  In this paper we consider the filtering of partially observed
multi-dimensional diffusion processes that are observed regularly at discrete
times. We assume that, for numerical reasons, one has to time-discretize the
diffusion process which typically leads to filtering that is subject to
discretization bias. The approach in [16] establishes that when only having
access to the time-discretized diffusion it is possible to remove the
discretization bias with an estimator of finite variance. We improve on the
method in [16] by introducing a modified estimator based on the recent work of
[17]. We show that this new estimator is unbiased and has finite variance.
Moreover, we conjecture and verify in numerical simulations that substantial
gains are obtained. That is, for a given mean square error (MSE) and a
particular class of multi-dimensional diffusion, the cost to achieve the said
MSE falls.
