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

该项目包括四个部分：（1）解决与几个最快检测问题相关联的最优停止问题。我们的目标是提供新的优化技术，这样的过程和相应的变分不等式的研究性质。(2)发展随机控制技术来研究个人（退休人员）应该如何将其财富投资于风险金融市场的问题，以最大限度地减少她失去财富的可能性，最大限度地减少她的终身短缺。 在这种情况下，我们的目标是发展L-无穷控制的早期结果，并分析相关的变分等式更现实的市场模型。(3)为不完全市场制定新的定价原则，目的是为不完全市场中的衍生证券定价和对冲提供新的见解。 这一部分还包括任何一维扩散（决策延迟）的脉冲和奇异控制问题的解决方案。 (4)建立半马尔可夫切换过程的随机极限定理，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、健康监测等领域都有应用。 改进的随机控制技术可以告知公众，财务规划师和立法者关于退休破产的风险。 这项工作将有利于个人在生活中面临的重要财务问题上做出决策：在共同基金中投资多少;应该购买多少保险;是否是借贷投资股票市场的好时机以及应该借贷多少;何时宣布破产是最好的;新定价原则的发展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。本部分最后一节的结果将有助于管理层为公司的福利做出更好的决策，这将有利于国家，因为可用资源将得到更有效的利用。该项目的最终目标是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。这对于创建良好的金融模型，有利于经济预测，投资和政策决策非常重要。

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.