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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 128625, 8 pages
http://dx.doi.org/10.1155/2013/128625
Research Article

Numerical Analysis for Stochastic Partial Differential Delay Equations with Jumps

Yan Li1,2 and Junhao Hu3

1Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
2College of Science, Huazhong Agriculture University, Wuhan 430074, China
3College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China

Received 3 January 2013; Accepted 21 March 2013

Academic Editor: Xuerong Mao

Copyright © 2013 Yan Li and Junhao Hu. 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. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  2. K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. View at MathSciNet
  3. Q. Luo, F. Deng, J. Bao, B. Zhao, and Y. Fu, “Stabilization of stochastic Hopfield neural network with distributed parameters,” Science in China F, vol. 47, no. 6, pp. 752–762, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Q. Luo, F. Deng, X. Mao, J. Bao, and Y. Zhang, “Theory and application of stability for stochastic reaction diffusion systems,” Science in China F, vol. 51, no. 2, pp. 158–170, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, UK, 2007. View at Zentralblatt MATH · View at MathSciNet
  6. Y. Shen and J. Wang, “An improved algebraic criterion for global exponential stability of recurrent neural networks with time-varying delays,” IEEE Transactions on Neural Networks, vol. 19, no. 3, pp. 528–531, 2008. View at Publisher · View at Google Scholar
  7. Y. Shen and J. Wang, “Almost sure exponential stability of recurrent neural networks with markovian switching,” IEEE Transactions on Neural Networks, vol. 20, no. 5, pp. 840–855, 2009. View at Publisher · View at Google Scholar
  8. I. Gyöngy and N. Krylov, “Accelerated finite difference schemes for linear stochastic partial differential equations in the whole space,” SIAM Journal on Mathematical Analysis, vol. 42, no. 5, pp. 2275–2296, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. Jentzen, P. E. Kloeden, and G. Winkel, “Efficient simulation of nonlinear parabolic SPDEs with additive noise,” The Annals of Applied Probability, vol. 21, no. 3, pp. 908–950, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. P. E. Kloeden, G. J. Lord, A. Neuenkirch, and T. Shardlow, “The exponential integrator scheme for stochastic partial differential equations: pathwise error bounds,” Journal of Computational and Applied Mathematics, vol. 235, no. 5, pp. 1245–1260, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. Bao, A. Truman, and C. Yuan, “Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps,” Proceedings of The Royal Society of London A, vol. 465, no. 2107, pp. 2111–2134, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. B. Boufoussi and S. Hajji, “Successive approximation of neutral functional stochastic differential equations with jumps,” Statistics and Probability Letters, vol. 80, no. 5-6, pp. 324–332, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. E. Hausenblas, “Finite element approximation of stochastic partial differential equations driven by Poisson random measures of jump type,” SIAM Journal on Numerical Analysis, vol. 46, no. 1, pp. 437–471, 2007/08. View at Publisher · View at Google Scholar · View at MathSciNet
  14. S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, vol. 113 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  15. M. Röckner and T. Zhang, “Stochastic evolution equations of jump type: existence, uniqueness and large deviation principles,” Potential Analysis, vol. 26, no. 3, pp. 255–279, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  16. J. Bao, B. Böttcher, X. Mao, and C. Yuan, “Convergence rate of numerical solutions to SFDEs with jumps,” Journal of Computational and Applied Mathematics, vol. 236, no. 2, pp. 119–131, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. N. Jacob, Y. Wang, and C. Yuan, “Numerical solutions of stochastic differential delay equations with jumps,” Stochastic Analysis and Applications, vol. 27, no. 4, pp. 825–853, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. View at Publisher · View at Google Scholar · View at MathSciNet