Asymptotic properties of an estimator for the drift coefficient of a stochastic differential equation with fractional Brownian motion
Authors:
E. I. Kasyts’ka and P. S. Knopov
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 79 (2009), 73-81
MSC (2000):
Primary 60H10; Secondary 62M05
DOI:
https://doi.org/10.1090/S0094-9000-09-00781-9
Published electronically:
December 29, 2009
MathSciNet review:
2494536
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: A stochastic differential equation with respect to fractional Brownian motion is considered. We study the maximum likelihood estimator for the drift coefficient. We assume that the coefficient belongs to a given compact set of functions and prove the strong consistency of the estimator and its asymptotic normality.
References
- A. Ya. Dorogovtsev, Teoriya otsenok parametrov sluchaĭ nykh protsessov, “Vishcha Shkola”, Kiev, 1982 (Russian). MR 668517
- M. Zähle, Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Related Fields 111 (1998), no. 3, 333–374. MR 1640795, DOI https://doi.org/10.1007/s004400050171
- Jean Mémin, Yulia Mishura, and Esko Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett. 51 (2001), no. 2, 197–206. MR 1822771, DOI https://doi.org/10.1016/S0167-7152%2800%2900157-7
- Yu. V. Krvavich and Yu. S. Mīshura, Some maximal inequalities for moments of Wiener integrals with respect to fractional Brownian motion, Teor. Ĭmovīr. Mat. Stat. 61 (1999), 72–83 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 61 (2000), 75–86 (2001). MR 1866968
- J. Pfanzagl, On the measurability and consistency of minimum contrast estimates, Metrika 14 (1969), 249–272.
- A. N. Kolmogorov and S. V. Fomin, Èlementy teorii funktsiĭ i funktsional′nogo analiza, Izdat. “Nauka”, Moscow, 1968 (Russian). Second edition, revised and augmented. MR 0234241
References
- A. Ya. Dorogovtsev, The Theory of Estimates of the Parameters of Random Processes, Vyshcha Shkola, Kiev, 1982. (Russian) MR 668517 (84h:62122)
- M. Zähle, Integration with respect to fractal functions and stochastic calculus, Probab. Theory Relat. Fields III (1998), 333–374. MR 1640795 (99j:60073)
- J. Mémin, Yu. Mishura, and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett. 51 (2001), 197–206. MR 1822771 (2002b:60096)
- Yu. V. Krvavych and Yu. S. Mishura, Some maximal inequalities for moments of Wiener integrals with respect to fractional Brownian motion, Teor. Imovir. Mat. Stat. 61 (1999), 72–83; English transl. in Theory Probab. Math. Statist. 61 (2000), 75–86. MR 1866968 (2002h:60072)
- J. Pfanzagl, On the measurability and consistency of minimum contrast estimates, Metrika 14 (1969), 249–272.
- A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, 1968; English transl. from the first (1960) Russian ed., Graylock Press, Albany, N.Y., 1961. MR 0234241 (38:2559)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
60H10,
62M05
Retrieve articles in all journals
with MSC (2000):
60H10,
62M05
Additional Information
E. I. Kasyts’ka
Affiliation:
Glushkov Institute for Cybernetics, National Academy of Sciences of Ukraine, Academician Glushkov Avenue, 03187 Kyiv, Ukraine
P. S. Knopov
Affiliation:
Glushkov Institute for Cybernetics, National Academy of Sciences of Ukraine, Academician Glushkov Avenue, 03187 Kyiv, Ukraine
Email:
knopov1@yahoo.com
Keywords:
Fractional Wiener process,
stochastic integral,
stochastic differential equation,
drift coefficient
Received by editor(s):
July 7, 2008
Published electronically:
December 29, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
