Geodesic
Filtering, smoothing and prediction of degenerate diffusion processes. Backward equations
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 4, pp. 725-737

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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 $X$. 
@article{TVP_1983_28_4_a9,
     author = {B. L. Rozovskiǐ},
     title = {Filtering, smoothing and prediction of degenerate diffusion processes. {Backward} equations},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {725--737},
     publisher = {mathdoc},
     volume = {28},
     number = {4},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_4_a9/}
}
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B. L. Rozovskiǐ. Filtering, smoothing and prediction of degenerate diffusion processes. Backward equations. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 4, pp. 725-737. http://geodesic.mathdoc.fr/item/TVP_1983_28_4_a9/