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Abstract and Applied Analysis
Volume 2012, Article ID 371239, 20 pages
http://dx.doi.org/10.1155/2012/371239
Research Article

Existence and Uniqueness of Solutions to Neutral Stochastic Functional Differential Equations with Poisson Jumps

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2Department of Mechanics, Tianjin University, Tianjin 300072, China

Received 9 March 2012; Revised 12 May 2012; Accepted 15 May 2012

Academic Editor: Márcia Federson

Copyright © 2012 Jianguo Tan 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.

Linked References

  1. X. Mao, “Razumikhin-type theorems on exponential stability of neutral stochastic functional-differential equations,” SIAM Journal on Mathematical Analysis, vol. 28, no. 2, pp. 389–401, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. V. Kolmanovskii, N. Koroleva, T. Maizenberg, X. Mao, and A. Matasov, “Neutral stochastic differential delay equations with Markovian switching,” Stochastic Analysis and Applications, vol. 21, no. 4, pp. 819–847, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. X. Mao, “Razumikhin-type theorems on exponential stability of neutral stochastic functional-differential equations,” SIAM Journal on Mathematical Analysis, vol. 28, no. 2, pp. 389–401, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. X. Mao, Y. Shen, and C. Yuan, “Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching,” Stochastic Processes and their Applications, vol. 118, no. 8, pp. 1385–1406, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. X. Mao, “Exponential stability in mean square of neutral stochastic differential-functional equations,” Systems & Control Letters, vol. 26, no. 4, pp. 245–251, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. X. R. Mao, Stochastic Differential Equations and Their Applications, Horwood, Chichester, UK, 1997.
  7. Q. Luo, X. Mao, and Y. Shen, “New criteria on exponential stability of neutral stochastic differential delay equations,” Systems & Control Letters, vol. 55, no. 10, pp. 826–834, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. S. Zhou and S. Hu, “Razumikhin-type theorems of neutral stochastic functional differential equations,” Acta Mathematica Scientia. Series B, vol. 29, no. 1, pp. 181–190, 2009. View at Publisher · View at Google Scholar
  9. J. Luo, “Fixed points and stability of neutral stochastic delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 431–440, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. L. S. Wang and H. Xue, “Convergence of numerical solutions to stochastic differential delay equations with Poisson jump and Markovian switching,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1161–1172, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. L. S. Wang, C. Mei, and H. Xue, “The semi-implicit Euler method for stochastic differential delay equations with jumps,” Applied Mathematics and Computation, vol. 192, no. 2, pp. 567–578, 2007. View at Publisher · View at Google Scholar
  12. L. Ronghua, M. Hongbing, and D. Yonghong, “Convergence of numerical solutions to stochastic delay differential equations with jumps,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 584–602, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. L. Ronghua and C. Zhaoguang, “Convergence of numerical solution to stochastic delay differential equation with Poisson jump and Markovian switching,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 451–463, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. J. W. Luo, “Comparison principle and stability of Ito stochastic differential delay equations with Poisson jump and Markovian switching,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 64, no. 2, pp. 253–262, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. J. Tan and H. Wang, “Mean-square stability of the Euler-Maruyama method for stochastic differential delay equations with jumps,” International Journal of Computer Mathematics, vol. 88, no. 2, pp. 421–429, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. D. Z. Liu, G. Yang, and W. Zhang, “The stability of neutral stochastic delay differential equations with Poisson jumps by fixed points,” Journal of Computational and Applied Mathematics, vol. 235, no. 10, pp. 3115–3120, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. J. Luo and T. Taniguchi, “The existence and uniqueness for non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps,” Stochastics and Dynamics, vol. 9, no. 1, pp. 135–152, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH