A bound for the distance between fractional Brownian motion and the space of Gaussian martingales on an interval

Authors:
O. L. Banna and Yu. S. Mishura

Translated by:
N. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **83** (2010).

Journal:
Theor. Probability and Math. Statist. **83** (2011), 13-25

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

Published electronically:
February 2, 2012

MathSciNet review:
2768845

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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.

**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,*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)**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)**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. MR**2494540**, 10.1090/S0094-9000-09-00773-X**5.**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**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.**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**8.**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, Academician Glushkov Avenue 2, Kiev 03127, 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, Academician Glushkov Avenue 2, Kiev 03127, Ukraine

Email:
myus@univ.kiev.ua

DOI:
http://dx.doi.org/10.1090/S0094-9000-2012-00838-7

Keywords:
Wiener process,
fractional Brownian motion,
Gaussian martingale,
approximation in a class of functions

Received by editor(s):
April 7, 2010

Published electronically:
February 2, 2012

Article copyright:
© Copyright 2012
American Mathematical Society