This article addresses the robust measurement of covariations in the context of solutions to stochastic evolution equations in Hilbert spaces using functional data analysis. For such equations, standard techniques for functional data based on cross-sectional covariances are often inadequate for identifying statistically relevant random drivers and detecting outliers since they overlook the interplay between cross-sectional and temporal structures. Therefore, we develop an estimation theory for the continuous quadratic covariation of the latent random driver of the equation instead of a static covariance of the observable solution process. We derive identifiability results under weak conditions, establish rates of convergence and a central limit theorem based on infill asymptotics, and provide long-time asymptotics for estimation of a static covariation of the latent driver. Applied to term structure data, our approach uncovers a fundamental alignment with scaling limits of covariations of specific short-term trading strategies, and an empirical study detects several jumps and indicates high-dimensional and time-varying covariations.

翻译：本文采用函数型数据分析方法，针对Hilbert空间中随机演化方程解的协变鲁棒测量问题展开研究。对于此类方程，基于截面协方差的传统函数型数据分析技术往往难以有效识别具有统计相关性的随机驱动因素和异常值，原因在于其忽略了截面结构与时间结构之间的相互作用。为此，我们建立了方程潜在随机驱动过程的连续二次协变估计理论，而非观测解过程的静态协方差。在弱条件下推导出可辨识性结果，建立了基于填充渐近的收敛速度与中心极限定理，并给出了潜变量驱动过程静态协变估计的长期渐近性。通过对期限结构数据的实证分析，本方法揭示了与特定短期交易策略协变尺度极限的根本一致性，实证研究检测到多次跳跃现象并呈现出高维时变协变特征。