  We present recent results on the existence of a continuous time limit for
Ensemble Kalman Filter algorithms. In the setting of continuous signal and
observation processes, we apply the original Ensemble Kalman Filter algorithm
proposed by [1] as well as a recent variant [2] to the respective
discretizations and show that in the limit of decreasing stepsize the filter
equations converge to an ensemble of interacting (stochastic) differential
equations in the ensemble-mean-square sense. Our analysis also allows for the
derivation of convergence rates with respect to the stepsize.
  An application of our analysis is the rigorous derivation of continuous
ensemble filtering algorithms consistent with discrete approximation schemes.
Conversely, the continuous time limit allows for a better qualitative and
quantitative analysis of the time-discrete counterparts using the rich theory
of dynamical systems in continuous time.
  [1] Burgers, G., van Leeuwen, P. J., Evensen, G. (1998). Analysis scheme in
the ensemble Kalman filter. Monthly weather review, 126(6), 1719-1724.
  [2] de Wiljes, J., Reich, S., Stannat, W. (2018). Long-Time Stability and
Accuracy of the Ensemble Kalman-Bucy Filter for Fully Observed Processes and
Small Measurement Noise. SIAM Journal on Applied Dynamical Systems, 17(2),
1152-1181.
