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

该方案包括四个部分：(1)求解与跳跃过程的几个最快检测问题相关的最优停止问题。其目的是为这类过程提供新的优化技术，并研究相应的变分不等式的性质。(2)发展随机控制技术来研究个人(退休人员)应如何将其财富投资于风险金融市场的问题，以最小化其寿命超过其财富的概率，最小化其一生的短缺。在这种情况下，目的是发展L无穷控制的早期结果，并分析与之相关的变分方程，以得到更真实的市场模型。(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衍生证券的定价和对冲提供新的见解。这一部分还包括为任何一维扩散(带有决策延迟)开发脉冲和奇异控制问题的解决方案。(4)发展了半马尔可夫切换过程的随机极限定理，阐明了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、机器监测、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能会让公众、财务规划师和立法者知道退休后破产的风险。这项工作将有利于个人在一生中面临的重大金融问题的决策：投资多少共同基金；应该购买多少保险；现在是不是借钱投资股市的好时机和借多少钱；什么时候最好宣布破产；什么时候应该退休等等。新定价原则的发展为市场上的衍生品提供了更好的定价机制，这将使处于经济中心的金融机构受益。这一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定，这将使国家受益，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从投资者的典型行为品质了解价格形成来洞察市场动态。这对于创建有益于经济预测、投资和政策决策的良好金融模型非常重要。

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.