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The Itô formula for fractional Brownian fields

Author(s): Yu. S. Mishura; S. A. Il'chenko
Translated by: Yulia Mishura
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, vipusk 69 (2003).
Journal: Theor. Probability and Math. Statist. No. 69 (2004), 153-166.
MSC (2000): Primary 60G60, 60H05
Posted: February 9, 2005
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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:

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A. Kamount, On the fractional anisotropic Wiener field, Probab. Math. Statist. 16 (1996), no. 1, 85-98. MR 1407935 (98a:60064)

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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)

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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

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M. Zähle, Integration with respect to fractional functions and stochastic calculus, Probab. Theory Related Fields 111 (1998), 333-374. MR 1640795 (99j:60073)

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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

DOI: 10.1090/S0094-9000-05-00622-8
PII: S 0094-9000(05)00622-8
Received by editor(s): 19/MAR/2003
Posted: February 9, 2005
Additional Notes: The first author is partially supported by the NATO grant PST.CLG.980408.
Copyright of article: Copyright 2005, American Mathematical Society

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