Geodesic
On the coincidence of $\sigma$-algebras in the filtration problem for diffusion processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 4, pp. 771-780

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Multidimensional diffusion process (1) with two components is considered. The main theorem states that two $\sigma$-algebras (one generated by the observable process $y_t$ and another generated by its innovation $\overline w_t$ and initial condition $y_0$) coincide. 
@article{TVP_1979_24_4_a6,
     author = {N. V. Krylov},
     title = {On the coincidence of $\sigma$-algebras in the filtration problem for diffusion processes},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {771--780},
     publisher = {mathdoc},
     volume = {24},
     number = {4},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_4_a6/}
}
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N. V. Krylov. On the coincidence of $\sigma$-algebras in the filtration problem for diffusion processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 4, pp. 771-780. http://geodesic.mathdoc.fr/item/TVP_1979_24_4_a6/