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Banna, O. L.; Mishura, Yu. S.

A bound for the distance between fractional Brownian motion and the space of Gaussian martingales on an interval. (English. Russian original) Zbl 1243.60034

Theory Probab. Math. Stat. 83, 13-25 (2011); translation from Teor. Jmovirn. Mat. Stat. 83, 12-21 (2010).
Summary: We obtain a lower bound for the distance between fractional Brownian motion and the space of Gaussian martingales on an interval. The distances between fractional Brownian motion and some subspaces of Gaussian martingales are compared. The upper and lower bounds are obtained for the constant in the representation of a fractional Brownian motion in terms of the Wiener process.

Cited in 3 Documents

MSC:

60G15 Gaussian processes
60G44 Martingales with continuous parameter
60G22 Fractional processes, including fractional Brownian motion
60E15 Inequalities; stochastic orderings

Keywords:

Wiener process; fractional Brownian motion; Gaussian martingale; approximation in a class of functions
PDFBibTeX XMLCite
\textit{O. L. Banna} and \textit{Yu. S. Mishura}, Theory Probab. Math. Stat. 83, 13--25 (2011; Zbl 1243.60034); translation from Teor. Jmovirn. Mat. Stat. 83, 12--21 (2010)
Full Text: DOI

References:

[1] T. O. Androshchuk, Approximation of a stochastic integral with respect to fractional Brownian motion by integrals with respect to absolutely continuous processes, Teor. Ĭmovīr. Mat. Stat. 73 (2005), 17 – 26 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 73 (2006), 19 – 29. · Zbl 1118.60049
[2] O. L. Banna, An approximation of the fractional Brownian motion whose Hurst index is near the unity by stochastic integrals with linear-power integrands, Applied Statistics. Actuarial and Finance Mathematics 1 (2007), 60-67. (Ukrainian) · Zbl 1164.60412
[3] 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) · Zbl 1199.60132
[4] 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. · Zbl 1224.60079
[5] Taras Androshchuk and Yuliya Mishura, Mixed Brownian – fractional Brownian model: absence of arbitrage and related topics, Stochastics 78 (2006), no. 5, 281 – 300. · Zbl 1115.60043 · doi:10.1080/17442500600859317
[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. · Zbl 1224.60074
[7] 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. · Zbl 0955.60034 · doi:10.2307/3318691
[8] Tran Hung Thao, A note on fractional Brownian motion, Vietnam J. Math. 31 (2003), no. 3, 255 – 260. · Zbl 1052.60030
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