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

摘要扩散过程的奇异控制及其在经济动力学模型中的应用本文的研究属于最优随机控制的范畴。它用时间泛函的奇异值方法研究扩散过程的最优控制问题。这种类型的控制通常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穷大的。这类问题的最优策略是从先验未知边界反映过程。这类作用也出现在加性控制问题中，作为“砰-砰”最优策略的近似。建议的研究包括发展具有梯度约束的相关偏微分方程组的理论，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红利优化和多维投资组合优化模型。同时也考虑了奇异随机控制理论在保险最优化问题中的应用。我们还将研究大型经济体的随机控制模型，并分析数学金融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，这项研究的应用将包括设计更好的模型来优化制造过程和产出。此外，我们的研究还将在数学、金融和保险领域开发新的优化模型。虽然许多非专业人士认为数学金融是一种快速盈利的工具，但实际上它是一种降低风险的“技术”。在这一点上，它与保险业的合并再自然不过了。在金融市场和保险业中，控制风险以及金融风险的保险方面的问题最近变得更加突出。在设计保险和金融机构使用的政策时，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，以及未能采用最佳程序，可能会产生严重的经济和社会影响。我们的研究涉及为金融和保险公司开发最优风险控制技术的数学模型，这也将使人们能够更好地了解上市金融公司应该坚持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。

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.