线性二次正倒向随机平均场理论及应用

近年来，由于广泛的出现在政治、经济、金融、工程等领域，受控随机大种群系统的平均场对策问题受到广泛的关注。该问题最大的特点在于可以忽略主体的存在，但是他们的集体行为将会给所有主体施加某种显著的影响。众所周知，诸多金融数学和随机最优控制领域中的问题需要利用正倒向随机微分方程分析并解决。本项目着重研究线性二次正倒向平均场随机系统系列问题。由于系统是由相互耦合的正倒向随机微分方程给出的，我们无法直接运用传统的随机分析方法得到相关结果。在本项目中，我们试图利用逼近技术、压缩映射、随机最大值原理、动态规划原理、正倒向随机微分方程解的相关结果以及部分信息和延迟等理论，得到原始问题的分散化和中心化控制以及近似Nash均衡点。本项目的研究，无论是从政治、经济、金融、工程等实际问题出发，还是从受控的平均场随机问题的理论发展方面，都具有重要的意义。

In recent years, mean field games for controlled stochastic large-population problems have attracted consistent and intense attentions by research communities. The most special feature of controlled large-population system lies in the existence of considerable insignificant agents who are individually negligible but their collective behaviors will impose some significant impact on all agents. As we all know, many problems in the fields of financial mathematics and stochastic optimal control need to be analyzed and solved by forward-backward stochastic differential equation. This program will focus on the series of linear quadratic stochastic large population problems which are driven by forward-backward stochastic differential equations. Since the systems are driven by coupled forward-backward stochastic differential equations, we cannot obtain the responding results by conditional stochastic analysis directly. In this program, we try to use approximation technique, contractive mapping, stochastic maximum principle, dynamic programming principle, related results of forward-backward stochastic differential equations, partial information and delay theory to obtain the decentralized strategy, centralized strategy and the approximate Nash equilibrium of the original problems. The research of this program is of great significance not only from the practical problems of politics, economy, finance and engineering, but also from the theoretical developments of controlled stochastic mean field problems.