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(\epsilon^2), \ \epsilon>0 with a cost of order O(\epsilon^{-2}\log(\epsilon)^2).  This implies a reduction in cost compared to the (single level) EnKBF which requires a cost of O(\epsilon^{-3}) to achieve an MSE of O(\epsilon^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 \sim  10 ^3 .
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 an ideal version of our multilevel EnKBF can achieve a mean square error (MSE) of O(\epsilon^2), \ \epsilon>0 with a cost of order O(\epsilon^{-2}\log(\epsilon)^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(\epsilon^{-3}) to achieve an MSE of O(\epsilon^2). We test our theory on a linear problem, which we motivate through high-dimensional examples of order \sim \mathcal{O 10 ^4) and \mathcal{O .
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Kalman--Bucy
Kalman-Bucy
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McKean--Vlasov-type
McKean-Vlasov-type
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ensemble Kalman filter
ensemble Kalman filter
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an ideal version of
A
In order to prove this result we provide a Monte Carlo convergence and approximation bounds associated to time-discretized EnKBFs.
D
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.
D
a relatively
R
example
examples
A
\mathcal{O
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^3
^4) and \mathcal{O