The local asymptotic normality of a family of measures generated by solutions of stochastic differential equations with a small fractional Brownian motion

Author:
T. Androshchuk

Translated by:
V. V. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **71** (2004).

Journal:
Theor. Probability and Math. Statist. **71** (2005), 1-15

MSC (2000):
Primary 62F12; Secondary 60G15, 60H10

DOI:
https://doi.org/10.1090/S0094-9000-05-00643-5

Published electronically:
December 30, 2005

MathSciNet review:
2144316

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A formula for the likelihood ratio of measures generated by solutions of a stochastic differential equation with a fractional Brownian motion is established in the paper. We find sufficient conditions that the family of measures generated by solutions of such an equation is locally asymptotically normal.

**1.**Yu. Kutoyants,*Identification of dynamical systems with small noise*, Mathematics and its Applications, vol. 300, Kluwer Academic Publishers Group, Dordrecht, 1994. MR**1332492****2.**I. A. Ibragimov and R. Z. Khas'minski,*Statistical Estimation. Asymptotic Theory*, ``Nauka'', Moscow, 1979; English transl., Springer-Verlag, New York-Berlin, 1981.MR**0620321 (82g:62006)****3.**M. Zähle,*Integration with respect to fractal functions and stochastic calculus. I*, Probab. Theory Related Fields**111**(1998), no. 3, 333–374. MR**1640795**, https://doi.org/10.1007/s004400050171**4.**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**, https://doi.org/10.2307/3318691**5.**David Nualart and Aurel Răşcanu,*Differential equations driven by fractional Brownian motion*, Collect. Math.**53**(2002), no. 1, 55–81. MR**1893308****6.**Yu. V. Krvavich and Yu. S. Mīshura,*Differentiability of fractional integrals whose kernels are defined by fractal Brownian motion*, Ukraïn. Mat. Zh.**53**(2001), no. 1, 30–40 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J.**53**(2001), no. 1, 35–47. MR**1834637**, https://doi.org/10.1023/A:1010432716012**7.**R. Sh. Liptser and A. N. Shiryaev,*\cyr Statistika sluchaĭnykh protsessov.*, Izdat. “Nauka”, Moscow, 1974 (Russian). \cyr Nelineĭnaya filtratsiya i smezhnye voprosy. [Nonlinear filtering and related problems]; Probability Theory and Mathematical Statistics, Vol. 15. MR**0431365**

Robert S. Liptser and Albert N. Shiryaev,*Statistics of random processes. I*, Second, revised and expanded edition, Applications of Mathematics (New York), vol. 5, Springer-Verlag, Berlin, 2001. General theory; Translated from the 1974 Russian original by A. B. Aries; Stochastic Modelling and Applied Probability. MR**1800857****8.**T. O. Androshchuk,*An estimate for higher moments of the deviation between a solution of a stochastic differential equation and its trend*, Visnyk Kyiv. Univ. Ser. Matem. Mech. (2004), no. 12, 60-62. (Ukrainian)

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2000):
62F12,
60G15,
60H10

Retrieve articles in all journals with MSC (2000): 62F12, 60G15, 60H10

Additional Information

**T. Androshchuk**

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

Email:
nutaras@univ.kiev.ua

DOI:
https://doi.org/10.1090/S0094-9000-05-00643-5

Keywords:
Fractional brownian motion,
local asymptotic normality of a system of measures,
dynamic systems with small noise

Received by editor(s):
March 12, 2004

Published electronically:
December 30, 2005

Article copyright:
© Copyright 2005
American Mathematical Society