The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman-Bucy filtering for continuous-time, linear-Gaussian signal and observation models. We present a system of equations that describe the flow of individual particles and the flow of the sample covariance and the sample mean in continuous-time ensemble filtering. We consider these equations and their characteristics in a number of popular ensemble Kalman filtering variants. Given these equations, we study their asymptotic convergence to the optimal Bayesian filter. We also study in detail some non-asymptotic time-uniform fluctuation, stability, and contraction results on the sample covariance and sample mean (or sample error track). We focus on testable signal/observation model conditions, and we accommodate fully unstable (latent) signal models. We discuss the relevance and importance of these results in characterising the filter's behaviour, e.g. it's signal tracking performance, and we contrast these results with those in classical studies of stability in Kalman-Bucy filtering.We also provide a novel (and negative) result proving that the bootstrap particle filter cannot track even the most basic unstable latent signal, in contrast with the ensemble Kalman filter (and the optimal filter). We provide intuition for how the main results extend to nonlinear signal models and comment on their consequence on some typical filter behaviours seen in practice, e.g. catastrophic divergence.

翻译：本综述旨在全面概述连续时间线性-高斯信号与观测模型下的集成卡尔曼-布西滤波理论。我们提出了一套描述连续时间集成滤波中粒子个体流动、样本协方差及样本均值流动的方程组，并考察了这些方程在若干主流集成卡尔曼滤波变体中的特性。基于这些方程，我们研究了其向最优贝叶斯滤波器的渐近收敛性，并详述了样本协方差、样本均值（或样本误差轨迹）的非渐近时间一致波动性、稳定性与收缩性结果。研究中聚焦于可检验的信号/观测模型条件，并涵盖了完全不稳定（潜在）信号模型。我们探讨了这些结果在刻画滤波器行为（如信号追踪性能）方面的相关性与重要性，并将其与经典卡尔曼-布西滤波稳定性研究的结果进行对比。此外，我们提出了一项新颖（且负面）的结论：与集成卡尔曼滤波器（及最优滤波器）不同，自举粒子滤波器即便对最基础的不稳定潜在信号也无法追踪。我们进一步阐释了主要结果向非线性信号模型推广的直觉，并评述了其对实践中常见滤波器行为（如灾难性发散）的影响。