Approximation of fractional Brownian motion by Wiener integrals

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

Translated by:
Oleg Klesov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **79** (2008).

Journal:
Theor. Probability and Math. Statist. **79** (2009), 107-116

MSC (2000):
Primary 60G15; Secondary 60G44

Published electronically:
December 28, 2009

MathSciNet review:
2494540

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We find an approximation in the space of a fractional Brownian motion by martingales of the form , where is a Wiener process, is a power function with a negative index, that is where , , and is the index of fractional Brownian motion.

**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 best approximation to the*, Visnyk Kyiv Shevchenko Univ., ser. matem. mekh. (2008), 38-43. (Ukrainian)**3.**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**4.**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**5.**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

**Yu. S. Mishura**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Email:
myus@univ.kiev.ua

**O. L. Banna**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Email:
bannaya@mail.univ.kiev.ua

DOI:
http://dx.doi.org/10.1090/S0094-9000-09-00773-X

Keywords:
Wiener integral,
fractional Brownian motion

Received by editor(s):
September 17, 2007

Published electronically:
December 28, 2009

Article copyright:
© Copyright 2009
American Mathematical Society