The Itô formula for fractional Brownian fields
Authors:
Yu. S. Mishura and S. A. Il’chenko
Translated by:
Yulia Mishura
Journal:
Theor. Probability and Math. Statist. 69 (2004), 153-166
MSC (2000):
Primary 60G60, 60H05
DOI:
https://doi.org/10.1090/S0094-9000-05-00622-8
Published electronically:
February 9, 2005
MathSciNet review:
2110913
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We prove the existence of the stochastic integral of the second kind constructed with respect to Hölder fields, in particular, with respect to fractional Brownian fields, and derive the Itô formula for a linear combination of fractional Brownian fields with different Hurst indices $H_i\in (\frac {1}{2},1)$, $i=1,2$.
References
- Anna Kamont, On the fractional anisotropic Wiener field, Probab. Math. Statist. 16 (1996), no. 1, 85–98. MR 1407935
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, “Nauka i Tekhnika”, Minsk, 1987 (Russian). Edited and with a preface by S. M. Nikol′skiĭ. MR 915556
- S. A. Īl′chenko and Yu. S. Mīshura, Generalized two-parameter Lebesgue-Stieltjes integrals and their applications to fractional Brownian fields, Ukraïn. Mat. Zh. 56 (2004), no. 4, 435–450 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 56 (2004), no. 4, 527–546. MR 2105898, DOI https://doi.org/10.1007/s11253-005-0065-2
- M. Zähle, Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Related Fields 111 (1998), no. 3, 333–374. MR 1640795, DOI https://doi.org/10.1007/s004400050171
References
- A. Kamount, On the fractional anisotropic Wiener field, Probab. Math. Statist. 16 (1996), no. 1, 85–98. MR 1407935 (98a:60064)
- 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, Yverdon, 1993. MR 0915556 (89a:26009); MR 1347689 (96d:26012)
- S. A. Ilchenko and Yu. S. Mishura, The generalized two-parameter Lebesgue–Stieltjes integrals and their application to fractional Brownian fields, Ukrain. Matem. Zh. 56 (2004), no. 4, 435–450; English transl. in Ukrain. Math. J. MR 2105898
- M. Zähle, Integration with respect to fractional functions and stochastic calculus, Probab. Theory Related Fields 111 (1998), 333–374. MR 1640795 (99j:60073)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
60G60,
60H05
Retrieve articles in all journals
with MSC (2000):
60G60,
60H05
Additional Information
Yu. S. Mishura
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Glushkova 6, Kyiv 03127, Ukraine
Email:
myus@univ.kiev.ua
S. A. Il’chenko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty of Mechanics and Mathematics, National Taras Shevchenko University, Glushkova 6, Kyiv 03127, Ukraine
Email:
ilchenko_sv@univ.kiev.ua
Received by editor(s):
March 19, 2003
Published electronically:
February 9, 2005
Additional Notes:
The first author is partially supported by the NATO grant PST.CLG.980408.
Article copyright:
© Copyright 2005
American Mathematical Society
