Kalman-Bucy Filter Overview

• The Kalman-Bucy filter is the continuous-time counterpart of the Kalman filter, delivering optimal recursive state estimation for linear systems with Gaussian noise.
• It underpins numerous extensions including ensemble, low-rank, and localized methods, which enhance performance in high-dimensional and nonlinear applications.
• Robust and uncertainty-aware adaptations address model imperfections and heavy-tailed noises, broadening its applicability in atmospheric, oceanographic, and engineering systems.

The Kalman-Bucy filter is the continuous-time analog of the Kalman filter, providing an optimal recursive solution for linear dynamical systems with Gaussian process and observation noise. It is foundational to continuous-time data assimilation and filtering in high-dimensional applications, including atmospheric, oceanographic, and engineering systems. Since its original formulation, the Kalman-Bucy filter has served as the basis for a diverse spectrum of modern extensions: ensemble-based variants for high-dimensional and nonlinear settings; robust and uncertainty-aware filters; low-rank and dynamical low-rank approximations; and multilevel, localized, and quantization-based computational strategies.

1. Classical Kalman-Bucy Filter: Formulation and Properties

For a linear continuous-time system described by

$$dx(t) = F(t)x(t)\,dt + G(t)\,dw(t), \quad dy(t) = H(t)x(t)\,dt + D(t)\,dv(t),$$

with $w(t)$, $v(t)$ independent Brownian motions, the Kalman-Bucy filter provides the exact finite-dimensional solution to the continuous-time filtering problem under Gaussian assumptions. The estimates evolve via

$$d\hat{x}(t) = F(t) \hat{x}(t)\,dt + P(t) H(t)^\top R^{-1} (dy(t) - H(t)\hat{x}(t)\,dt)$$

$$\dot{P}(t) = F(t)P(t) + P(t)F(t)^\top + G(t)QG(t)^\top - P(t)H(t)^\top R^{-1} H(t)P(t)$$

where $P(t)$ is the error covariance matrix, and the Kalman gain is $K(t) = P(t)H(t)^\top R^{-1}$. This Riccati equation admits strong contraction and comparison inequalities. For time-invariant systems, the steady-state error covariance is the stabilizing solution of the algebraic Riccati equation, and can also be expressed in a frequency-domain integral involving the plant dynamics and the Bode integral, explicitly capturing fundamental estimation limits via system zeros, poles, and unstable modes (Fang et al., 2018).

Stability and contraction results guarantee that, under observability and controllability, filter estimates "forget" initialization exponentially fast and the filter remains stable, even for unstable signals, due to the dissipative effect of the measurement update (Bishop et al., 2016). For noise-free systems, stability holds under uniform complete observability, with the estimation error converging to zero and extensions existing for non-Gaussian initial data and small process noise (Reddy et al., 2019).

2. Ensemble and Particle-Based Kalman-Bucy Extensions

Direct implementation of the classical Kalman-Bucy filter is computationally prohibitive for high-dimensional systems, motivating ensemble (particle-based) approximations.

Ensemble Kalman-Bucy Filter (EnKBF): The state estimate is represented with an ensemble of $M$ particles. Each member evolves according to system dynamics, with correction terms involving empirical covariances. For large $M$, the ensemble mean and covariance converge in probability to the Kalman-Bucy solution (mean-field limit and "propagation of chaos"). The filtering SDE for ensemble members is: $$dX_t^{(i)} = f\left(X_t^{(i)}\right)dt + Q^{1/2}dW_t^{(i)} + K_t^M \left( dY_t - H X_t^{(i)}dt \right)$$ with $K_t^M$ computed from the ensemble sample covariances. Uniform-in-time $L^p$ convergence and stability are established under observability conditions; with $N$ ensemble members,

$$\sup_t \mathbb{E} \left\|P_t^{(N)} - P_t \right\|_F^n \leq C(n)/\sqrt{N}$$

(Moral et al., 2016). Under full observations and small measurement noise, mean-square estimation error scales as $\mathcal{O}(\epsilon^{1/2})$ (Wiljes et al., 2016).

