基于时间不一致性和约束的保险公司最优决策研究

发布时间：2018-07-10 16:05

本文选题：均值-方差准则 + 时间一致性　；参考：《兰州大学》2014年博士论文

【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.
[Abstract]:Stochastic optimal control or dynamic programming is a powerful tool for solving many dynamic optimization problems in economy and finance. With the continuous enrichment and development of economic and financial theory, there are many problems of time inconsistency stochastic control. The so-called time inconsistency is that it is not satisfied with the Bellman optimality principle, so that the dynamic programming method is not satisfied. Therefore, it is necessary to study the time consistency stochastic control problem, especially the time consistency control or strategy. The common time inconsistencies stochastic control problem has the famous dynamic mean variance model, the hyperbolic discounted optimal investment consumption problem and so on. In addition, many economic and financial models. We need to consider some practical constraints, therefore, the stochastic optimization problem with constraints is also a very important and challenging field of research. This paper mainly considers two time inconsistent random control problems and a constrained stochastic optimization problem. First, the time induced reinsurance strategy is studied. Secondly, the time is discussed. Consistency dividend strategy. Finally, the dividend optimization problem with solvency constraints is discussed.
In the first chapter, the background of time inconsistency stochastic control problem and constrained stochastic optimization problem is introduced. Secondly, the main work of this paper is introduced. Finally, some preparatory knowledge to solve the problem of time inconsistent random control are listed.
In the second chapter, the time consistent investment reinsurance strategy is considered when the risk aversion depends on the state. It is assumed that the surplus process is a diffusion process, and the insurance company can buy proportional reinsurance and invest in the financial market. The financial market consists of a riskless asset and a variety of risky assets, in which the price of the risk assets obeys geometric Brown. In this case, we consider two optimization problems, one is the investment reinsurance problem, and the other is the only case of investment. In particular, when the risk aversion is dynamically dependent on the current wealth, the model is more realistic. We use the method developed by Bjork and Murgoci (2010), through the corresponding expansion of H The JB equation derives the time consistency strategy for these two problems. The results show that our time consistency strategy also depends on the current wealth, which is more reasonable than the case of risk aversion as a constant.
In the third chapter, we study the time consistency dividend strategy with non exponential discounted function under dual model. A company should pay dividends to shareholders, the discounted function is non exponential, and the company's wealth process is described with a dual model. The purpose is to find a dividend policy to maximize the company's bankruptcy and pay the shareholders. The non exponential discounted function will cause the problem to be inconsistent with time. But we have to look for the time consistency strategy and consider our problem as a non cooperative game. The derived equilibrium strategy is time consistent. We give an extended Hamilton-Jacobi-Bellman equation system and validation. The equilibrium strategy and the equilibrium value function are derived. For the case of a pseudo exponential function, we give the equilibrium strategy and the analytic expression of the equilibrium value function. In addition, the numerical results are given to illustrate our results and to analyze the effect of the parameters on the results.
The fourth chapter studies the problem of dividend optimization with the solvency constraint under the jump diffusion model. It is assumed that the surplus of the insurance company follows the jump diffusion model and considers the company's optimal bonus problem under the solvency constraint or the ruin probability constraint. The strategy is obstacle bonus strategy. When the ruin probability constraint is added, the problem of dividend optimization becomes more complex and difficult to solve. We use the method and technique of the stochastic analysis and partial differential integral equation to consider a constrained dividend optimization problem. By analyzing some properties of the ruin probability, the optimal problem of the dividend optimization is given. Strategy and optimal value function.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：F840.31;F224

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