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Stochastic integrals and stochastic differential equations with respect to the fractional Brownian field
Author(s):
Yu.
S.
Mishura;
S.
A.
Il'chenko
Translated by:
O. I. Klesov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika,
vipusk 75
(2006).
Journal:
Theor. Probability and Math. Statist.
No. 75
(2007),
93-108.
MSC (2000):
Primary 60H10, 60H05, 60G15
Posted:
January 24, 2008
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Abstract:
Stochastic differential equations on the plane are considered with respect to the fractional Brownian field. We prove the existence and uniqueness of a solution for such equations. These results are based on new estimates obtained for norms in the Besov type spaces for the two-parameter stochastic integral considered with respect to the fractional Brownian field.
References:
-
- 1.
- S. A. Il'chenko and Yu. S. Mishura, Generalized two-parameter Lebesgue-Stieltjes integrals and their applications to fractional Brownian fields, Ukrain. Mat. Zh. 56 (2004), no. 4, 435-450; English transl. in Ukrainian Math. J. 56 (2004), no. 4, 527-546. MR 2105898 (2005i:60068)
- 2.
- D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55-81. MR 1893308 (2003f:60105)
- 3.
- Yu. S. Mishura and S. A. Il'chenko, Some estimates for two-parameter generalized stochastic Lebesgue-Stieltjes integrals, Theory Stochastic Processes 9(25) (2003), no. 3-4, 87-100. MR 2306063
- 4.
- F. Russo and P. Vallois, The generalized covariation process and Itô formula, Stochastic Process. Appl. 59 (1995), 81-104. MR 1350257 (96f:60089)
- 5.
- A. Kamont, On the fractional anisotropic Wiener field, Probab. Math. Statist. 16 (1996), no. 1, 85-98. MR 1407935 (98a:60064)
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Additional Information:
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
S.
A.
Il'chenko
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:
ilchenko_sv@univ.kiev.ua
DOI:
10.1090/S0094-9000-08-00717-5
PII:
S 0094-9000(08)00717-5
Received by editor(s):
17/OCT/2005
Posted:
January 24, 2008
Additional Notes:
The first author is supported by the grant NATO PST.CLG 980408
Copyright of article:
Copyright
2008,
American Mathematical Society
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