Sequential filtering of components of a Markov chain in the case of singular diffusion matrix

Let $(\theta_n,\xi_n)$, $n=0,\Delta,\dots$ $(\Delta>0)$ be a $k+l$-dimensional Markov chain satisfying 1) where $\xi_n$ is the observable component and $\theta_n$ is the unobservable one. In this paper, we obtain the recurrent relations (2) for the conditional expectations and covariance matrix which define the optimal mean square estimates and errors. The results remain valid also in the case when the diffusion matrix is singular.