Multilevel Ensemble Kalman-Bucy Filters
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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 ensemble Kalman filter, which has proven to be successful due to its applicability and computational cost. We prove that our multilevel EnKBF can achieve a mean square error (MSE) of 𝒪(ϵ^2), ϵ>0 with a cost of order 𝒪(ϵ^-2log(ϵ)^2). This implies a reduction in cost compared to the (single level) EnKBF which requires a cost of 𝒪(ϵ^-3) to achieve an MSE of 𝒪(ϵ^2). In order to prove this result we provide a Monte Carlo convergence and approximation bounds associated to time-discretized EnKBFs. To the best of our knowledge, these are the first set of Monte-Carlo type results associated with the discretized EnKBF. We test our theory on a linear problem, which we motivate through a relatively high-dimensional example of order ∼ 10^3.
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