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Quasi-linear stochastic differential equations with a fractional Brownian component
Author(s):
Yu.
S.
Mishura
Translated by:
the author
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika,
vipusk 68
(2003).
Journal:
Theor. Probability and Math. Statist.
No. 68
(2004),
103-115.
MSC (2000):
Primary 60H10
Posted:
June 10, 2004
MathSciNet review:
2000399
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Abstract:
The paper is devoted to stochastic differential equations with a fractional Brownian component. The fractional Brownian motion is constructed on the white noise space with the help of ``forward'' and ``backward'' fractional integrals. The fractional white noise and Wick products are considered. A similar construction for the ``complete'' fractional integral is considered by Elliott and van der Hoek. We consider two possible approaches to the existence and uniqueness of solutions of stochastic differential equation with a fractional Brownian motion.
References:
-
- 1.
- R. Elliott and J. van der Hoek, A General Fractional White Noise Theory and Applications to Finance, Preprint, University of Adelaide, 1999.
- 2.
- R. Elliott and J. van der Hoek, Fractional Brownian motion and financial modelling, Trends in Mathematics. Mathematical Finance, Birkhäuser, Basel, 2001, pp. 140-151.
- 3.
- H. Giessing, Wick Calculus with Applications to Anticipating Stochastic Differential Equations, Manuscript, University of Bergen, 1994.
- 4.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach, Sci. Publ., Yverdon, 1993. MR 96d:26012
- 5.
- H. Holden, B. Øksendal, J. Ubøe, and T. Zhang, Stochastic Partial Differential Equations, Birkhäuser, Boston, 1996. MR 98f:60124
- 6.
- G. Vage, A general existence and uniqueness theorem for Wick-SDEs in
 , Stochastics and Stochastic Reports 58 (1996), 259-284. MR 97j:60111
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Additional Information:
Yu.
S.
Mishura
Affiliation:
Department of Mathematical Analysis, Faculty for Mathematics and Mechanics, Kyiv National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv--127 03127, Ukraine
Email:
myus@mail.univ.kiev.ua
DOI:
10.1090/S0094-9000-04-00608-8
PII:
S 0094-9000(04)00608-8
Received by editor(s):
29/MAR/2002
Posted:
June 10, 2004
Additional Notes:
The work was supported by the project INTAS-99-00016.
Copyright of article:
Copyright
2004,
American Mathematical Society
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