The importance of optimal filtration problem for Markov chain with two states observed in Gaussian white noise (GWN) for a lot of concrete technical problems is well known. The equation for a posterior probability $\pi(t)$ of one of the states was obtained many years ago. The aim of this paper is to study a simple filtration method. It is shown that this simplified filtration is asymptotically efficient in some sense if the diffusion constant of the GWN goes to 0. Some advantages of this procedure are discussed.
"On some Filtration Procedure for Jump Markov Process Observed in White Gaussian Noise." Ann. Statist. 20 (4) 2153 - 2160, December, 1992. https://doi.org/10.1214/aos/1176348909