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
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
MathSciNet review:
2368740
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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.
References
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- Yu. V. Kozachenko and Yu. S. MÄ«shura, Maximum upper bounds for the moments of stochastic integrals and solutions of stochastic differential equations that have fractional Brownian motion with Hurst index $H<1/2$. I, Teor. ÄŹmovÄ«r. Mat. Stat. 75 (2006), 45â56 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 75 (2007), 51â64. MR 2321180, DOI https://doi.org/10.1090/S0094-9000-08-00713-8
- 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. NikolâČskiÄ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
- X. Fernique, RegularitĂ© des trajectoires des fonctions alĂ©atoires gaussiennes, Ăcole dâĂtĂ© de ProbabilitĂ©s de Saint-Flour, IV-1974, Springer, Berlin, 1975, pp. 1â96. Lecture Notes in Math., Vol. 480 (French). MR 0413238
- Jean MĂ©min, Yulia Mishura, and Esko Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett. 51 (2001), no. 2, 197â206. MR 1822771, DOI https://doi.org/10.1016/S0167-7152%2800%2900157-7
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References
- D. Nualart and Y. Ouknine, Regularization of differential equations by fractional noise, Stoch. Process. Appl. 102 (2002), 103â116. MR 1934157 (2004b:60151)
- 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 $H<1/2$. I, Teor. Imovirnost. ta Matem. Statyst. 75 (2006), 45â56; English transl. in Theory Probab. Math. Statist. 75 (2007). MR 2321180 (2008g:60167)
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, âNauka i tekhnikaâ, Minsk, 1987; English transl., Gordon and Breach Science Publishers, New York, 1993. MR 1347689 (96d:26012)
- X. Fernique, RegularitĂ© des trajectoires des fonctions alĂ©atoires gaussiennes. Ăcole dâĂtĂ© de ProbabilitĂ©s de Saint-Flour IV, Lecture Notes in Mathematics 480, Springer, Berlin, 1975, 2â95. MR 0413238 (54:1355)
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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,
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
