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

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **87** (2012).

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

Abstract | References | Similar Articles | 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.

**1.**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**2.**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)**3.**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)**4.**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)**5.**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**, https://doi.org/10.1090/S0094-9000-09-00773-X**6.**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****7.**O. L. Banna and Yu. S. Mīshura,*A bound for the distance between fractional Brownian motion and the space of Gaussian martingales on an interval*, Teor. Ĭmovīr. Mat. Stat.**83**(2010), 12–21 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist.**83**(2011), 13–25. MR**2768845**, https://doi.org/10.1090/S0094-9000-2012-00838-7**8.**N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel'kov,*Numerical Methods*, fourth edition, ``Binom'', Moscow, 2006. (Russian)

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2010):
60G15,
60G22,
60G44

Retrieve articles in all journals with MSC (2010): 60G15, 60G22, 60G44

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

DOI:
https://doi.org/10.1090/S0094-9000-2014-00903-5

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