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# Theory of Probability and Mathematical Statistics

## Approximation of a Wiener process by integrals with respect to the fractional Brownian motion of power functions of a given exponent

### O. L. Banna, Yu. S. Mishura, S. V. Shklyar

Abstract: The best uniform approximation of a Wiener process by integrals of the form ∫_{0}^{t}f(s)dB_{s}^{H} is established in the space L∞([0,T];L_2(Ω)), where {B_{t}^{H}, t\in[0,T]} is the fractional Brownian motion with the Hurst index H and f(s)=k•s^{α}, s\in[0,T], for k>0 and \$ α=H-1/2.

Keywords: Wiener process, fractional Brownian motion, integral with respect to the fractional Brownian motion, an approximation in a class of functions

Bibliography:
1. T. O. Androshchuk, Approximation of a stochastic integral with respect to fractional Brownian motion by integrals with respect to absolutely continuous processes, Teor. Imovirnost. Matem. Statist. 73 (2005), 17-26; English transl. in Theor. Probability and Math. Statist. 73 (2006), 19-29.
2. O. L. Banna and Yu. S. Mishura, The simplest martingales for the approximation of the fractional Brownian motion, Visnyk Kyiv National Taras Shevchenko University. Mathematics and Mechanics 19 (2008), 38-43. (Ukrainian)
3. 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. Imovirnost. Matem. Statist. 83 (2010), 12-21; English transl. in Theor. Probability and Math. Statist. 83 (2011), 13-25.
4. V. V. Doroshenko, Yu. S. Mishura, and O. L. Banna, The distance between fractional Brownian motion and the subspace of martingales with similar'' kernels, Teor. Imovirnost. Matem. Statist. 87 (2012), 38-45; English transl. in Theor. Probability and Math. Statist. 87 (2013), 41-49.
5. Yu. S. Mishura, O. L. Banna, and V. V. Doroshenko, The distance between the fractional Brownian motion and the subspace of Gaussian martingales, Visnyk Kyiv National Taras Shevchenko University. Mathematics and Mechanics 1 (2013), 53-60. (Ukrainian)
6. Yu. S. Mishura and O. L. Banna, Approximation of fractional Brownian motion by Wiener integrals, Teor. Imovirnost. Matem. Statist. 79 (2008), 106-115; English transl. in Theor. Probability and Math. Statist. 79 (2009), 107-116.
7. T. Androshchuk and Yu. S. Mishura, Mixed Brownian-fractional Brownian model: absence of arbitrage and related topics, Stochastics 78 (2006), 281-300.
8. O. Banna and Y. 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.
9. A. Le Breton, Filtering and parameter estimation in a simple linear system driven by a fractional Brownian motion, Stat. Probab. Lett. 38 (1998), 263-274.
10. Yu. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Math., vol. 1929, Springer, Berlin, 2008.
11. I. Norros, E. Valkeila, and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 5(4) (1999), 571-587.
12. S. Shklyar, G. Shevchenko, Yu. Mishura, V. Doroshenko, and O. Banna, Approximation of fractional Brownian motion by martingales, Methodol. Comput. Appl. Probab. 16 (2014), no. 3, 539-560.
13. T. H. Thao, A note on fractional Brownian motion, Vietnam J. Math. 31 (2003), no. 3, 255-260.