Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

Request Permissions   Purchase Content 
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The best uniform approximation of a Wiener process by integrals of the form

$\displaystyle \int _{0}^{t}f(s)\,dB_{s}^{H}$

is established in the space $ L_{\infty } ([0,T];L_{2} (\Omega ))$, where $ \{ B_{t}^{H}, t\in [0, T]\} $ is the fractional Brownian motion with the Hurst index $ H$ and $ f(s)=k\cdot s^{\alpha }$, $ s\in [0,T]$, for $ k>0$ and $ \alpha =H-1/2$.

References [Enhancements On Off] (What's this?)

  • 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)

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 60G15, 60G44

Retrieve articles in all journals with MSC (2010): 60G15, 60G44


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

American Mathematical Society