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
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 Free Access
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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.
References
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References
- Yu. A. Kutoyants, Identification of Dynamical Systems with Small Noise, Mathematics and Its Applications, vol. 300, Kluwer, Dordrecht, 1994. MR 1332492 (97b:93093)
- I. A. Ibragimov and R. Z. Khas’minskiĭ, Statistical Estimation. Asymptotic Theory, “Nauka”, Moscow, 1979; English transl., Springer-Verlag, New York–Berlin, 1981. MR 0620321 (82g:62006)
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- I. Norros, E. Valkeila, and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on \fBms, Bernoulli 55 (1999), 571–587. MR 1704556 (2000f:60053)
- D. Nualart and A. Rǎscanu, Differential equations driven by \fBm, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308 (2003f:60105)
- Yu. V. Krvavich and Yu. S. Mishura, Differentiability of fractional integrals whose kernels are defined by fractal Brownian motion, Ukrain. Mat. Zh. 53 (2001), no. 1, 30–40; English transl. in Ukrainian Math. J. 53 (2001), no. 1, 35–47. MR 1834637 (2002d:60046)
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- 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)
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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
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
