Journal of Applied Mathematics

A Class of Stochastic Nonlinear Delay System with Jumps

Ling Bai, Kai Zhang, and Wenju Zhao

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We consider stochastic suppression and stabilization for nonlinear delay differential system. The system is assumed to satisfy local Lipschitz condition and one-side polynomial growth condition. Since the system may explode in a finite time, we stochastically perturb this system by introducing independent Brownian noises and Lévy noise feedbacks. The contributions of this paper are as follows. (a) We show that Brownian noises or Lévy noise may suppress potential explosion of the solution for some appropriate parameters. (b) Using the exponential martingale inequality with jumps, we discuss the fact that the sample Lyapunov exponent is nonpositive. (c) Considering linear Lévy processes, by the strong law of large number for local martingale, sufficient conditions for a.s. exponentially stability are investigated in Theorem 13.

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J. Appl. Math., Volume 2014 (2014), Article ID 458306, 11 pages.

First available in Project Euclid: 26 March 2014

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Bai, Ling; Zhang, Kai; Zhao, Wenju. A Class of Stochastic Nonlinear Delay System with Jumps. J. Appl. Math. 2014 (2014), Article ID 458306, 11 pages. doi:10.1155/2014/458306.

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