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Mean Square Stability and Dissipativity of Split-Step Theta Method for Stochastic Delay Differential Equations with Poisson White Noise Excitations

  • Haiyan Yuan
  • Jihong Shen
  • Cheng Song
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 466)

Abstract

In this paper, a split-step theta (SST) method is introduced and analyzed for nonlinear neutral stochastic differential delay equations (NSDDEs). The asymptotic mean square stability of the split-step theta (SST) method is considered for nonlinear neutral stochastic differential equations. It is proved that, under the one-sided Lipschitz condition and the linear growth condition, for all positive stepsizes, the split-step theta method with \( \theta \in (1/2,1] \) is asymptotically mean square stable. The stability for the method with \( \theta \in [0,1/2] \) is also obtained under a stronger assumption. It further studies the mean square dissipativity of the split-step theta method with \( \theta \in (1/2,1] \) and proves that the method possesses a bounded absorbing set in mean square independent of initial data.

Keywords

Split-step theta method Nonlinear neutral stochastic differential delay equations Mean square stability Dissipativity 

Notes

Acknowledgements

This work was supported by the Natural Science Foundation of Heilongjiang Province (A201418) and the Creative Talent Project Foundation of Heilongjiang Province Education Department (UNPYSCT-2015102).

Declare. The authors declare that there is no conflict of interests regarding the publication of this article.

References

  1. 1.
    Baker, C.T.H., Buckwar, E.: Exponential stability in p-th mean of solutions, and of convergent euler-type solutions, of stochastic delay differential equations. J. Comput. Appl. Math. 184, 404–427 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Buckwar, E.: The θ-maruyama scheme for stochastic functional differential equation with distributed memory term. Monte Carlo Method Appl. 10, 235–244 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Buckwar, E.: One-step approximations for stochastic functional differential equations. J. Appl. Numer. Math. 56, 667–681 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hu, Y., Mohammed, S.A., Yan, F.: Discrete-time approximations of stochastic delay equations: The milstein scheme. Ann. Probab. 32, 265–314 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Mingzhu, L., Wanrong, C., Zhe, F.: Convergence and stability of the semi-implicit euler method for a linear stochastic differential delay equation. J. Comput. Appl. Math. 170, 255–268 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Xuerong, M.: Exponential stability of equidistant euler-maruyama approximations of stochastic differential delay equations. J. Comput. Appl. Math. 200, 297–316 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kolmanovskii V. B., Myshkis A.: Applied Theory of Functional Differential Equations Kluwer Academic, Norwell, MA(1992)Google Scholar
  8. 8.
    Randjelovic, J., Jankovic, S.: On the pth moment exponential stability criteria of neutral stochastic functional differential equations. J. Math. Anal. Appl. 326, 266 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jing. L.: Exponential stability for stochastic neutral partial functional differential equations. J. Math. Anal. Appl. 355, 414 (2009)Google Scholar
  10. 10.
    Li, X., Wanrong, C.: On mean-square stability of two-step Maruyama methods for nonlinear neutral stochastic delay differential equations. J. Appl. Math. Comput. 261, 373–381 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fuke, W., Chengming, H.: Exponential mean square stability of the theta approximations for neutral stochastic differential delay equations. J. Comput. Appl. Math. 286,172–185 (2015)Google Scholar
  12. 12.
    Chengming, H.: Exponential mean square stability of numerical methods for systems of stochastic differential equations. J. Comput. Appl. Math. 236, 4016–4026 (2012)MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2016

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Authors and Affiliations

  1. 1.College of AutomationHarbin Engineering UniversityHarbinChina
  2. 2.Department of MathematicsHeilongjiang Institute of TechnologyHarbinChina
  3. 3.School of ManagementHarbin Institute of TechnologyHarbinChina

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