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

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

Published electronically:
August 6, 2015

MathSciNet review:
3241857

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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. Imovirnost. Matem. Statist.**73**(2005), 17-26; English transl. in Theor. Probability and Math. Statist.**73**(2006), 19-29. MR**2213333 (2006m:60072)****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. MR**2768845 (2012c:60108)****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. MR**3241445****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. MR**2494540 (2010b:60113)****7.**T. Androshchuk and Yu. S. Mishura,*Mixed Brownian-fractional Brownian model: absence of arbitrage and related topics*, Stochastics**78**(2006), 281-300. MR**2270939 (2007k:60198)****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. MR**2498600 (2010d:60099)****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. MR**1629915 (99c:60088)****10.**Yu. Mishura,*Stochastic calculus for fractional Brownian motion and related processes*, Lecture Notes in Math., vol. 1929, Springer, Berlin, 2008. MR**2378138 (2008m:60064)****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. MR**1704556 (2000f:60053)****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. MR**3239808****13.**T. H. Thao,*A note on fractional Brownian motion*, Vietnam J. Math.**31**(2003), no. 3, 255-260. MR**2010525 (2004j:60081)**

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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