## Abstract and Applied Analysis

### Lyapunov Techniques for Stochastic Differential Equations Driven by Fractional Brownian Motion

#### Abstract

Little seems to be known about evaluating the stochastic stability of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) via stochastic Lyapunov technique. The objective of this paper is to work with stochastic stability criterions for such systems. By defining a new derivative operator and constructing some suitable stochastic Lyapunov function, we establish some sufficient conditions for two types of stability, that is, stability in probability and moment exponential stability of a class of nonlinear SDEs driven by fBm. We will also give an example to illustrate our theory. Specifically, the obtained results open a possible way to stochastic stabilization and destabilization problem associated with nonlinear SDEs driven by fBm.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 292653, 9 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412606757

Digital Object Identifier
doi:10.1155/2014/292653

Mathematical Reviews number (MathSciNet)
MR3182272

#### Citation

Zeng, Caibin; Yang, Qigui; Chen, YangQuan. Lyapunov Techniques for Stochastic Differential Equations Driven by Fractional Brownian Motion. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 292653, 9 pages. doi:10.1155/2014/292653. https://projecteuclid.org/euclid.aaa/1412606757

#### References

• A. N. Kolmogorov, “The Wiener spiral and some other interesting curves in Hilbert space,” Doklady Akademii Nauk SSSR, vol. 26, pp. 115–118, 1940.
• H. E. Hurst, “Long-term storage capacity in reservoirs,” Transactions of the American Society of Civil Engineers, vol. 116, pp. 400–410, 1951.
• H. E. Hurst, “Methods of using long-term storage in reservoirs,” Proceedings of the Institution of Civil Engineers Part I, chapter 5, pp. 519–590, 1956.
• B. B. Mandelbrot and J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Review, vol. 10, pp. 422–437, 1968.
• L. Decreusefond and A. S. Üstünel, “Fractional brownian motion: theory and applications,” ESAIM Proceedings, vol. 5, pp. 75–86, 1998.
• F. Biagini, Y. Hu, B. ${\text{\O}}$ksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, Berlin, Germany, 2008.
• Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, vol. 1929, Springer, Berlin, Germany, 2008.
• L. C. G. Rogers, “Arbitrage with fractional Brownian motion,” Mathematical Finance, vol. 7, no. 1, pp. 95–105, 1997.
• C. Dellacherie and P. A. Meyer, Probabilities and Potential B, vol. 72, North Holland Publishing, Amsterdam, The Netherlands, 1982.
• L. Decreusefond and A. S. Üstünel, “Stochastic analysis of the fractional Brownian motion,” Potential Analysis, vol. 10, no. 2, pp. 177–214, 1999.
• S. J. Lin, “Stochastic analysis of fractional Brownian motions,” Stochastics and Stochastics Reports, vol. 55, no. 1-2, pp. 121–140, 1995.
• W. Dai and C. C. Heyde, “Itô's formula with respect to fractional Brownian motion and its application,” Journal of Applied Mathematics and Stochastic Analysis, vol. 9, no. 4, pp. 439–448, 1996.
• T. E. Duncan, Y. Hu, and B. Pasik-Duncan, “Stochastic calculus for fractional Brownian motion. I. Theory,” SIAM Journal on Control and Optimization, vol. 38, no. 2, pp. 582–612, 2000.
• E. Alòs, O. Mazet, and D. Nualart, “Stochastic calculus with respect to Gaussian processes,” The Annals of Probability, vol. 29, no. 2, pp. 766–801, 2001.
• P. Carmona, L. Coutin, and G. Montseny, “Stochastic integration with respect to fractional Brownian motion,” Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol. 39, no. 1, pp. 27–68, 2003.
• R. J. Elliott and J. van der Hoek, “A general fractional white noise theory and applications to finance,” Mathematical Finance, vol. 13, no. 2, pp. 301–330, 2003.
• F. Biagini, B. ${\text{\O}}$ksendal, A. Sulem, and N. Wallner, “An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion,” Proceedings of The Royal Society of London A, vol. 460, no. 2041, pp. 347–372, 2004.
• M. Jolis, “On the Wiener integral with respect to the fractional Brownian motion on an interval,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2, pp. 1115–1127, 2007.
• E. Alòs and D. Nualart, “Stochastic integration with respect to the fractional Brownian motion,” Stochastics and Stochastics Reports, vol. 75, no. 3, pp. 129–152, 2003.
• M. Gradinaru, I. Nourdin, F. Russo, and P. Vallois, “$m$-order integrals and generalized Itô's formula: the case of a fractional Brownian motion with any Hurst index,” Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol. 41, no. 4, pp. 781–806, 2005.
• C. Bender, “An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter,” Stochastic Processes and Their Applications, vol. 104, no. 1, pp. 81–106, 2003.
• C. Zeng, Q. Yang, and Y. Q. Chen, “Solving nonlinear stochastic differential equations with fractional Brownian motion using reducibility approach,” Nonlinear Dynamics, vol. 67, no. 4, pp. 2719–2726, 2012.
• C. Zeng, Y. Chen, and Q. Yang, “Almost sure and moment stability properties of fractional order Black-Scholes model,” Fractional Calculus and Applied Analysis, vol. 16, no. 2, pp. 317–331, 2013.
• X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, UK, 1997.
• C. Zeng, Y. Chen, and Q. Yang, “The fBm-driven Ornstein-Uhlenbeck process: probability density function and anomalous diffusion,” Fractional Calculus and Applied Analysis, vol. 15, no. 3, pp. 479–492, 2012. \endinput