Maximal upper bounds for the moments of stochastic integrals and solutions of stochastic differential equations with respect to fractional Brownian motion with Hurst index . 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

Published electronically:
July 14, 2008

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Abstract | References | Similar Articles | Additional Information

Abstract: We study stochastic differential equations with Wiener integral considered with respect to fractional Brownian motion with Hurst index . 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:
http://dx.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