Abstract
Ensemble Kalman–Bucy filters (EnKBFs) are an important tool in data assimilation that aim to approximate the posterior distribution for continuous time filtering problems using an ensemble of interacting particles. In this work we extend a previously derived unifying framework for consistent representations of the posterior distribution to correlated observation noise and use these representations to derive an EnKBF suitable for this setting as a constant gain approximation of these optimal filters. Existence and uniqueness results for both the EnKBF and its mean field limit are provided. The existence and uniqueness of solutions to its limiting McKean-Vlasov equation does not seem to be covered by the existing literature. In the correlated noise case the evolution of the ensemble depends also on the pseudoinverse of its empirical covariance matrix, which has to be controlled for global well-posedness. These bounds may also be of independent interest. Finally the convergence to the mean field limit is proven. The results can also be extended to other versions of EnKBFs.
Funding Statement
Sebastian Ertel is supported by Deutsche Forschungsgemeinschaft through IRTG 2544—Stochastic Analysis in Interaction.
The research of Wilhelm Stannat has been partially funded by Deutsche Forschungsgemeinschaft (DFG)—Project-ID 318763901—SFB 1294.
Acknowledgments
The authors would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper.
The second author is also affiliated with Bernstein Center for Computational Neuroscience, Berlin, Germany.
Citation
Sebastian W. Ertel. Wilhelm Stannat. "Analysis of the ensemble Kalman–Bucy filter for correlated observation noise." Ann. Appl. Probab. 34 (1B) 1072 - 1107, February 2024. https://doi.org/10.1214/23-AAP1985
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