Approximation of fractional Brownian motion by Wiener integrals
Authors:
Yu. S. Mishura and O. L. Banna
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 79 (2009), 107-116
MSC (2000):
Primary 60G15; Secondary 60G44
DOI:
https://doi.org/10.1090/S0094-9000-09-00773-X
Published electronically:
December 28, 2009
MathSciNet review:
2494540
Full-text PDF Free Access
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Abstract: We find an approximation in the space $L_\infty ([0,T];L_2(\Omega ))$ of a fractional Brownian motion by martingales of the form $\int _0^ta(s) dW_s$, where $W$ is a Wiener process, $a(s)$ is a power function with a negative index, that is $a(s)=k\cdot s^{-\alpha }$ where $k>0$, $\alpha =H-1/2$, and $H$ is the index of fractional Brownian motion.
References
- 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, DOI https://doi.org/10.1090/S0094-9000-07-00678-3
- O. L. Banna and Yu. S. Mishura, The simplest martingales for the best approximation to the \fBm, Visnyk Kyiv Shevchenko Univ., ser. matem. mekh. (2008), 38–43. (Ukrainian)
- 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
- 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. 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), 11–20; English transl. in Theory Probab. Math. Statist. 73 (2006), 19–29. MR 2213333 (2006m:60072)
- O. L. Banna and Yu. S. Mishura, The simplest martingales for the best approximation to the \fBm, Visnyk Kyiv Shevchenko Univ., ser. matem. mekh. (2008), 38–43. (Ukrainian)
- 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)
- 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
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
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