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Theory of Probability and Mathematical Statistics

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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
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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$.

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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, 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

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