<p style='text-indent:20px;'>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 <i>indefinite</i> 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 <i>fully-coupled</i> variational analysis under the person-by-person optimality principle, and the 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 <i>embedding representation</i>. 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.</p>

中文翻译：

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社会最优的线性二次高斯平均场控制

<p style='text-indent:20px;'>本文研究了一类统一随机线性-二次-高斯（LQG）社会最优问题，在广义设置下涉及大量弱耦合交互代理。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，并且控制权重在成本泛函中可能是<i>不确定的</i>。该设置具有创新性，在数学金融中的应用（如均值变量模型中的投资组合选择）具有重要的理论和现实意义。在逐人最优原则下使用一些<i>全耦合</i>变分分析，以及平均场近似方法，分散的社会控制是由一类新型的一致性条件（CC）系统为典型的代表代理人推导出来的。这种CC系统是一些平均场正反向随机微分方程（MF-FBSDE）结合<i>嵌入表示</i>。仔细检查了这种前后随机微分方程（FBSDE）系统的适定性。相关的社会渐近最优性与一系列弱耦合反向随机微分方程（BSDE）的平均值的收敛有关。它们通过一些 Lyapunov 方程得到验证。</p> 仔细检查了这种前后随机微分方程（FBSDE）系统的适定性。相关的社会渐近最优性与一系列弱耦合反向随机微分方程（BSDE）的平均值的收敛有关。它们通过一些 Lyapunov 方程得到验证。</p> 仔细检查了这种前后随机微分方程（FBSDE）系统的适定性。相关的社会渐近最优性与一系列弱耦合反向随机微分方程（BSDE）的平均值的收敛有关。它们通过一些 Lyapunov 方程得到验证。</p>