Problems in Stochastic Control, Incomplete Markets, and Stochastic Limit Theorems

随机控制、不完全市场和随机极限定理中的问题

• 批准号: 0604491
• 负责人: Erhan Bayraktar
• 金额: $ 8.87万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604491&HistoricalAwards=false
• 关键词: Problems; Stochastic; Control; Incomplete; Markets

The proposed project consists of four parts: (1) Solving the optimal stopping problems associated with several quickest detection problems for processes that jump. The goal is to provide novel optimization techniques for such processes and study properties of the corresponding variational inequalities. (2) Developing stochastic control techniques to study the problem of how an individual (retiree) should invest her wealth in a risky financial market in order to minimize the probability that she outlives her wealth, minimizes her life-time shortfall. In this case, the aim is to develop the earlier results of L-infinity control and to analyze the associated variational equalities for more realistic market models. (3) Developing new pricing principles for incomplete markets, with the objective of providing new insights into pricing and hedging derivative securities in incomplete markets. This part also includes developing solutions of impulse and singular control problems for any one-dimensional diffusion (with decision making delay). (4) Developing stochastic limit theorems for processes with semi-Markov switching to elucidate the impact of common characteristics of the investors on the aggregate quantities like the market prices.Optimal stopping problems have applications in the areas of seismology, machine monitoring, finance, insurance, health surveillance among others. Improved stochastic control techniques may inform the public, financial planners and legislators about the risk of ruin in retirement. This work will benefit individuals' decision making on important financial matters they face during their lives: How much to invest in mutual funds; how much insurance one should buy; whether it is a good time to borrow to invest in the stock market and how much one should borrow; when it is best to declare bankruptcy; when one should retire, etc. The development of new pricing principles provides better pricing mechanisms for derivative products in the markets, which will benefit the financial institutions which are at the center of the economy. The results in the final section of this part will help the management make better decisions for the welfare of their companies which will benefit the nation since available resources will be used more efficiently. The final objective of the project will obtain insights into the market dynamics by understanding the price formation from typical behavioral qualities of investors. This is important in creating good financial models that benefit economic forecast, investment and policy decisions.

拟议的项目由四个部分组成：（1）解决与跳跃过程的几个最快检测问题相关的最佳停止问题。目标是为此类过程提供新颖的优化技术并研究相应变分不等式的性质。 （2）开发随机控制技术来研究个人（退休人员）如何将其财富投资于有风险的金融市场的问题，以尽量减少她耗尽财富的可能性，尽量减少她一生的损失。 在本例中，目的是开发 L-无穷大控制的早期结果，并分析相关的变分方程以获得更现实的市场模型。 （3）针对不完全市场制定新的定价原则，旨在为不完全市场中的衍生证券定价和对冲提供新的见解。 这部分还包括为任何一维扩散（具有决策延迟）开发脉冲和奇异控制问题的解决方案。 (4) 开发半马尔可夫切换过程的随机极限定理，以阐明投资者的共同特征对市场价格等总量的影响。最优停止问题在地震学、机器监控、金融、保险等领域有应用、健康监测等。 改进的随机控制技术可以让公众、财务规划者和立法者了解退休后破产的风险。 这项工作将有利于个人在一生中面临的重要财务问题上做出决策：投资多少共同基金；应该购买多少保险；现在是否是借钱投资股市的好时机以及应该借多少钱；什么时候是最好宣布破产的时候；新定价原则的制定为市场上的衍生产品提供了更好的定价机制，这将有利于处于经济中心的金融机构。本部分最后一部分的结果将帮助管理层为公司的福利做出更好的决策，这将造福国家，因为可用资源将得到更有效的利用。该项目的最终目标将通过了解投资者典型行为品质的价格形成来深入了解市场动态。这对于创建有利于经济预测、投资和政策决策的良好金融模型非常重要。