Ensemble Transform Kalman-Bucy Filters (ETKBF and DETKBF): ETKBFs embed the continuous-time analysis step in "pseudo-time", updating ensemble perturbations (BGR09) or the full ensemble (BR10) via ODEs. Transform versions recast the update in ensemble space (lower-dimensional weights) and introduce diagonally semi-implicit (DSI) integration to handle stiffness for large background-to-observational error covariance ratios, ensuring stability for infrequent observations. ETKBF and DETKBF offer accuracy equivalent to local ETKF with strong computational advantages, especially under proper DSI integration (Amezcua et al., 2011).

Multilevel and Localized Methods: Multilevel EnKBF (MLEnKBF) and its localized variant (MLLEnKBF) combine multilevel Monte Carlo (MLMC) and covariance localization, yielding further computational gains and reducing spurious correlations (which arise with small ensembles in high-dimensions). The telescoping MLMC estimator for expectation $\eta_T(\varphi)$ writes: $$\eta_T^{(\mathrm{ML})}(\varphi) = \eta_T^{N_0,0}(\varphi) + \sum_{l=1}^L \left(\eta_T^{N_l,l}(\varphi) - \eta_T^{N_l,l-1}(\varphi)\right)$$ Cost is reduced from $\mathcal{O}(\epsilon^{-3})$ for single-level to $\mathcal{O}(\epsilon^{-2} \log(\epsilon)^2)$ for the ML estimator for a prescribed MSE $\mathcal{O}(\epsilon^2)$ (Chada et al., 2020, Ruzayqat et al., 2021, Chada, 24 Feb 2025).

3. Extensions to Uncertainty, Robustness, and Heavy-Tailed Noise

Robust Kalman-Bucy Filtering: Robust extensions generalize the Kalman-Bucy problem to uncertainty in system dynamics, noise covariances, and even to model uncertainty represented by a family of probability measures. The robust filter is formulated as a minimax estimator under a sublinear or convex expectation, often implemented via Girsanov transformation and the minimax theorem. The optimal estimator solves a Kalman-Bucy-type SDE under a new measure, with a decomposition into a nominal estimator and an explicit uncertainty correction term. Error bounds are derived in terms of the system uncertainty variance (Ji et al., 2019, Ji et al., 2020, Kunisch et al., 31 May 2025).

Observation Noise with Infinite Second Moment: For systems with Lévy-driven (integrable, infinite-variance) observation noise, the classical Kalman-Bucy filter is extended via approximation by truncating large jumps in the Lévy process. The limiting filter discards infinite variance components, becoming robust to impulsive noise and behaving as a best linear estimator in $L^1$. When all observation noise components have infinite variance, the update term vanishes, and the filter reduces to a deterministic ODE for the state estimate (Applebaum et al., 2013).

Uncertain Dynamical Systems: Deterministic Kalman filters for systems with parametric uncertainty (e.g., in A, Γ, R, Q) can be combined using various strategies: a filter built from expected (averaged) system parameters ("expected system"), a plain ensemble average over all possible models, or an energy/minimum expected Mahalanobis distance estimator (precision-weighted mean). Explicit energy-based error bounds are derived as a function of matrix uncertainty (Kunisch et al., 31 May 2025).

4. Operator-Theoretic, Low-Rank, and Dynamical Low-Rank Approximations

For large-scale systems, low-rank and dynamically low-rank approximations to the Kalman-Bucy filter dramatically reduce computational cost.

Oja Flow Low-Rank Approximation: The full covariance is approximated as $\widetilde{P}(t) = U(t)\widetilde{R}(t)U(t)^\top$, with $U(t)$ evolving on the Stiefel manifold via Oja's flow,

$dU/dt = (I_n - UU^\top)AU$

and $\widetilde{R}(t)$ solving a reduced r-dimensional Riccati equation. Stability, convergence, and conditions on the reduced subspace (minimal rank capturing all unstable modes of A) are rigorously analyzed (Tsuzuki et al., 5 Mar 2024).

