  In the following article we consider the non-linear filtering problem in
continuous-time and in particular the solution to Zakai's equation or the
normalizing constant. We develop a methodology to produce finite variance,
almost surely unbiased estimators of the solution to Zakai's equation. That is,
given access to only a first order discretization of solution to the Zakai
equation, we present a method which can remove this discretization bias. The
approach, under assumptions, is proved to have finite variance and is
numerically compared to using a particular multilevel Monte Carlo method.
