Path properties of multifractal Brownian motion
Authors:
K. V. Ral’chenko and G. M. Shevchenko
Translated by:
N. N. Semenov
Journal:
Theor. Probability and Math. Statist. 80 (2010), 119-130
MSC (2000):
Primary 60G15, 60G17; Secondary 60G18
DOI:
https://doi.org/10.1090/S0094-9000-2010-00799-X
Published electronically:
August 19, 2010
MathSciNet review:
2541957
Full-text PDF Free Access
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Additional Information
Abstract: A new generalization of fractional Brownian motion (called multifractal Brownian motion) is considered for the case where the Hürst index $H$ is a function of time $t$. The pathwise continuity of multifractal Brownian motion is proved. Global and local Hölder properties are also studied.
References
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- Yuliya S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. MR 2378138
- 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
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References
- A. Ayache and J. Lévy Véhel, Generalized multifractional Brownian motion, Fractals: Theory and Applications in Engineering, Springer, London, 1999, pp. 17–32. MR 1726365 (2001d:60037)
- J. M. Bardet and P. Bertran, Identification of the multiscale fractional Brownian motion with biomechanical applications, J. Time Series Anal. 28 (2007), no. 1, 1–52. MR 2332850 (2008e:60095)
- A. M. Garsia, E. Rodemich, and H. Rumsey, Jr., A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J. 20 (1970), no. 6, 565–578. MR 0267632 (42:2534)
- C. Lacaux, Real harmonizable multifractional Lévy motions, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 3, 259–277. MR 2060453 (2005b:60103)
- Yu. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lect. Notes Math., vol. 1929, Springer, Berlin, 2008. MR 2378138 (2008m:60064)
- I. Norros, E. Valkeila, and J. 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 (2000f:60053)
- A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed., McGraw-Hill Book Company, New York, 1991. MR 0176501 (31:773)
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Additional Information
K. V. Ral’chenko
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Volodymyrska 64, Kyiv 01601, Ukraine
Email:
k.ralchenko@gmail.com
G. M. Shevchenko
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Volodymyrska 64, Kyiv 01601, Ukraine
Email:
zhora@univ.kiev.ua
Keywords:
Gaussian process,
fractional Brownian motion,
multifractal Brownian motion,
Hürst index
Received by editor(s):
March 12, 2009
Published electronically:
August 19, 2010
Additional Notes:
The authors are grateful to the European Commission for support of their investigations in the framework of the program “Marie Curie Actions”, grant PIRSES-GA-2008-230804
Article copyright:
© Copyright 2010
American Mathematical Society
