Skip to main content
SpringerLink
Log in

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

  • Conference paper
  • First Online:
Automation Control Theory Perspectives in Intelligent Systems (CSOC 2016)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 466))

Included in the following conference series:

  • 909 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Buckwar, E.: The θ-maruyama scheme for stochastic functional differential equation with distributed memory term. Monte Carlo Method Appl. 10, 235–244 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Buckwar, E.: One-step approximations for stochastic functional differential equations. J. Appl. Numer. Math. 56, 667–681 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hu, Y., Mohammed, S.A., Yan, F.: Discrete-time approximations of stochastic delay equations: The milstein scheme. Ann. Probab. 32, 265–314 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Xuerong, M.: Exponential stability of equidistant euler-maruyama approximations of stochastic differential delay equations. J. Comput. Appl. Math. 200, 297–316 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kolmanovskii V. B., Myshkis A.: Applied Theory of Functional Differential Equations Kluwer Academic, Norwell, MA(1992)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jing. L.: Exponential stability for stochastic neutral partial functional differential equations. J. Math. Anal. Appl. 355, 414 (2009)

    Google Scholar 

  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)

    Google Scholar 

  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. Chengming, H.: Exponential mean square stability of numerical methods for systems of stochastic differential equations. J. Comput. Appl. Math. 236, 4016–4026 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. College of Automation, Harbin Engineering University, Harbin, 150001, China

    Haiyan Yuan & Jihong Shen

  2. Department of Mathematics, Heilongjiang Institute of Technology, Harbin, 150050, China

    Haiyan Yuan

  3. School of Management, Harbin Institute of Technology, Harbin, 150001, China

    Cheng Song

Authors
  1. Haiyan Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jihong Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cheng Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haiyan Yuan .

Editor information

Editors and Affiliations

  1. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Radek Silhavy

  2. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Roman Senkerik

  3. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Zuzana Kominkova Oplatkova

  4. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Petr Silhavy

  5. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Zdenka Prokopova

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Yuan, H., Shen, J., Song, C. (2016). Mean Square Stability and Dissipativity of Split-Step Theta Method for Stochastic Delay Differential Equations with Poisson White Noise Excitations. In: Silhavy, R., Senkerik, R., Oplatkova, Z.K., Silhavy, P., Prokopova, Z. (eds) Automation Control Theory Perspectives in Intelligent Systems. CSOC 2016. Advances in Intelligent Systems and Computing, vol 466. Springer, Cham. https://doi.org/10.1007/978-3-319-33389-2_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-33389-2_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-33387-8

  • Online ISBN: 978-3-319-33389-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation