On the mathematical theory of ensemble (linear-Gaussian) Kalman–Bucy filtering

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