We discuss the main mathematical features of the ensemble Kalman filter applied to filtering problems, both in discrete and continuous time. Results concerning stability and accuray will be presented in the case of fully observed processes and small measurement noise. To assess consistency, we also derive the dynamical mean-field equation for the empirical distribution in the infinite ensemble limit and compare it with the Kushner-Stratonovich equation for the posterior distributionof  the optimal filter. Major open questions in the partially observed case as well as generalizations to second order accurate transform filters will be discussed.

Reference: J. de Wiljes, S. Reich, W. Stannat: Long-time stability and accuracy of
the ensemble Kalman-Bucy filter for fully observed processes and small
measurement noise, arXiv:1612.06065