ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY MULTIVARIATE FRACTIONAL BROWNIAN MOTIONS
https://doi.org/10.29235/1561-8323-2018-62-4-398-405
Abstract
In this article, n-dimensional stochastic differential equations driven by multivariate fractional Brownian motions with the Hurst indices greater than 1/3 and a drift term are considered. We have obtained an expansion of expectations Ptg (x) = Eg (Xxt) for small t, where Xxt denotes the solution of the mentioned equation with an initial value x, and g: Rn → R is a sufficiently smooth function.
About the Authors
Maxim M. Vaskouski
Belarusian State University
Belarus
Belarus
Vaskouski Maxim Mikhailovich – Ph. D. (Physics and Mathematics), Associate professor
4, Nezavisimosti Ave., 220030, Minsk
Ilya V. Kachan
Belarusian State University
Belarus
Belarus
Kachan Ilya Vadimovich – Undergraduate, Assistant of the Department
4, Nezavisimosti Ave., 220030, Minsk
References
1. Baudoin F., Coutin L. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Processes and their Applications, 2007, vol. 117, no. 5, pp. 550–574. https://doi.org/10.1016/j. spa.2006.09.004
2. Neuenkirch A., Nourdin I., Rößler A., Tindel S. Trees and asymptotic expansions for fractional stochastic differential equations. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 2009, vol. 45, no. 1, pp. 157–174. https://doi. org/10.1214/07-aihp159
3. Friz P., Hairer M. A Course on Rough Paths with an introduction to regularity structures. Springer International Publishing Switzerland, 2014. 263 p.
4. Gubinelli M. Controlling rough paths. Journal of Functional Analysis, 2004, vol. 216, no. 1, pp. 86–140. https://doi. org/10.1016/j.jfa.2004.01.002
5. Vaskouski M. M., Kachan I. V. An analogue of the Ito formula for stochastic differential equations with fractional Brownian motions having different Hurst indices greater than 1/3. Analiticheskie i chislennye metody modelirovaniya estestvenno-nauchnykh i sotsial’nykh problem [Analytical and numerical methods of modeling of natural-scientific and social problems]. Penza, 2017, pp. 12–16 (in Russian).
6. Nualart D., Rascanu A. Differential equations driven by fractional Brownian motion. Collectanea Mathematica, 2002, vol. 53, no. 1, pp. 55–81.
7. Levakov A. A. Stochastic Differential Equations. Minsk, Belarusian State University, 2009. 231 p. (in Russian) .
8. Oksendal B. Stochastic differential equations. An introduction with applications. Berlin, Heidelberg, New York, Springer-Verlag, 2003. 379 p.
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