The distance between fractional Brownian motion and the subspace of martingales with “similar” kernels
Authors:
V. Doroshenko, Yu. Mishura and O. Banna
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 87 (2013), 41-49
MSC (2010):
Primary 60G15, 60G22; Secondary 60G44
DOI:
https://doi.org/10.1090/S0094-9000-2014-00903-5
Published electronically:
March 21, 2014
MathSciNet review:
3241445
Full-text PDF Free Access
Abstract |
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Additional Information
Abstract: We study the problem of approximation of a fractional Brownian motion with the help of Gaussian martingales that can be represented as the integrals with respect to a Wiener process and with nonrandom integrands being “similar” to the kernel of the fractional Brownian motion. The “similarity” is understood in the sense that an integrand is the value of the kernel at some point. We establish analytically and evaluate numerically the upper and lower bounds for the distance between the fractional Brownian motion and the space of Gaussian martingales.
References
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- S. Shklyar, G. Shevchenko, Yu. Mishura, V. Doroshenko, and O. Banna, The approximation of fractional Brownian motion by martingales, Methodology and Computing in Applied Probability. (to appear)
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- Yu. S. Mishura and O. L. Banna, The simplest martingales for the best approximation of a fractional Brownian motion, Visnyk Kyiv Univ., ser. fiz. mat. nauk (2008), no. 19, 38–43. (Ukrainian)
- Yu. S. Mīshura and O. L. Banna, Approximation of fractional Brownian motion by Wiener integrals, Teor. Ĭmovīr. Mat. Stat. 79 (2008), 96–104 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 79 (2009), 107–116. MR 2494540, DOI https://doi.org/10.1090/S0094-9000-09-00773-X
- Oksana Banna and Yuliya Mishura, Approximation of fractional Brownian motion with associated Hurst index separated from 1 by stochastic integrals of linear power functions, Theory Stoch. Process. 14 (2008), no. 3-4, 1–16. MR 2498600
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- N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods, fourth edition, “Binom”, Moscow, 2006. (Russian)
References
- I. Norros, E. Valkeila, and J. 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 (2000f:60053)
- S. Shklyar, G. Shevchenko, Yu. Mishura, V. Doroshenko, and O. Banna, The approximation of fractional Brownian motion by martingales, Methodology and Computing in Applied Probability. (to appear)
- O. L. Banna, An approximation of a fractional Brownian motion whose Hurst index is close to unity by stochastic integrals with linearly-exponential integrands, Applied Statistics, Actuarial and Financial Mathematics (2007), no. 1, 60–67. (Ukrainian)
- Yu. S. Mishura and O. L. Banna, The simplest martingales for the best approximation of a fractional Brownian motion, Visnyk Kyiv Univ., ser. fiz. mat. nauk (2008), no. 19, 38–43. (Ukrainian)
- Yu. S. Mishura and O. L. Banna, Approximation of fractional Brownian motion by Wiener integrals, Teor. Imovir. Mat. Stat. 79 (2008), 96–104; English transl. in Theory Probab. Math. Statist. 79 (2009), 107–116. MR 2494540 (2010b:60113)
- O. L. Banna and Yu. S. Mishura, Approximation of fractional Brownian motion with associated Hurst index separated from 1 by stochastic integrals of linear power functions, Theory Stoch. Process. 14 (30) (2008), no. 3–4, 1–16. MR 2498600 (2010d:60099)
- O. L. Banna and Yu. S. Mishura, A bound for the distance between fractional Brownian motion and the space of Gaussian martingales on an interval, Teor. Imovir. Mat. Stat. 83 (2010), 12–21; English transl. in Theory Probab. Math. Statist. 83 (2011), 13–25. MR 2768845 (2012c:60108)
- N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods, fourth edition, “Binom”, Moscow, 2006. (Russian)
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Additional Information
V. Doroshenko
Affiliation:
Faculty of Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Volodymyrska 64, 01601 Kyiv, Ukraine
Email:
vadym.doroshenko@gmail.com
Yu. Mishura
Affiliation:
Faculty of Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Volodymyrska 64, 01601 Kyiv, Ukraine
Email:
yumishura@gmail.com
O. Banna
Affiliation:
Faculty of Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Volodymyrska 64, 01601 Kyiv, Ukraine
Keywords:
Wiener process,
fractional Brownian process,
Gaussian martingale,
approximation of fractional Brownian motion by Gaussian martingales
Received by editor(s):
June 1, 2012
Published electronically:
March 21, 2014
Article copyright:
© Copyright 2014
American Mathematical Society
