dependent risk aversion. It is assumed that the surplus process is approximated by a diffusion process. The insurer can purchase propor-tional reinsurance and invest in a financial market which consists of one risk-free asset and multiple risky assets whose price processes follow geometric Brownian motions. Under these, we consider two optimization problems, an investment-reinsurance problem and an investment-only problem. In particular, when the risk aversion depends dynamically on current wealth, the model is more realistic. Using the approach developed by Bjork and Murgoci (2010), the time-consistent strategies for the two problems are derived by means of corresponding extension of the Hamilton-Jacobi-Bellman equation. The time-consistent strategies are de-pendent on current wealth, this case thus is more reasonable than the one with constant risk aversion.
In Chapter3, we study the time-consistent dividend strategy with non-exponential discounting in a dual model. We consider a dividend-payment problem for a com-pany with non-exponential discounting, whose surplus process is described by a dual model. The target is to find a dividend strategy that maximizes the expected discounted value of dividends which are paid to the shareholders until the ruin time of the company. The non-exponential discount function results in the problem of being time-inconsistent. But we seek only the time-consistent strategy, which is an equilibrium strategy derived by taking our problem as a non-cooperate game. Extended Hamilton-Jacobi-Bellman equation system and verification theorem are provided to derive the equilibrium strategy and the equilibrium value function. For the case of pseudo-exponential discount functions, closed-form expressions for the equilibrium strategy and the equilibrium value function are derived. In addition, some numerical illustrations of our results are showed.
In Chapter4, we investigate the dividend optimization for jump-diffusion with solvency constraint. Assuming that the surplus of insurer follow a jump diffusion model, we consider a dividend optimization problem under the solven-cy constraints or ruin probability constraint. Prom, the existing literature, if not considering bankruptcy probability constraint, the optimal dividend policy under jump diffusion model is a barrier strategy. When the bankruptcy probability con-straint is considered, the dividend optimization problem is difficult to solve. We use stochastic analysis and partial differential-integral equation to consider a div-idend optimization problem with constraint, by analyzing some properties of the ruin probability, we present the optimal strategy and optimal value function of the dividend optimization problem.
第一章首先介绍了时间不一致性随机控制问题及约束随机优化问题产生的背景；其次介绍了本文的主要工作；最后,列出了一些解决时间不一致随机控制问题的预备知识.
第二章考虑了风险厌恶依赖于状态时的时间一致性投资再保险策略.假设盈余过程为扩散过程,保险公司可以购买比例再保险并且在金融市场投资.金融市场由一个无风险资产和多种风险资产组成,其中风险资产的价格服从几何布朗运动.在此情形下,我们分别考虑两个优化问题,其中一个是投资-再保险问题,另外一个是只有投资的情形.特别地,当考虑风险厌恶动态地依赖于当前财富时,模型更符合现实.我们使用由Bjork和Murgoci(2010)所发展的方法,通过相应的扩展HJB方程导出了这两个问题的时间一致性策略.结果表明我们的时间一致性策略也依赖于当前财富,这比当风险厌恶为常数的情形更合理.
第三章研究了对偶模型下具有非指数折现函数的时间一致性分红策略.一个公司要分红给股东,折现函数是非指数的,并且公司的财富过程用一个对偶模型来描述.我们的目的是寻找一个分红策略来最大化到公司破产为止支付给股东的红利的期望折现值.非指数折现函数会导致这个问题是时间不一致的.但是我们要寻找时间一致性策略,把我们的问题看做一个非合作博弈,导出的均衡策略就是时间一致的.我们给出一个扩展的Hamilton-Jacobi-Bellman方程系统和验证定理来导出均衡策略和均衡值函数.对一种伪指数函数的情形,我们给出了均衡策略和均衡值函数的解析表达式.另外,给出了数值结果来例证我们的结果并且分析参数对结果的影响.
