Multilevel Estimation of Normalization Constants using Ensemble Kalman–Bucy filters

In this article we consider the application of multilevel Monte Carlo, for the estimation of normalizing constants. In particular we will make use of the filtering algorithm, the ensemble Kalman–Bucy filter (EnKBF), which is an N-particle representation of the Kalman–Bucy filter (KBF). The EnKBF is of interest as it coincides with the optimal filter in the continuous-linear setting, i.e. the KBF. This motivates our particular setup in the linear setting. The resulting methodology we will use is the multilevel ensemble Kalman–Bucy filter (MLEnKBF). We provide an analysis based on deriving Lq-bounds for the normalizing constants using both the single-level, and the multilevel algorithms, which is largely based on previous work deriving the MLEnKBF Chada et al. (2022). Our results will be highlighted through numerical results, where we firstly demonstrate the error-to-cost rates of the MLEnKBFs comparing it to the EnKBF on a linear Gaussian model. Our analysis will be specific to one variant of the MLEnKBF, whereas the numerics will be tested on different variants. We also exploit this methodology for parameter estimation, where we test this on the models arising in atmospheric sciences, such as the stochastic Lorenz 63 and 96 model.