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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 458306, 11 pages
http://dx.doi.org/10.1155/2014/458306
Research Article

A Class of Stochastic Nonlinear Delay System with Jumps

1College of Mathematics, Jilin University, Changchun 130061, China
2Florida State University, Department of Scientific Computing, Tallahassee, FL 32306, USA

Received 30 August 2013; Revised 25 November 2013; Accepted 9 December 2013; Published 30 January 2014

Academic Editor: Chong Lin

Copyright © 2014 Ling Bai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.