Singular Control of Diffusion Processes and its Applications to the Models of Economic Dynamics

扩散过程的奇异控制及其在经济动态模型中的应用

• 批准号: 0072388
• 负责人: Michael Taksar
• 金额: $ 9万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072388&HistoricalAwards=false
• 关键词: Singular; Control; Diffusion; Processes; its

ABSTRACTSINGULAR CONTROL OF DIFFUSION PROCESSESAND ITS APPLICATIONS TO THE MODELS OF ECONOMIC DYNAMICSThe proposed research lies within the area of optimalstochastic control. It deals with optimal control ofdiffusion processes by means of singular with respectto time functionals. This type of control naturallyappears in the problems in which there is no naturalrestriction on the rates of control and as a resultthe optimal control rate is infinite. The optimalpolicy in these types of problems is to reflect theprocess from an a priori unknown boundary. Such typeof action also arises in problems with additive control,as an approximation for a "bang-bang" optimal policy.The proposed research consists of developing the theoryof the related Partial Differential Equations with gradientconstraints, studying optimal reflecting barriers andoptimal policies in one dimensional and multidimensionalcases. Applications include stochastic control modelsof flexible manufacturing systems as well as dividendoptimization and multidimensional portfolio optimizationmodels. It is also intended to consider application ofthe singular stochastic control theory to optimization problems in insurance. We will also study stochasticcontrol models of large economies and analysis of therelationship between the general framework of the noarbitrage asset pricing in mathematical finance and theequilibrium paths in those models.In addition to developing new stochastic control and stochasticprocesses theory, the applications of this research wouldinclude devising better models for optimization of themanufacturing processes and outputs. In addition our researchwould also result in developing new optimization modelsin mathematical finance and insurance. While perceived bymany nonspecialists as a tool for making a fast profit,mathematical finance in fact is a "technology" for riskreduction. In this regard its merge with insurance is onlytoo natural. The issues of controlling the risk as well asinsurance aspects of the financial risk has loomed largerecently in both financial markets as well as in the insuranceindustry. The importance of optimization in devise of thepolicies employed by insurance and financial institutionsis hard to overestimate. Exposure to unnecessary economic and financial risks and failure to employ the optimal proceduresmay have serious economic and social impacts. Our researchdeals with development of mathematical models of optimalrisk control techniques for financial and insurance corporations.This will also enable one to get a better insight into thenature of the optimal risk reduction techniques a publiclytraded financial corporation should adhere to, as well as theoptimal dividend distribution policy it should follow.

本文的研究属于最优随机控制的范畴。利用时间泛函的奇异性来研究扩散过程的最优控制问题。 这种类型的控制自然出现在对控制率没有自然限制的问题中，因此最优控制率是无限的。这类问题的最优策略是从一个先验未知的边界反映过程。 本文的研究内容包括发展相关的梯度约束偏微分方程理论，研究一维和多维情况下的最优反射障碍和最优策略。 应用包括柔性制造系统的随机控制模型以及股息优化和多维投资组合优化模型。本文还讨论了奇异随机控制理论在最优化中的应用 保险的问题。我们还将研究大型经济体的随机控制模型，并分析数理金融学中无套利资产定价的一般框架与这些模型中均衡路径之间的关系，除了发展新的随机控制和随机过程理论外，这项研究的应用还包括设计更好的模型来优化制造过程和产出。此外，我们的研究还将为数学金融和保险领域开发新的优化模型提供参考. 虽然被许多非专业人士视为快速获利的工具，但数理金融实际上是一种降低风险的“技术”。在这方面，它与保险的合并是再自然不过的了。控制风险的问题以及金融风险的保险方面最近在金融市场和保险业都显得很重要。保险和金融机构所采用的政策设计的优化的重要性很难被高估。暴露于不必要的经济和 金融风险和未能采用最佳程序可能会产生严重的经济和社会影响。我们的研究涉及金融和保险公司最优风险控制技术的数学模型的发展，这也将使人们能够更好地了解上市金融公司应该遵循的最优风险降低技术的性质，以及它应该遵循的最优股息分配政策。