第四章研究了跳扩散模型下带偿付能力约束的分红优化问题.假设保险公司的盈余遵从跳扩散模型,考虑公司在偿付能力约束或破产概率约束下的最优分红问题.由已有的文献知道,当不考虑破产概率约束时,跳扩散模型下的最优分红策略为障碍分红策略.附加了破产概率约束后,分红优化问题变得比较复杂,甚至很难求解.我们利用随机分析和偏微分积分方程的方法和技巧来考虑一个约束分红优化问题,通过分析破产概率的一些性质,给出了分红优化问题的最优策略及最优值函数.
Stochastic optimal control or dynamic programming method is a powerful tool to solve many dynamic optimization problems in the economics and finance. With the economic and financial theory are unceasingly rich and the development, a lot of time-inconsistent stochastic control problems appears. The time-inconsistency means that the Bellman Optimality Principle does not hold, as a consequence, dynamic programming cannot be applied. Therefore, it is very necessary to s-tudy the time-inconsistent stochastic control problems, especially, to study their time-consistent control or strategy. The celebrated dynamic mean-variance model and optimal investment and consumption under hyperbolic discounting are both time-inconsistent stochastic control problems. In addition, many of the economic and financial models need to consider some realistic restrictions. As a result, the stochastic optimization problems with constraints are also very important and are challenging research fields. This thesis considers two time-inconsistent stochastic control problems and a stochastic optimization problem with the constraint. First, we study the time-consistent investment-reinsurance strategy. Second, we consider the time-consistent dividend-payment strategy. Finally, we discuss the dividend optimization problem with solvency constraint.
In Chapter1, we firstly introduce the background of the time-inconsistent stochastic control problems and stochastic optimization problems with constraints. Then state the main results of this thesis. Finally, we list some preliminaries to solve the time-inconsistent stochastic control problems.
Chapter2is devoted to studying the time-consistent investment and reinsur-ance strategies with state dependent risk aversion. It is assumed that the surplus process is approximated by a diffusion process. The insurer can purchase propor-tional reinsurance and invest in a financial market which consists of one risk-free asset and multiple risky assets whose price processes follow geometric Brownian motions. Under these, we consider two optimization problems, an investment-reinsurance problem and an investment-only problem. In particular, when the risk aversion depends dynamically on current wealth, the model is more realistic. Using the approach developed by Bjork and Murgoci (2010), the time-consistent strategies for the two problems are derived by means of corresponding extension of the Hamilton-Jacobi-Bellman equation. The time-consistent strategies are de-pendent on current wealth, this case thus is more reasonable than the one with constant risk aversion.
In Chapter3, we study the time-consistent dividend strategy with non-exponential discounting in a dual model. We consider a dividend-payment problem for a com-pany with non-exponential discounting, whose surplus process is described by a dual model. The target is to find a dividend strategy that maximizes the expected discounted value of dividends which are paid to the shareholders until the ruin time of the company. The non-exponential discount function results in the problem of being time-inconsistent. But we seek only the time-consistent strategy, which is an equilibrium strategy derived by taking our problem as a non-cooperate game. Extended Hamilton-Jacobi-Bellman equation system and verification theorem are provided to derive the equilibrium strategy and the equilibrium value function. For the case of pseudo-exponential discount functions, closed-form expressions for the equilibrium strategy and the equilibrium value function are derived. In addition, some numerical illustrations of our results are showed.
In Chapter4, we investigate the dividend optimization for jump-diffusion with solvency constraint. Assuming that the surplus of insurer follow a jump diffusion model, we consider a dividend optimization problem under the solven-cy constraints or ruin probability constraint. Prom, the existing literature, if not considering bankruptcy probability constraint, the optimal dividend policy under jump diffusion model is a barrier strategy. When the bankruptcy probability con-straint is considered, the dividend optimization problem is difficult to solve. We use stochastic analysis and partial differential-integral equation to consider a div-idend optimization problem with constraint, by analyzing some properties of the ruin probability, we present the optimal strategy and optimal value function of the dividend optimization problem.
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