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Theory of Probability and Mathematical Statistics

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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$. II


Authors: Yu. V. Kozachenko and Yu. S. Mishura
Translated by: O. I. Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 76 (2007).
Journal: Theor. Probability and Math. Statist. 76 (2008), 59-76
MSC (2000): Primary 60G15; Secondary 60H05, 60H10
DOI: https://doi.org/10.1090/S0094-9000-08-00732-1
Published electronically: July 14, 2008
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Abstract: We study stochastic differential equations with Wiener integral considered with respect to fractional Brownian motion with Hurst index $ H<1/2$. We prove the existence and uniqueness of a strong solution of the equations and find maximal upper bounds for moments of a solution and its increments. We obtain estimates for the distribution of the supremum of a solution on an arbitrary interval. The modulus of continuity of solutions is found and estimates for the distributions of the norms of solutions are obtained in some Lipschitz spaces.


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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

DOI: https://doi.org/10.1090/S0094-9000-08-00732-1
Keywords: Fractional Brownian motion, Wiener integral, stochastic differential equation, moment estimates
Received by editor(s): October 2, 2006
Published electronically: July 14, 2008
Additional Notes: Research is partially supported by the NATO grant PST.CLG.9804
Article copyright: © Copyright 2008 American Mathematical Society

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