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

DOI:
https://doi.org/10.1090/S0094-9000-08-00732-1

Published electronically:
July 14, 2008

Full-text PDF

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.

**1.**D. Nualart and Y. Ouknine,*Regularization of differential equations by fractional noise*, Stoch. Process. Appl.**102**(2002), 103-116. MR**1934157 (2004b:60151)****2.**Yu. V. Kozachenko and Yu. S. Mishura,*Maximal upper bounds for moments of stochastic integrals and solutions of stochastic differential equations with respect to the fractional Brownian motion with Hurst index . I*, Teor. Imovirnost. ta Matem. Statyst.**75**(2006), 45-56; English transl. in Theory Probab. Math. Statist.**75**(2007). MR**2321180 (2008g:60167)****3.**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)****4.**X. Fernique,*Regularité des trajectoires des fonctions aléatoires gaussiennes. École d'Été de Probabilités de Saint-Flour IV*, Lecture Notes in Mathematics**480**, Springer, Berlin, 1975, 2-95. MR**0413238 (54:1355)****5.**J. Mémin, Yu. Mishura, and E. Valkeila,*Inequalities for the moments of Wiener integrals with respect to fractional Brownian motion*, Stat. Prob. Letters**51**(2001), 197-206. MR**1822771 (2002b:60096)****6.**V. V. Buldygin and Yu. V. Kozachenko,*Metric Characterization of Random Variables and Random Processes*, American Mathematical Society, Providence, Rhode Island, 2000. MR**1743716 (2001g:60089)**

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2000):
60G15,
60H05,
60H10

Retrieve articles in all journals with MSC (2000): 60G15, 60H05, 60H10

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