Approximation of multifractional Brownian motion by absolutely continuous processes
Author:
K. V. Ral’chenko
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 82 (2011), 115-127
MSC (2010):
Primary 60G15; Secondary 60H10, 65C30
DOI:
https://doi.org/10.1090/S0094-9000-2011-00831-9
Published electronically:
August 4, 2011
MathSciNet review:
2790487
Full-text PDF Free Access
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Additional Information
Abstract: We consider absolutely continuous stochastic processes that converge to multifractional Brownian motion in Besov-type spaces. We prove that solutions of stochastic differential equations with these processes converge to the solution of the equation with multifractional Brownian motion.
References
- Robert J. Adler, An introduction to continuity, extrema, and related topics for general Gaussian processes, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 12, Institute of Mathematical Statistics, Hayward, CA, 1990. MR 1088478
- Albert Benassi, Stéphane Jaffard, and Daniel Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoamericana 13 (1997), no. 1, 19–90 (English, with English and French summaries). MR 1462329, DOI https://doi.org/10.4171/RMI/217
- Brahim Boufoussi, Marco Dozzi, and Renaud Marty, Local time and Tanaka formula for a Volterra-type multifractional Gaussian process, Bernoulli 16 (2010), no. 4, 1294–1311. MR 2759180, DOI https://doi.org/10.3150/10-BEJ261
- Serge Cohen, From self-similarity to local self-similarity: the estimation problem, Fractals: theory and applications in engineering, Springer, London, 1999, pp. 3–16. MR 1726364
- 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 (1970/71), 565–578. MR 267632, DOI https://doi.org/10.1512/iumj.1970.20.20046
- Yuliya S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. MR 2378138
- David Nualart and Aurel Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308
- R. F. Peltier and J. Lévy Véhel, Multifractional Brownian motion: definition and preliminary results, INRIA research report, vol. 2645, 1995.
- K. V. Ral′chenko and G. M. Shevchenko, Properties of the paths of a multifractal Brownian motion, Teor. Ĭmovīr. Mat. Stat. 80 (2009), 106–116 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 80 (2010), 119–130. MR 2541957, DOI https://doi.org/10.1090/S0094-9000-2010-00799-X
- Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
References
- R. J. Adler, An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, IMS Lecture Notes–Monograph Series, vol. 12, Institute of Mathematical Statistics, Hayward, CA, 1990. MR 1088478 (92g:60053)
- A. Benassi, S. Jaffard, and D. Roux, Gaussian processes and pseudodifferential elliptic operators, Revista Mathematica Iberoamericana 13 (1997), no. 1, 19–89. MR 1462329 (98k:60056)
- B. Boufoussi, M. Dozzi, and R. Marty, Local time and Tanaka formula for a Volterra-type multifractional Gaussian process, http://hal.archives-ouvertes.fr/hal-00389740/en/. Bernoulli 16 (2010), no. 4, 1294–1311. MR 2759180
- S. Cohen, From self-similarity to local self-similarity: The estimation problem, Fractals: Theory and Applications in Engineering, Springer, London, 1999, pp. 3–16. MR 1726364 (2001c:60063)
- 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)
- Yu. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes in Mathematics, vol. 1929, Springer, Berlin, 2008. MR 2378138 (2008m:60064)
- D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308 (2003f:60105)
- R. F. Peltier and J. Lévy Véhel, Multifractional Brownian motion: definition and preliminary results, INRIA research report, vol. 2645, 1995.
- K. V. Ral’chenko and G. M. Shevchenko, Properties of the paths of a multifractal Brownian motion, Teor. Imovir. Mat. Stat. 80 (2009), 106–116; English transl. in Theory Probab. Math. Statist. 80 (2010), 119–130. MR 2541957 (2010h:60122)
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives. Theory and applications, Nauka i tekhnika, Minsk, 1987; English transl., Gordon and Breach Science Publishers, Yverdon, 1993. MR 1347689 (96d:26012)
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Additional Information
K. V. Ral’chenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
k.ralchenko@gmail.com
Keywords:
Gaussian process,
fractional Brownian motion,
multifractional Brownian motion,
stochastic differential equation,
Young integral
Received by editor(s):
December 8, 2009
Published electronically:
August 4, 2011
Additional Notes:
The first author is indebted to the European Commission for support in the framework of the “Marie Curie Actions” program, grant PIRSES-GA-2008-230804
Article copyright:
© Copyright 2011
American Mathematical Society
