# On the continuous time limit of the Ensemble Kalman Filter

## TL;DR

This paper establishes the mathematical foundation for the continuous time limit of the Ensemble Kalman Filter, showing convergence of discrete algorithms to stochastic differential equations and enabling better analysis of filtering methods.

## Contribution

It provides a rigorous proof of the continuous time limit for ensemble Kalman filters and derives convergence rates, bridging discrete algorithms with continuous stochastic processes.

## Key findings

- Discrete ensemble Kalman filters converge to stochastic differential equations in the continuous limit.
- Convergence rates with respect to stepsize are derived.
- The analysis facilitates better qualitative and quantitative understanding of filtering algorithms.

## Abstract

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.