The filtering problem: an application of weak approximations of SDEs

Abstract
We present here an alternative view of the continuous time filtering problem, namely the problem is considered as a special case within the theory of weak approximations of stochastic differential equations (SDEs). The class of algorithms arising from this new perspective on the filtering problem estimate the conditional distribution of the signal by first employing an approximation result due to Picard [Lecture Notes in Control and Inform. Sci., 61, Springer, Berlin, 1984.] and then weakly approximating the resulting SDE. As a specific example, Lyons-Victoir cubature on Wiener space is presented. The main characteristics of these algorithms along with a convergence result for the entire class are briefly discussed.