  In this paper, we consider the filtering problem for partially observed
diffusions, which are regularly observed at discrete times. We are concerned
with the case when one must resort to time-discretization of the diffusion
process if the transition density is not available in an appropriate form. In
such cases, one must resort to advanced numerical algorithms such as particle
filters to consistently estimate the filter. It is also well known that the
particle filter can be enhanced by considering hierarchies of discretizations
and the multilevel Monte Carlo (MLMC) method, in the sense of reducing the
computational effort to achieve a given mean square error (MSE). A variety of
multilevel particle filters (MLPF) have been suggested in the literature, e.g.,
in Jasra et al., SIAM J, Numer. Anal., 55, 3068--3096. Here we introduce a new
alternative that involves a resampling step based on the optimal Wasserstein
coupling. We prove a central limit theorem (CLT) for the new method. On
considering the asymptotic variance, we establish that in some scenarios, there
is a reduction, relative to the approach in the aforementioned paper by Jasra
et al., in computational effort to achieve a given MSE. These findings are
confirmed in numerical examples. We also consider filtering diffusions with
unstable dynamics; we empirically show that in such cases a change of measure
technique seems to be required to maintain our findings.
