论文标题
线性二次高斯均值控制社会Optima
Linear Quadratic Gaussian Mean-Field Controls of Social Optima
论文作者
论文摘要
本文研究了一类统一的随机线性二次高斯（LQG）社会最佳问题问题，这些问题涉及{广义}设置下的大量弱耦合互动剂。对于每个单独的代理，控制和状态过程在其线性动力学中同时进入扩散和漂移项，并且控制权重的成本功能可能是\ emph {Indefinite}。该设置是{创新的，并且具有出色的理论和现实意义}，作为其在数学金融中的应用{（例如，在均值差异模型中选择投资组合）}。使用某些\ emph {完全耦合的}在人为最佳原理和均值场近似方法下的变异分析，分散的社会控制由典型代表性代理的一类新类型一致性条件（CC）系统得出。这种CC系统是一些均值的前后前回向随机微分方程（MF-FBSDE）与\ emph {嵌入表示}相结合。仔细检查了这种前向后的随机微分方程（FBSDE）系统的适应性。相关的社会渐近型最优性与一系列弱耦合的向后随机微分方程（BSDE）的平均值有关。它们通过一些Lyapunov方程进行了验证。
This paper investigates a class of unified stochastic linear quadratic Gaussian (LQG) social optima problems involving a large number of weakly-coupled interactive agents under a {generalized} setting. For each individual agent, the control and state process enters both diffusion and drift terms in its linear dynamics, and the control weight might be \emph{indefinite} in cost functional. This setup is {innovative and has great theoretical and realistic significance} as its applications in mathematical finance {(e.g., portfolio selection in mean-variation model)}. Using some \emph{fully-coupled} variational analysis under person-by-person optimality principle, and mean-field approximation method, the decentralized social control is derived by a class of new type consistency condition (CC) system for typical representative agent. Such CC system is some mean-field forward-backward stochastic differential equation (MF-FBSDE) combined with \emph{embedding representation}. The well-posedness of such forward-backward stochastic differential equation (FBSDE) system is carefully examined. The related social asymptotic optimality is related to the convergence of the average of a series of weakly-coupled backward stochastic differential equation (BSDE). They are verified through some Lyapunov equations.
