Unbiased estimation of the solution to Zakai's equation

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.