Maximal upper bounds for the moments of stochastic integrals and solutions of stochastic differential equations with respect to fractional Brownian motion with Hurst index $H<1/2$. I
Authors:
Yu. V. Kozachenko and Yu. S. Mishura
Translated by:
O. I. Klesov
Journal:
Theor. Probability and Math. Statist. 75 (2007), 51-64
MSC (2000):
Primary 60G15, 60H05
DOI:
https://doi.org/10.1090/S0094-9000-08-00713-8
Published electronically:
January 23, 2008
MathSciNet review:
2321180
Full-text PDF Free Access
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Additional Information
Abstract: Upper moment bounds and maximal upper moment bounds are obtained for Wiener integrals considered with respect to a fractional Brownian motion with Hurst index $H<1/2$. Maximal bounds are derived from new maximal inequalities for Gaussian random variables and stochastic processes.
References
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- C. Bender, Integration with Respect to a Fractional Brownian Motion and Related Market Models, Ph.D. Thesis, Konstanz University, 2003.
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- 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
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- M. A. Lifshits, Gaussian random functions, Mathematics and its Applications, vol. 322, Kluwer Academic Publishers, Dordrecht, 1995. MR 1472736
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References
- R. S. Liptser and A. N. Shiryaev, Theory of Martingales, “Nauka”, Moscow, 1986; English transl., Kluwer Academic Publishers, Dordrecht, 1989. MR 1022664 (90j:60046)
- C. Bender, Integration with Respect to a Fractional Brownian Motion and Related Market Models, Ph.D. Thesis, Konstanz University, 2003.
- J. Memin, Yu. Mishura, and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to fractional Brownian motions, Statist. Probab. Lett. 51 (2001), no. 2, 197–206. MR 1822771 (2002b:60096)
- Yu. V. Krvavich and Yu. S. Mishura, Some maximal inequalities for moments of Wiener integrals with respect to fractional Brownian motion, Teor. Imovir. Mat. Stat. 61 (1999), 72–83; English transl. in Theory Probab. Math. Statist. 61 (2000), 75–86. MR 1866968 (2002h:60072)
- 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, New York, 1993. MR 1347689 (96d:26012)
- D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308 (2003f:60105)
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Additional Information
Yu. V. Kozachenko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
yvk@univ.kiev.ua
Yu. S. Mishura
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
myus@univ.kiev.ua
Keywords:
Fractional Brownian motion,
Wiener integral,
moment inequalities,
Gaussian stochastic processes
Received by editor(s):
December 1, 2005
Published electronically:
January 23, 2008
Additional Notes:
This work is partially supported by the NATO grant PST.CLG.980408
Article copyright:
© Copyright 2008
American Mathematical Society
