Multilevel Ensemble Kalman-Bucy Filters

In this article we consider the linear filtering problem in continuous-time. We develop and apply multilevel Monte Carlo (MLMC) strategies for ensemble Kalman-Bucy filters (EnKBFs). These filters can be viewed as approximations of conditional McKean-Vlasov-type diffusion processes. They are also interpreted as the continuous-time analogue of the \textit{ensemble Kalman filter}, which has proven to be successful due to its applicability and computational cost. We prove that an ideal version of our multilevel EnKBF can achieve a mean square error (MSE) of O(?(2)), ?> 0 with a cost of order O(?(-2)log(?)(2)). In order to prove this result we provide a Monte Carlo convergence and approximation bounds associated to time-discretized EnKBFs. This implies a reduction in cost compared to the (single level) EnKBF which requires a cost of O(?(-3)) to achieve an MSE of O(?(2)). We test our theory on a linear problem, which we motivate through high-dimensional examples of order similar to O(10(4)) and O(10(5)), where we also numerically test an alternativedeterministic counterpart of the EnKBF.