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

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被处理。首先，通过先验估计和“桥”的概念，即将FBSDE与具有约束的Riccati微分不等式联系起来的映射，建立具有混合初始和终端条件的FBSDE的适定性。其次，开发了FBSDEs的峰值变化技术，并利用该技术推导了FBSDEs最优控制所满足的Pontryagin型极大值原理。最后，将引入解耦技术研究具有随机系数和混合初始和末端条件的耦合线性FBSDEs，并利用随机Riccati方程求解相关的线性二次最优控制问题。本课题所导出的理论将大大丰富FBSDEs的现有理论，并将对经典随机最优控制理论进行深入扩展。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），收益最大化和风险最小化是非常普遍的。众所周知，高回报伴随着高风险。仔细研究表明，人们对回报/风险的偏好通常不是线性的。著名的阿莱/埃尔斯伯格式悖论（Allais/Ellsberg type paradox）是很好的反例，它表明在高风险情况下做出的决策与预期效用理论不一致。为了弥补这一点，可以通过所谓的后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、保险债权、期权等）、有大投资者的市场（包括金融市场、能源市场等）（如一些大型对冲基金，其主要特点是价格过程依赖于大投资者的头寸和交易策略）等等。本项目建立的理论将帮助我们至少部分地回答以下问题：非线性偏好如何影响投资的最优交易策略？当一些或有债权被允许交易时，最优的交易策略是什么？大投资者是如何影响市场的？

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?