Multilevel Localized Ensemble Kalman Bucy Filters

In this article we propose and develop a new methodology which is inspired from Kalman filtering and multilevel Monte Carlo (MLMC), entitle the multilevel localized ensemble Kalman--Bucy Filter (MLLEnKBF). Based on the work of Chada et al. \cite{CJY20}, we provide an important extension on this which is to include the technique of covariance localization. Localization is important as it can induce stability and remove long spurious correlations, particularly with a small ensemble size. Our resulting algorithm is used for both state and parameter estimation, for the later we exploit our method for normalizing constant estimation. As of yet, MLMC has only been applied to localized data assimilation methods in a discrete-time setting, therefore this work acts as a first in the continuous-time setting. Numerical results indicate its performance, and benefit through a range of model problems, which include a linear Ornstein--Uhlenbeck process, of moderately high dimension, and the Lorenz 96 model, for parameter estimation. Our results demonstrate improved stability, and that with MLMC, one can reduce the computational complexity to attain an order is MSE $\mathcal{O}(\epsilon^2)$, for $\epsilon>0$.