Approximation of random variables by functionals of the increments of a fractional Brownian motion
Authors:
G. M. Shevchenko and T. O. Shalaiko
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 87 (2013), 199-208
MSC (2010):
Primary 60G22; Secondary 60G15, 65C30
DOI:
https://doi.org/10.1090/S0094-9000-2014-00913-8
Published electronically:
March 21, 2014
MathSciNet review:
3241456
Full-text PDF Free Access
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Additional Information
Abstract: A lower estimate is given for the accuracy of approximation of random variables by functionals of the increments of a fractional Brownian motion with Hurst index $H>\frac 12$.
References
- Francesca Biagini, Yaozhong Hu, Bernt Øksendal, and Tusheng Zhang, Stochastic calculus for fractional Brownian motion and applications, Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2008. MR 2387368
- J. M. C. Clark and R. J. Cameron, The maximum rate of convergence of discrete approximations for stochastic differential equations, Stochastic differential systems (Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978) Lecture Notes in Control and Information Sci., vol. 25, Springer, Berlin-New York, 1980, pp. 162–171. MR 609181
- A. Deya, A. Neuenkirch, and S. Tindel, A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion, Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 2, 518–550 (English, with English and French summaries). MR 2954265, DOI https://doi.org/10.1214/10-AIHP392
- A. B. Dieker and M. Mandjes, On spectral simulation of fractional Brownian motion, Probab. Engrg. Inform. Sci. 17 (2003), no. 3, 417–434. MR 1984656, DOI https://doi.org/10.1017/S0269964803173081
- Ulf Grenander and Gábor Szegő, Toeplitz forms and their applications, 2nd ed., Chelsea Publishing Co., New York, 1984. MR 890515
- Yu. Mishura and G. Shevchenko, The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion, Stochastics 80 (2008), no. 5, 489–511. MR 2456334, DOI https://doi.org/10.1080/17442500802024892
- Andreas Neuenkirch and Ivan Nourdin, Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion, J. Theoret. Probab. 20 (2007), no. 4, 871–899. MR 2359060, DOI https://doi.org/10.1007/s10959-007-0083-0
- Andreas Neuenkirch, Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion, Stochastic Process. Appl. 118 (2008), no. 12, 2294–2333. MR 2474352, DOI https://doi.org/10.1016/j.spa.2008.01.002
References
- F. Biagini, Y. Hu, B. Oksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, Berlin, 2008. MR 2387368 (2010a:60145)
- J. M. C. Clark and R. J. Cameron, The maximum rate of convergence of discrete approximations for stochastic differential equations, Stochastic Differential Systems Filtering and Control, Lecture Notes in Control and Information Sciences, vol. 25, 1980, no. 1, pp. 162–171. MR 609181 (82f:60133)
- A. Deya, A. Neuenkirch, and S. Tindel, A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion, Ann. Inst. Henri Poincaré, Probab. Stat. 48 (2012), no. 2, 518–550. MR 2954265
- T. Dieker and M. Mandjes, On spectral simulation of fractional Brownian motion, Probab. Engrg. Inform. Sci. 17 (2003), no. 3, 417–434. MR 1984656 (2004c:60231)
- U. Grenander and G. Szegö, Toeplitz Forms and their Applications, Chelsea Publishing Company, New York, 1984. MR 890515 (88b:42031)
- Yu. Mishura and G. Shevchenko, The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion, Stochastics 80 (2008), no. 5, 489–511. MR 2456334 (2009k:60138)
- I. Nourdin and A. Neuenkirch, Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion, J. Theoret. Probab. 20 (2007), no. 4, 871–899. MR 2359060 (2009e:60150)
- A. Neuenkirch, Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion, Stoch. Process. Appl. 118 (2008), no. 12, 2294–2333. MR 2474352 (2010d:60152)
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Additional Information
G. M. Shevchenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine
Email:
zhora@univ.kiev.ua
T. O. Shalaiko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine
Email:
tarasenya@gmail.com
Keywords:
Fractional Brownian motion,
Itô–Wiener expansion,
an accuracy of approximation
Received by editor(s):
November 23, 2011
Published electronically:
March 21, 2014
Article copyright:
© Copyright 2014
American Mathematical Society