Dynamical Low-Rank Kalman-Bucy Process (DLR-KBP): The filtering distribution is projected onto a time-varying low-dimensional manifold using the ansatz

$X_t = U^0_t + U_t Y_t^\top$

The evolving modes $U_t$, mean term $U^0_t$, and stochastic coefficients $Y_t^\top$ are governed by coupled SDEs, with the orthogonality condition ensuring uniqueness. The low-rank filter achieves substantial speed-up and, when filter uncertainty is concentrated along a low-dimensional subspace (small process/observation noise), approaches the accuracy of the full-order filter. This approach also extends to ensemble formulations (DLR-ENKF), extracting further computational benefits (Nobile et al., 14 Sep 2025).

5. Filter Adaptations for Non-Standard System Classes

Observation Noise Correlations: Extensions to correlated observation noise require modification of the gain term and introduce additional transport-type corrections. The mean-field EnKBF for correlated noise is a McKean–Vlasov SDE with coefficients depending on the law of the process. For such systems, rigorous a priori bounds, existence/uniqueness, and propagation of chaos properties can be established with careful control of pseudoinverses of empirical covariance matrices (Ertel et al., 2022).

Jump Linear and Semi-Markov Systems: For systems driven by semi-Markov jump processes, the Kalman-Bucy filter is approximated via optimal quantization of sojourn times and pre-computation of Riccati flows along typical trajectories. This enables real-time filtering where the set of system matrices switches according to a semi-Markov process, with rigorous Lipschitz error bounds and convergence results (with error scaling in quantization and time-discretization step). Application to magnetic levitation systems demonstrates practical feasibility (Saporta et al., 2014).

6. Comparative Performance and Computational Strategies

The following table summarizes key method classes, their defining properties, and computational implications:

Class	Key Properties	Computational Notes
Classical Kalman-Bucy	Optimal for linear, Gaussian, continuous-time systems	Requires full matrix Riccati propagation
ETKBF, DETKBF	Ensemble / pseudo-continuous-time analysis ODE; transform in ensemble space	Avoids high-dimensional inverses; DSI
Multilevel, Localized EnKBF	Combines localization for small ensembles and MLMC for error/cost trade-off	Cost $\sim \mathcal{O}(\epsilon^{-2}\log^2 \epsilon)$
Low-rank, DLR approximations	Evolution on low-dimensional principal (time-varying) subspaces; Oja flow; DLR ansatz	Cost scales with subspace dimension
Robust, Uncertainty-Aware	Estimation under model uncertainty, convex/sublinear expectation, minimax, g-expectation	Additional uncertainty correction required
Lévy Noise Extensions	Handles integrable, infinite-variance noise via jump truncation and limiting operations	Limit may trivialize filter

In practical large-scale geophysical models, the balance between accuracy, computational tractability, and data assimilation requirements leads to widespread adoption of ensemble, multilevel, and low-rank Kalman-Bucy variants.

7. Outlook and Implications

Advances in the Kalman-Bucy framework and its variants have enabled scalable, accurate continuous-time filtering in high-dimensional, nonlinear, and uncertainty-dominated regimes. The unifying feature is the transition from deterministic filter equations to stochastic, ensemble-based, or operator-theoretic formulations ensuring tractable propagation of state uncertainty and efficient incorporation of observational data. Open challenges include numerical implementation for highly nonlinear systems, adaptive dimension reduction, robustification under heavy-tailed or ambiguous noise, and further theoretical characterization of ensemble propagation, stability, and convergence properties in high dimensions and extended dynamical regimes.

The Kalman-Bucy filter and its descendants remain at the heart of continuous-time filtering theory and practice, with deep ties to control, probability, matrix differential equations, and modern computational science.