Skip to main content
SpringerLink
Log in

Strong Convergence and Asymptotic Exponential Stability of Modified Truncated EM Method for Neutral Stochastic Differential Equations with Time-dependent Delay

  • Research Article
  • Published:
Frontiers of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider asymptotic exponential stability of the exact solution and the corresponding modified truncated EM (MTEM) method for neutral stochastic differential equations (NSDEs) with time-dependent delay. We obtain sufficient conditions under which the MTEM method replicates the exponential stability of the exact solution no matter time-dependent delay δ(t) is bounded or not. To make sure that the stability conclusions are meaningful, we will obtain the strong convergence of the MTEM method at first. Two examples are presented to illustrate our conclusions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Gan S.Q., Schurz H., Zhang H.M., Mean-square convergence of stochastic θ-methods for nonlinear neutral stochastic differential delay equations. Int. J. Numer. Anal. Model. Ser. B, 2011, 8(2): 201–213

    MathSciNet  MATH  Google Scholar 

  2. Hu Y.Z., Wu F.K., Huang C.M., General decay pathwise stability of neutral stochastic differential equations with unbounded delay. Acta Math. Sin. (Engl. Ser.), 2011, 27(11): 2153–2168

    Article  MathSciNet  MATH  Google Scholar 

  3. Ikeda N., Watanabe S., Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library, Vol. 24, Amsterdam: North-Holland Publishing Co., 1981

    Book  MATH  Google Scholar 

  4. Kloeden P.E., Platen E., Numerical Solution of Stochastic Differential Equations. Appl. Math. (N. Y.), Vol. 23, Berlin: Springer-Verlag, 1992

    Book  MATH  Google Scholar 

  5. Lakshmikantham V., Wen L.Z., Zhang B.G., Theory of Differential Equations with Unbounded Delay. Math. Appl., Vol. 298, Dordrecht: Kluwer Academic Publishers Group, 1994

    Book  MATH  Google Scholar 

  6. Lan G.Q., Asymptotic exponential stability of modified truncated EM method for neutral stochastic differential delay equations. J. Comput. Appl. Math., 2018, 340: 334–341

    Article  MathSciNet  MATH  Google Scholar 

  7. Lan G.Q., Wang Q.S., Strong convergence rates of modified truncated EM methods for neutral stochastic differential delay equations. J. Comput. Appl. Math., 2019, 362: 83–98

    Article  MathSciNet  MATH  Google Scholar 

  8. Lan G.Q., Xia F., Strong convergence rates of modified truncated EM method for stochastic differential equations. J. Comput. Appl. Math., 2018, 334: 1–17

    Article  MathSciNet  MATH  Google Scholar 

  9. Lan G.Q., Xia F., Wang Q.S., Polynomial stability of exact solution and a numerical method for stochastic differential equations with time-dependent delay. J. Comput. Appl. Math., 2019, 346: 340–356

    Article  MathSciNet  MATH  Google Scholar 

  10. Lan G.Q., Yuan C.G., Exponential stability of the exact solutions and θ-EM approximations to neutral SDDEs with Markov switching. J. Comput. Appl. Math., 2015, 285: 230–242

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu L.N., Zhu Q.X., Mean-square stability of two classes of theta method for neutral stochastic differential delay equations. J. Comput. Appl. Math., 2016, 305: 55–67

    Article  MathSciNet  MATH  Google Scholar 

  12. Mao X.R., Stochastic Differential Equations and Applications, 2nd Edition. Chichester: Horwood Publishing Limited, 2007

    MATH  Google Scholar 

  13. Mao X.R., The truncated Euler-Maruyama method for stochastic differential equations. J. Comput. Appl. Math., 2015, 290: 370–384

    Article  MathSciNet  MATH  Google Scholar 

  14. Mao X.R., Yuan C.G., Stochastic Differential Equations with Markovian Switching. London: Imperial College Press, 2006

    Book  MATH  Google Scholar 

  15. Milosevic M., Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama method. Math. Comput. Modelling, 2011, 54(9–10): 2235–2251

    Article  MathSciNet  MATH  Google Scholar 

  16. Milošević M., Almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama approximation. Math. Comput. Modelling, 2013, 57(3–4): 887–899

    Article  MathSciNet  MATH  Google Scholar 

  17. Mo H.Y., Zhao X.Y., Deng F.Q., Mean-square stability of the backward Euler-Maruyama method for neutral stochastic delay differential equations with jumps. Math. Methods Appl. Sci., 2017, 40(5): 1794–1803

    Article  MathSciNet  MATH  Google Scholar 

  18. Obradović M., Milošević M., Stability of a class of neutral stochastic differential equations with unbounded delay and Markovian switching and the Euler–Maruyama method. J. Comput. Appl. Math., 2017, 309: 244–266

    Article  MathSciNet  MATH  Google Scholar 

  19. Tan L., Yuan C.G., Convergence rates of theta-method for neutral SDDEs under non-globally Lipschitz continuous coefficients. 2017, arXiv:1701.00223

  20. Tan L., Yuan C.G., Strong convergence of a tamed theta scheme for NSDDEs with onesided Lipschitz drift. Appl. Math. Comput., 2018, 338: 607–623

    MathSciNet  MATH  Google Scholar 

  21. Wang W.Q., Chen Y.P., Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations. Appl. Numer. Math., 2011, 61(5): 696–701

    Article  MathSciNet  MATH  Google Scholar 

  22. Wu F.K., Hu S.G., Mao X.R., Razumikhin-type theorem for neutral stochastic functional differential equations with unbounded delay. Acta Math. Sci. Ser. B Engl. Ed., 2011, 31(4): 1245–1258

    MathSciNet  MATH  Google Scholar 

  23. Yue C., Zhao L.B., Strong convergence of the split-step backward Euler method for stochastic delay differential equations with a nonlinear diffusion coefficient. J. Comput. Appl. Math., 2021, 382: Paper No. 113087, 17 pp.

  24. Zhang H.M., Gan S.Q., Mean-square convergence of one-step methods for neutral stochastic differential delay equations. Appl. Math. Comput., 2008, 204(2): 884–890

    MathSciNet  MATH  Google Scholar 

  25. Zhang W., Song M.H., Liu M.Z., Strong convergence of the partially truncated Euler–Maruyama method for a class of stochastic differential delay equations. J. Comput. Appl. Math., 2018, 335: 114–128

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhao J.J., Yi Y.L., Xu Y., Strong convergence and stability of the split-step theta method for highly nonlinear neutral stochastic delay integro differential equation. Appl. Numer. Math., 2020, 157: 385–404

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhu Q.X., Stability analysis of stochastic delay differential equations with Levy noise. Systems Control Lett., 2018, 118: 62–68

    Article  MathSciNet  MATH  Google Scholar 

  28. Zong X.F., Wu F.K., Huang C.M., Exponential mean square stability of the theta approximations for neutral stochastic differential delay equations. J. Comput. Appl. Math., 2015, 286: 172–185

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank anonymous referees for their very useful comments and suggestions. This work was supported by Beijing Municipal NSF (No. 1192013).

Author information

Authors and Affiliations

  1. College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, 100029, China

    Guangqiang Lan & Ge Wang

Authors
  1. Guangqiang Lan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ge Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guangqiang Lan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lan, G., Wang, G. Strong Convergence and Asymptotic Exponential Stability of Modified Truncated EM Method for Neutral Stochastic Differential Equations with Time-dependent Delay. Front. Math 18, 1479–1504 (2023). https://doi.org/10.1007/s11464-021-0246-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-021-0246-9

Keywords

MSC2020

Search

Navigation