Optimal filtration for systems with fractional Brownian noises
Author:
S. V. Posashkov
Translated by:
V. Semenov
Journal:
Theor. Probability and Math. Statist. 72 (2006), 135-144
MSC (2000):
Primary 60G35; Secondary 60G15, 60H05, 60J65
DOI:
https://doi.org/10.1090/S0094-9000-06-00671-5
Published electronically:
September 5, 2006
MathSciNet review:
2168143
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Abstract: We consider the problem of the optimal filtration for systems with the noise being a multivariate fractional Brownian motion. We partially solve the problem of the optimal filtration for nonlinear systems. The system of equations for the optimal filtration is obtained in the case of linear systems.
References
- R. Sh. Liptser and A. N. Shiryaev, Statistika sluchaĭ nykh protsessov, Izdat. “Nauka”, Moscow, 1974 (Russian). Nelineĭ naya filtratsiya i smezhnye voprosy. [Nonlinear filtering and related problems]; Probability Theory and Mathematical Statistics, Vol. 15. MR 0431365
- Gopinath Kallianpur, Stochastic filtering theory, Applications of Mathematics, vol. 13, Springer-Verlag, New York-Berlin, 1980. MR 583435
- M. L. Kleptsyna, A. Le Breton, and M. C. Roubaud, An elementary approach to filtering in systems with fractional Brownian observation noise, Probability Theory and Mathematical Statistics, Proceeding of the 7th Vilnius Conference, VSP/TEV, Utrecht/Vilnius, 2000, 373–392.
- M. L. Kleptsyna, A. Le Breton, and M.-C. Roubaud, General approach to filtering with fractional Brownian noises—application to linear systems, Stochastics Stochastics Rep. 71 (2000), no. 1-2, 119–140. MR 1813509
- Ilkka Norros, Esko Valkeila, and Jorma Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 5 (1999), no. 4, 571–587. MR 1704556, DOI https://doi.org/10.2307/3318691
References
- R. Sh. Liptser and A. N. Shiryaev, Statistics of Random Processes, “Nauka”, Moscow, 1974; English transl., Springer-Verlag, Berlin, 1977. MR 0431365 (55:4365)
- G. Kallianpur, Stochastic Filtering Theory, Springer-Verlag, Berlin, 1980. MR 0583435 (82f:60089)
- M. L. Kleptsyna, A. Le Breton, and M. C. Roubaud, An elementary approach to filtering in systems with fractional Brownian observation noise, Probability Theory and Mathematical Statistics, Proceeding of the 7th Vilnius Conference, VSP/TEV, Utrecht/Vilnius, 2000, 373–392.
- ---, General approach to filtering with fractional Brownian noises—application to linear systems, Stoch. Stoch. Rep. 71 (2000), 119–140. MR 1813509 (2001k:93131)
- I. Norros, E. Valkeila, and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 5(4) (1999), 571–587. MR 1704556 (2000f:60053)
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Additional Information
S. V. Posashkov
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Keywords:
Problem of filtration,
fractional Brownian motion
Received by editor(s):
July 2, 2004
Published electronically:
September 5, 2006
Article copyright:
© Copyright 2006
American Mathematical Society
