通胀不确定下最优消费-投资组合和退休选择问题研究

A model of the optimal consumption-portfolio and retirement choice is studied when an agent is facing with the inflation uncertainty and the financial market with a bond and several stocks. Firstly，the inflation rates and the prices of assets are modelled by the stochastic differential equations, and the voluntary retiree’ utility optimization is characterized. Secondly, the related stochastic control systems with Markovian regime switching disturbed by Brownian motions and/or Poisson processes are mathematically analyzed, and the optimum principles of the systems are derived. Thirdly, through using the dynamic optimization theory, backward stochastic differential equations and stochastic calculus, an ambiguity investor’s optimal making-decision is deduced under inflation and regime switching, moreover the effects of inflation on an investor’s optimal policies with different preferences are expounded. Next, by using martingale theory, dynamic programming and stopping technique, the optimal portfolio, consumption–leisure, insurance and retirement choices with CES or Cobb-Douglas utility under inflation are solved by Hamilton-Jacobi-Bellman equations. Finally, by utilizing Monte Karlo technique and the financial market data，the analyses of calibration or simulation for the models are provided. The obtained results of the stochastic theory provide new mathematical methods for the other related fields as the financial risk measurement and control, and the obtained financial consequences which characterizes the behavior of a voluntary retirement investor are not only valuable for the theory of the modern financial investment decision-making but also really helpful for an investor’s investment in a financial market.

研究通胀下最优消费-投资组合和退休问题。首先，通胀率和风险资产价格由随机微分方程建模，自愿退休投资者效用优化得以刻画；其次，对由布朗运动或Poisson跳过程扰动的Markov转换随机控制系统进行数理分析并建立最优控制的最优性原理等；进而，利用动态优化理论、倒向随机微分方程和随机分析等解决通胀和机制转换下模型不确定投资者最优决策的一般理论，并在投资者不同风险偏好下分析通胀对最优决策的影响；接下来，利用鞅方法、动态规划和停时技术刻画通胀下具有CES或Cobb-Douglas 效用投资者最优消费-闲暇、投资、保险和退休所满足的HJB方程并求解；最后使用蒙特卡洛方法等结合金融市场数据进行模拟以评价模型的性能和结论的经济意义。所得随机优化等理论为金融风险度量与控制等同类研究提供新的数学方法；所得模型结果刻画了投资者行为特征，是对现代金融投资决策理论的有益探索，具有较高的理论价值和实际经济指导意义。

通胀影响着人们的消费和投资行为，而现实世界中通胀的变化不仅具有随机性，往往表现为Markov机制转移性。在研究投资者最优消费-投资组合和退休选择问题时必须要考虑通胀不确定因素，与此同时，金融市场中的模型不确定、Markov 机制转换也会对投资者的策略带来影响。为此，首先建立必要的基于机制转换随机过程和随机控制相关理论。其次，对于机制转换或模型不确定下带通胀的最优消费-投资组合和退休选择模型进行理论分析。接下来，利用相关随机动态优化理论与方法寻求最优投资策略的解析解（或近似解）；基于变分法和鞅方法探讨通胀不确定下最优消费-投资和退休选择问题；提出随机动态规划理论在解决通胀不确定下最优消费和投资组合和退休决策中的一般思路和方法，包括HJB方程的建立与求解。最后形成通胀不确定下最优消费-投资组合和退休决策模型的分析与求解技术；求解结果的性能评价方法与技术；建立通胀不确定下最优消费-投资组合和退休决策中随机动态优化模型相关理论的理论体系。并且利用统计分析方法与蒙特卡罗模拟技术对理论结果进行检验的同时，为投资者的决策提供理论上的支持。所得结论具有重要的经济理论价值和现实意义。.具体来说，本项目首先获得了以下结果：通胀下带有生存消费约束的最优消费、投资与自愿退休模型；通胀条件下高管的股权激励和工作努力策略分析；通胀风险下的企业家投资–消费和对冲模型；通胀不确定下的最优动态资产配置；在部分信息下带通胀的最优交易选择等。.接下来，还得到下列结果：具有汇率风险的国际投资模型；道德风险下带有 Knight 不确定的最优动态契约设计；Knight 不确定与随机汇率下外商投资决策分析；考虑汇率变动的跨国直接投资和税收政策策略；CIR利率模型下带有随机劳动收入的最优消费投资策略等。