Filtering, smoothing and prediction of degenerate diffusion processes. Backward equations

Let $Z(t)=(X(t),Y(t))$ be a multidimensional degenerate Markov diffusion process and $f$ be a real function such that $\mathbf Mf^2(X(t))<\infty$. Equations for
$$ \mathbf M_{z,3}[f(X(t))\mid Y(r),\,r\in[s,\tau]] $$
with respect to $z$, $s$($t$, $x$ are fixed) are presented. We apply these equations to the problem of backward smoothing of $X$.