Optimal Control for Forward-Backward Stochastic Differential Equations and Related Topics

前向-后向随机微分方程的最优控制及相关主题

• 批准号: 0604309
• 负责人: Jiongmin Yong
• 金额: $ 15万
• 依托单位: The University of Central Florida Board of Trustees
• 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 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604309&HistoricalAwards=false
• 关键词: Optimal; Control; Forward; Backward; Stochastic

An optimal control theory framework will be established for forward-backward stochastic differential equations (FBSDEs). A number of fundamental mathematical questions will be addressed. First, well-posedness for FBSDEs with mixed initial and terminal conditions will be established by means of a priori estimates together with the notion of a "bridge", i.e., a map which relates the FBSDE to a Riccati differential inequality with constraints. Secondly, a spike variation technique for FBSDEs will be developed and used to derive a Pontryagin type maximum principle that is satisfied by optimal controls of FBSDEs. Finally, coupled linear FBSDEs with random coefficients and mixed initial and terminal conditions will be studied by introducing decoupling techniques and related linear-quadratic optimal control problems will be solved using stochastic Riccati equations. The theory derived in this project will substantially enrich the existing theory of FBSDEs, and will deeply extend the classical stochastic optimal control theory. Maximizing return and minimizing risks are very common in the world of investment (including stock market, mutual funds, retirement accounts, insurance, social security, etc.). It is well understood that high return is associated with high risks. Careful study shows that people's preferences towards return/risks are usually not linear. The well-known Allais/Ellsberg type paradoxes, which show that decisions made in the presence of high risk are inconsistent with expected utility theory, are excellent counterexamples for this. To compensate for this, some nonlinear preferences (also called nonlinear expectation) can be introduced via the so-called backward stochastic differential equations. Therefore, when such kind of expectation is used, optimal investment problem naturally becomes an optimal control problem for forward-backward stochastic differential equations. A similar situation happens for investment involving contingent claims (such as discount bonds, insurance claims, options, etc.), markets (including financial markets, energy markets, etc.) with large investors (such as some big hedge funds, a main feature of which is the dependence of the price processes on the positions and trading strategies of the large investors), and so on. The theory established in this project will help us to answer, at least in part, the following types of questions: How do the nonlinear preferences affect the optimal trading strategies for an investment? What will be the optimal trading strategies when some contingent claims are allowed to trade? How do the large investors influence the market?

建立了正倒向随机微分方程的最优控制理论框架。 一些基本的数学问题将得到解决。 首先，具有混合初始和终止条件的FBSDE的适定性将通过先验估计以及“桥”的概念来建立，即，一个映射，它将FBbundle映射到一个带约束的Riccati微分不等式。 其次，尖峰变化技术FBSDES将被开发和使用，以获得满足FBSDES的最优控制的庞特里亚金型最大值原理。 最后，耦合线性FBSDES随机系数和混合的初始和终端条件将研究通过引入解耦技术和相关的线性二次型最优控制问题将使用随机Riccati方程求解。 本课题的研究成果将极大地丰富和发展现有的随机最优控制理论，并将对经典的随机最优控制理论进行深入的拓展。最大化回报和最小化风险在投资领域（包括股票市场，共同基金，退休账户，保险，社会保障等）非常常见。 众所周知，高回报伴随着高风险。 仔细的研究表明，人们对回报/风险的偏好通常不是线性的。 著名的阿莱斯/埃尔斯伯格型悖论（Allais/Ellsberg type paradoxies）是一个很好的反例，它表明在高风险的情况下做出的决策与期望效用理论不一致。 为了弥补这一点，可以通过所谓的倒向随机微分方程引入一些非线性偏好（也称为非线性期望）。 因此，当使用这种期望时，最优投资问题自然就变成了正倒向随机微分方程的最优控制问题。 类似的情况也发生在涉及或有债权的投资（如贴现债券、保险索赔、期权等），市场（包括金融市场、能源市场等）与大型投资者（如一些大型对冲基金，其主要特征是价格过程对大型投资者的头寸和交易策略的依赖性）等等。在这个项目中建立的理论将帮助我们回答，至少部分回答以下类型的问题：非线性偏好如何影响投资的最优交易策略？ 当允许某些未定权益交易时，最优交易策略是什么？ 大型投资者如何影响市场？