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
Journal:
Theor. Probability and Math. Statist. 83 (2011), 13-25
MSC (2010):
Primary 60G15; Secondary 60G44
DOI:
https://doi.org/10.1090/S0094-9000-2012-00838-7
Published electronically:
February 2, 2012
MathSciNet review:
2768845
Full-text PDF Free Access
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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.
References
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- 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)
- 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)
- 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, DOI https://doi.org/10.1090/S0094-9000-09-00773-X
- 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, DOI https://doi.org/10.1080/17442500600859317
- 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
- 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, DOI https://doi.org/10.2307/3318691
- Tran Hung Thao, A note on fractional Brownian motion, Vietnam J. Math. 31 (2003), no. 3, 255–260. MR 2010525
References
- T. O. Androshchuk, Approximation of a stochastic integral with respect to fractional Brownian motion by integrals with respect to absolutely continuous processes, Teor. Imovir. Mat. Stat. 73 (2005), 17–26; English transl. in Theory Probab. Math. Statist. 73 (2006), 19–29. MR 2213333 (2006m:60072)
- 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)
- 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)
- Yu. S. Mishura and O. L. Banna, Approximation of fractional Brownian motion by Wiener integrals, Teor. Imovir. Mat. Stat. 79 (2008), 96–104; English transl. in Theory Probab. Math. Statist. 79 (2009), 107–116. MR 2494540 (2010b:60113)
- T. Androshchuk and Y. S. Mishura, Mixed Brownian–fractional Brownian model: absence of arbitrage and related topics, Stochastics 78 (2006), 281–300. MR 2270939 (2007k:60198)
- 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. Processes 14(30) (2008), no. 3–4, 1–16. MR 2498600 (2010d:60099)
- 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)
- 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, 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
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
