发表机构
Reykjavik University（雷克雅未克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对带共同噪声的异质线性二次平均场控制问题，推导有限总体精确Riccati系统并与连续统模型比较，给出定量收敛估计，并构造最优性差距消失的分散策略。
AI 中文摘要
我们研究了带共同噪声和随机系数的异质线性二次平均场控制问题的有限总体逼近。从包含 $N$ 个不可交换智能体的集中式社会规划者问题出发，我们推导出其精确的有限维随机 Riccati 系统，并识别其对角、非对角、仿射和标量分量的缩放。然后，我们将该系统与连续统模型的 Hilbert 空间值 Riccati 系统进行比较。我们的主要估计给出了完整后向系统（包括其共同噪声鞅被积函数）的定量收敛性。误差分解为系数和核一致性、共同噪声逼近、界面效应、精确对角修正以及离散对角带上极限相互作用核的质量。我们将这些估计传播到反馈增益，并通过逐单元耦合传播到最优状态、控制和初始社会价值。最后，对极限代表智能体反馈的逐单元投影产生一种分散式有限总体策略，其最优性差距消失。在所述正则性假设下，误差界通过显式的逼近和集中模量表示。这些模量的定量界产生代数速率，包括有限类型情形下后向系统的 $N^{-1/2}$ 速率；最优轨迹的速率还反映了可用的条件矩界。
英文摘要
We study the finite-population approximation of a heterogeneous linear--quadratic mean-field control problem with common noise and random coefficients. Starting from the centralized social planner problem for $N$ non-exchangeable agents, we derive its exact finite-dimensional stochastic Riccati system and identify the scaling of its diagonal, off-diagonal, affine, and scalar components. We then compare this system with the Hilbert-space-valued Riccati system of the continuum model. Our main estimates give quantitative convergence of the complete backward system, including its common-noise martingale integrands. The error separates coefficient and kernel consistency, common-noise approximation, interface effects, exact diagonal corrections, and the mass of the limiting interaction kernel on the discrete diagonal band. We propagate these estimates to the feedback gains and, by a cellwise coupling, to the optimal states, controls, and initial social values. Finally, a cellwise projection of the limiting representative-agent feedback yields a decentralized finite-population strategy whose optimality gap vanishes. Under the stated regularity assumptions, the error bounds are expressed through explicit approximation and concentration moduli. Quantitative bounds on these moduli yield algebraic rates, including an $N^{-1/2}$ rate for the backward system in finite-type regimes; the rates for optimal trajectories additionally reflect the available conditional moment bounds.
Comments49 pages, including appendices
