Extrapolation of multidimensional Markov processes from incomplete data

Let $(\theta_t,\eta_t)$, $t\ge0$, be a Markov process, where $\eta_t$ is the observable component and $\theta_t$ is the unobservable one. Put
$$ \pi_\beta(\tau,t)=\mathbf P(\theta_\tau=\beta\mid\eta_s,\ s\le t),\quad\tau\ge t, $$
if $\theta_t$ takes discrete values and
$$ \pi_\beta(\tau,t)=\frac{\partial\mathbf P(\theta_t\le\beta\mid\eta_s,\ s\le t)}{\partial\beta},\quad\tau\ge t, $$
if $\theta_\tau$ takes continuous values. When $\theta_t$, $t\ge0$, is a purely discontinuous Markov process and $\eta_t$ has the stochastic differential (5), in § 1 equations in $t$ and $\tau$ for $\pi_\beta(\tau,t)$ are deduced. In § 2 equations for the density $\pi_\beta(\tau,t)$ are obtained under the supposition that $(\theta_t,\eta_t)$ be a diffusion Markov process. In § 3 some cases of effective solving of extrapolation problems for processes regarded in § 2 are considered.