Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
È.M. Asadullin and F. S. Nasyrov, “On the Solution of a Nonlinear Filtration Problem for One-Dimensional Diffusion Processes,” Vestnik UGATU 12 (1) (30), 161 (2009).
È. M. Asadullin and F. S. Nasyrov, “About filtering problem of diffusion processes,” Ufa Math. J. 3 (2), 3 (2011).
R. E. Kalman and R. S. Bucy, “New results in linear filtering and prediction theory,” Trans. ASME Ser. D., J. Basic Engineering 83, 95 (1961).
A. N. Kolmogorov, “Interpolation and extrapolation of stationary random sequences,” Bull. Acad. Sci.URSS Ser. Math. [Izvestia Akad. Nauk. SSSR] 5 (1), 3 (1941).
R. S. Liptser and A. N. Shiryayev, Statistics of Random Processes (Nauka, Moscow, 1974; Springer-Verlag, New York–Heidelberg, Vol. I: 1977; Vol. II: 1978).
F. S. Nasyrov, Local Times, Symmetric Integrals, and Stochastic Analysis (Fizmatlit, Moscow, 2011) [in Russian].
F. S. Nasyrov, “On integration of systems of stochastic differential equations,” Mat. Tr. 19 (2), 158 (2016) [Sib. Adv. Math. 27, 187 (2017)].
B. Oksendal, Stochastic Differential Equations. An Introduction with Applications (Springer, Berlin, 2003; Mir;Moscow, 2003).
B. L. Rozovskiĭ, Stochastic Evolution Systems (Nauka, Moscow, 1983; Kluwer, Dordrecht, 1990).
A. N. Shiryaev, Probability. Vol. 1. (MTsNMO,Moscow, 2004; Springer, New York, 2016).
N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications (JohnWiley & Sons, New York, 1949).
M. Zakai, “On the optimal filtering of diffusion processes,” Z.Wahrsch. Verw. Gebiete 11, 230 (1969).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © G.R. Kagirova and F.S. Nasyrov, 2017, published in Matematicheskie Trudy, 2017, Vol. 20, No. 2, pp. 35–51.
About this article
Cite this article
Kagirova, G.R., Nasyrov, F.S. On an Optimal Filtration Problem for One-Dimensional Diffusion Processes. Sib. Adv. Math. 28, 155–165 (2018). https://doi.org/10.3103/S105513441803001X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S105513441803001X