The solution to the continuous time filtering problem boils down to compute the expectation of certain functionals of the signal, which are parametrized by the observations in a continuous manner. In general, an explicit solution is not available and one has to rely on numerical approximations.
The main idea is to approximate the law of the signal along a time partition. On the other hand, we also have to approximate the functionals, as we will only have at our disposal a finite (but usually large) number of observations. The classical approach by Picard gives a discretization error or order one with respect to the partition mesh used to discretize the law of the signal. The aim of this talk is to introduce a new approximation approach that is of order two. We will take advantage of the high frequency stream of observations. Our method is also superior to Picard's filter if compared in terms of computational costs.