  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 Kalma-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. Our results will be highlighted through numerical
results, where we firstly demonstrate the error-to-cost rates of the MLEnKBF
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.
