部分可观测平均场随机系统的最优控制理论及其应用

Mean-field stochastic systems have wild application prospects in the fields of statistical mechanics, quantum chemistry,large-population multi-agent systems,finance and econimics.Then it is worthy of making a thorough research on the corresponding control theory.This program is concerned with a kind of partially observable optimal control problem, where the drift coefficient of the observation equation is unbouded, and the observation noise is correlated to the state noise.Making use of some theoretical tools arising from stochastic control,stochastic differential equation, Sobolev space, Malliavin calculus,L^p convergence,the approach of the backward seperation is futhermore developed. A Zakai equation of the corresopinding adjoint processes is established. Some properties of the (week) solutions of the Zakai equation and mean-field stochastic differential equations are studied. Maximum principle and verification theorem are derived. Also, feedback solutions of certain linear-quadratic optimal control problems are addressed. To shed light on the appliction of the theoretical results obtained in this program, some important financial mathematics problems including mean-variance portfolio selection and risk measure are investigated.

平均场随机系统在统计力学、量子化学、大种群多智能体系统、金融经济领域应用前景广阔，其控制理论值得深入研究。本项目关注部分可观测信息下由均方场随机微分方程驱动的最优控制问题，在观测方程漂移项系数无界，且状态噪声和观测噪声相关假定下，利用随机控制、随机微分方程、Sobolev空间、Malliavin变分、L^p收敛等理论工具，进一步发展"倒向分离"方法，建立对偶过程满足的Zakai方程，研究均方场随机微分方程与Zakai方程（弱）解的性质，推出最优控制满足的最大值原理与验证定理，寻求线性二次最优控制问题的反馈解，并探讨在均值-方差投资组合、风险度量等重要金融问题中的应用。

平均场随机系统在统计力学、量子化学、大种群多智能体系统、金融经济领域应用前景广阔，其控制理论值得深入研究。本项目主要研究了部分可观测平均场正倒向随机系统最优控制、部分可观测正倒向随机系统最优控制，利用变分技术、随机滤波等工具，进一步发展“倒向分离”方法，推出最优控制满足的最大值原理与验证定理，并将得到的理论结果应用于线性二次控制以及金融问题。此外，还研究了不对称信息下的线性二次主从随机微分博弈和非零和微分博弈，运用最大值原理，分别得到了追随者与领导者的最优策略以及均衡点的显式解。