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

Authors:
O. L. Banna, Yu. S. Mishura and S. V. Shklyar

Translated by:
N. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **90** (2014).

Journal:
Theor. Probability and Math. Statist. **90** (2015), 13-22

MSC (2010):
Primary 60G15; Secondary 60G44

Published electronically:
August 6, 2015

MathSciNet review:
3241857

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The best uniform approximation of a Wiener process by integrals of the form

**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. MR**2213333**, 10.1090/S0094-9000-07-00678-3**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. 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**, 10.1090/S0094-9000-2012-00838-7**4.**V. Doroshenko, Yu. Mishura, and O. Banna,*The distance between fractional Brownian motion and the subspace of martingales with “similar” kernels*, Theory Probab. Math. Statist.**87**(2013), 41–49. MR**3241445**, 10.1090/S0094-9000-2014-00903-5**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. 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**, 10.1090/S0094-9000-09-00773-X**7.**Taras Androshchuk and Yuliya Mishura,*Mixed Brownian–fractional Brownian model: absence of arbitrage and related topics*, Stochastics**78**(2006), no. 5, 281–300. MR**2270939**, 10.1080/17442500600859317**8.**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****9.**Alain Le Breton,*Filtering and parameter estimation in a simple linear system driven by a fractional Brownian motion*, Statist. Probab. Lett.**38**(1998), no. 3, 263–274. MR**1629915**, 10.1016/S0167-7152(98)00029-7**10.**Yuliya S. Mishura,*Stochastic calculus for fractional Brownian motion and related processes*, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. MR**2378138****11.**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**, 10.2307/3318691**12.**Sergiy Shklyar, Georgiy Shevchenko, Yuliya Mishura, Vadym Doroshenko, and Oksana Banna,*Approximation of fractional Brownian motion by martingales*, Methodol. Comput. Appl. Probab.**16**(2014), no. 3, 539–560. MR**3239808**, 10.1007/s11009-012-9313-8**13.**Tran Hung Thao,*A note on fractional Brownian motion*, Vietnam J. Math.**31**(2003), no. 3, 255–260. MR**2010525**

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Additional Information

**O. L. Banna**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine

Email:
bannaya@mail.univ.kiev.ua

**Yu. S. Mishura**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine

Email:
myus@univ.kiev.ua

**S. V. Shklyar**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine

Email:
shklyar@univ.kiev.ua

DOI:
https://doi.org/10.1090/tpms/946

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

Received by editor(s):
March 16, 2014

Published electronically:
August 6, 2015

Article copyright:
© Copyright 2015
American Mathematical Society