Optimal filtration in systems with noise modeled by a polynomial of fractional Brownian motion

Authors:
Yu. S. Mishura and S. V. Posashkov

Translated by:
Oleg Klesov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **73** (2005).

Journal:
Theor. Probability and Math. Statist. **73** (2006), 117-124

MSC (2000):
Primary 60G35; Secondary 60G15, 60H05, 60G65

Published electronically:
January 17, 2007

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of optimal filtration in systems with noise modeled by a polynomial of fractional Brownian motion is partially solved by representing fractional Brownian motion in terms of standard Brownian motion.

**1.**R. S. Liptser and A. N. Shiryayev,*Statistics of random processes. I*, Springer-Verlag, New York-Heidelberg, 1977. General theory; Translated by A. B. Aries; Applications of Mathematics, Vol. 5. MR**0474486****2.**Gopinath Kallianpur,*Stochastic filtering theory*, Applications of Mathematics, vol. 13, Springer-Verlag, New York-Berlin, 1980. MR**583435****3.**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, pp. 373-392.**4.**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****5.**S. V. Posashkov,*Optimal filtering in systems with fractional Brownian noise*, Teor. Ĭmovīr. Mat. Stat.**72**(2005), 120–128 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**72**(2006), 135–144. MR**2168143**, 10.1090/S0094-9000-06-00671-5**6.**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**, 10.2307/3318691

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Additional Information

**Yu. S. Mishura**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Email:
myus@univ.kiev.ua

**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

Email:
corlagon@mail.univ.kiev.au

DOI:
https://doi.org/10.1090/S0094-9000-07-00686-2

Keywords:
Problem of filtration,
fractional Brownian motion

Received by editor(s):
October 4, 2004

Published electronically:
January 17, 2007

Additional Notes:
The first author is supported in part by the grant NATO PST.CLG 890408

Article copyright:
© Copyright 2007
American Mathematical Society