Quasi-linear stochastic differential equations with a fractional Brownian component
Author:
Yu. S. Mishura
Translated by:
the author
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 68 (2003).
Journal:
Theor. Probability and Math. Statist. 68 (2004), 103-115
MSC (2000):
Primary 60H10
DOI:
https://doi.org/10.1090/S0094-9000-04-00608-8
Published electronically:
June 10, 2004
MathSciNet review:
2000399
Full-text PDF Free Access
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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.
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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.
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- Gjermund Våge, A general existence and uniqueness theorem for Wick-SDEs in $({\scr S})^n_{-1,k}$, Stochastics Stochastics Rep. 58 (1996), no. 3-4, 259–284. MR 1424695, DOI https://doi.org/10.1080/17442509608834077
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.
5 H. Holden, B. Øksendal, J. Ubøe, and T. Zhang, Stochastic Partial Differential Equations, Birkhäuser, Boston, 1996.
6 G. Vage, A general existence and uniqueness theorem for Wick-SDEs in $(Y)^n_{-1,k}$, Stochastics and Stochastic Reports 58 (1996), 259–284.
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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
Received by editor(s):
March 29, 2002
Published electronically:
June 10, 2004
Additional Notes:
The work was supported by the project INTAS-99-00016.
Article copyright:
© Copyright 2004
American Mathematical Society
