  This work embeds a multilevel Monte Carlo sampling strategy into the Monte
Carlo step of the ensemble Kalman filter (EnKF) in the setting of finite
dimensional signal evolution and noisy discrete-time observations. The signal
dynamics is assumed to be governed by a stochastic differential equation (SDE),
and a hierarchy of time grids is introduced for multilevel numerical
integration of that SDE. The resulting multilevel EnKF is proved to
asymptotically outperform EnKF in terms of computational cost versus
approximation accuracy. The theoretical results are illustrated numerically.
