Stochastic integrals and stochastic differential equations with respect to the fractional Brownian field
Authors:
Yu. S. Mishura and S. A. Il’chenko
Translated by:
O. I. Klesov
Journal:
Theor. Probability and Math. Statist. 75 (2007), 93-108
MSC (2000):
Primary 60H10, 60H05, 60G15
DOI:
https://doi.org/10.1090/S0094-9000-08-00717-5
Published electronically:
January 24, 2008
MathSciNet review:
2321184
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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
- 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
- David Nualart and Aurel Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308
- Yuliya S. Mishura and Svetlana A. Ilchenko, Some estimates for two-parameter generalized stochastic Lebesgue-Stieltjes integrals, Theory Stoch. Process. 9 (2003), no. 3-4, 87–100. MR 2306063
- Francesco Russo and Pierre Vallois, The generalized covariation process and Itô formula, Stochastic Process. Appl. 59 (1995), no. 1, 81–104. MR 1350257, DOI https://doi.org/10.1016/0304-4149%2895%2993237-A
- Anna Kamont, On the fractional anisotropic Wiener field, Probab. Math. Statist. 16 (1996), no. 1, 85–98. MR 1407935
References
- 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)
- D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308 (2003f:60105)
- 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
- F. Russo and P. Vallois, The generalized covariation process and Itô formula, Stochastic Process. Appl. 59 (1995), 81–104. MR 1350257 (96f:60089)
- A. Kamont, On the fractional anisotropic Wiener field, Probab. Math. Statist. 16 (1996), no. 1, 85–98. MR 1407935 (98a:60064)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
60H10,
60H05,
60G15
Retrieve articles in all journals
with MSC (2000):
60H10,
60H05,
60G15
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
Received by editor(s):
October 17, 2005
Published electronically:
January 24, 2008
Additional Notes:
The first author is supported by the grant NATO PST.CLG 980408
Article copyright:
© Copyright 2008
American Mathematical Society
