This paper develops a statistical framework for goodness-of-fit testing of volatility functions in McKean-Vlasov stochastic differential equations, which describe large systems of interacting particles with distribution-dependent dynamics. While integrated volatility estimation in classical SDEs is now well established, formal model validation and goodness-of-fit testing for McKean-Vlasov systems remain largely unexplored, particularly in regimes with both large particle limits and high-frequency sampling. We propose a test statistic based on discrete observations of particle systems, analysed in a joint regime where both the number of particles and the sampling frequency increase. The estimators involved are proven to be consistent, and the test statistic is shown to satisfy a central limit theorem, converging in distribution to a centred Gaussian law.

翻译：本文针对描述具有分布依赖动力学特性的大型相互作用粒子系统的McKean-Vlasov随机微分方程，建立了一套用于波动率函数拟合优度检验的统计框架。尽管经典随机微分方程中的积分波动率估计方法现已较为成熟，但McKean-Vlasov系统的形式化模型验证与拟合优度检验研究仍基本处于空白状态，特别是在同时涉及大粒子极限与高频采样的机制中。我们基于粒子系统的离散观测数据提出了一种检验统计量，该统计量在粒子数量与采样频率同时增长的联合机制下进行分析。研究证明所涉及的估计量具有一致性，且该检验统计量满足中心极限定理，依分布收敛于中心化高斯分布。