Geodesic
On an optimal filtration problem for one-dimensional diffusion processes
Matematičeskie trudy, Tome 20 (2017) no. 2, pp. 35-51.

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We find a method that reduces the solution of a problem of nonlinear filtration of one-dimensional diffusion processes to the solution of a linear parabolic equation with constant diffusion coefficients whose remaining coefficients are random and depend on the trajectory of the observable process. The method consists in reducing the initial filtration problem to a simpler problem with identity diffusion matrix and subsequently reducing the solution of the parabolic Itô equation for the filtered density to solving the above-mentioned parabolic equation. In addition, the filtered densities of both problems are connected by a sufficiently simple formula. 
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G. R. Kagirova; F. S. Nasyrov. On an optimal filtration problem for one-dimensional diffusion processes. Matematičeskie trudy, Tome 20 (2017) no. 2, pp. 35-51. http://geodesic.mathdoc.fr/item/MT_2017_20_2_a1/

[1] Asadullin E. M., Nasyrov F. S., “O reshenii zadachi nelineinoi filtratsii odnomernykh diffuzionnykh protsessov”, Vestnik UGATU, 12:1(30) (2009), 161–165

[2] Asadullin E. M., Nasyrov F. S., “O zadache filtratsii diffuzionnykh protsessov”, Ufimsk. matem. zhurn., 3:2 (2011), 3–9

[3] Kolmogorov A. N., “Interpolirovanie i ekstrapolirovanie statsionarnykh sluchainykh posledovatelnostei”, Izv. AN SSSR. Ser. matem., 5:1 (1941), 3–14

[4] Liptser R. Sh., Shiryaev A. N., Statistika sluchainykh protsessov, Nauka, M., 1974

[5] Nasyrov F. S., Lokalnye vremena, simmetrichnye integraly i stokhasticheskii analiz, Fizmatlit, M., 2011

[6] Nasyrov F. S., “Ob integrirovanii sistem stokhasticheskikh differentsialnykh uravnenii”, Matem. tr., 19:2 (2016), 158–169

[7] Oksendal B., Stokhasticheskie differentsialnye uravneniya. Vvedenie v teoriyu i prilozheniya, Mir, M., 2003

[8] Rozovskii B. L., Evolyutsionnye stokhasticheskie sistemy, Nauka, M., 1983

[9] Shiryaev A. N., Veroyatnost, v. 1, MTsNMO, M., 2004

[10] Kalman R. E., Bucy R. S., “New results in linear filtering and prediction theory”, Trans. ASME Ser. D., J. Basic Engineering, 83 (1961), 95–108 | DOI | MR

[11] Wiener N., Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications, John Wiley Sons, New York, 1949 | MR | Zbl

[12] Zakai M., “On the optimal filtering of diffusion processes”, Z. Wahrsch. Verw. Gebiete, 11 (1969), 230–243 | DOI | MR | Zbl