Skip to main content
SpringerLink
Log in

Almost sure stability with general decay rate of exact and numerical solutions for stochastic pantograph differential equations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The exponential stability and the polynomial stability have been well studied for the exact and numerical solutions of the stochastic pantograph differential equations (SPDEs), while the result on the general decay stabilities is insufficient for SPDEs. In this paper, the sufficient conditions of the almost sure stability with general decay rate of the exact and numerical solutions for SPDEs have been considered. The stability of two different theta numerical methods, namely the split–step theta method and the stochastic linear theta method, have been discussed respectively. From the conditions, we established for the two theta approximations to reproduce the stability of the exact solution, and we see that the conditions for \(\theta \in \left [0,\frac {1}{2}\right ]\) are stronger than those for \(\theta \in \left (\frac {1}{2},1\right ]\). The bound of the Lyapunov exponent of the split-step theta method is little bigger than that of the stochastic linear theta method for sufficiently small step size. To illustrate the theoretical results, we give two examples to examine the almost sure exponential stability.

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

Similar content being viewed by others

References

  1. Appleby, J.A.D., Buckwar, E.: Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation. Electronic Journal of Qualitative Theory of Differential Equations. (2), 1–32 (2016)

  2. Fan, Z., Liu, M., Cao, W.: Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations. J. Math. Anal. Appl. 325(2), 1142–1159 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Gard, T.C.: Introduction to Stochastic Differential Equations. Dekker, New York (1988)

    MATH  Google Scholar 

  4. Guo, P., Li, C.: Almost sure exponential stability of numerical solutions for the stochastic pantograph differential equations. J. Math. Anal. Appl. 460, 411–424 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Haghighi, A., Hosseini, S.M.: A class of split-step balanced methods for stiff stochastic differential equations. Numerical Algorithms 61(1), 141–162 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Higham, D.J., Mao, X., Stuart, A.M.: Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40(3), 1041–1063 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Higham, D.J., Mao, X., Yuan, C.: Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SIAM J. Numer. Anal. 45(2), 592–609 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hu, L., Mao, X., Shen, Y.: Stability and boundedness of nonlinear hybrid stochastic differential delay equations. Syst. Control Lett. 62(2), 178–187 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, C.: Exponential mean square stability of numerical methods for systems of stochastic differential equations. J. Comput. Appl. Math. 236(16), 4016–4026 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Huang, C.: Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations. J. Comput. Appl. Math. 259, 77–86 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Janković, S., Randjelović, J., Jovanović, M.: Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations. J. Math. Anal. Appl. 355(2), 811–820 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Khasminskii, R.: Stochastic stability of differential equations. Springer Science & Business Media, Berlin (2011)

    Google Scholar 

  13. Liu, W., Mao, X.: Almost sure stability of the Euler-Maruyama method with random variable stepsize for stochastic differential equations. Numerical Algorithms 74(2), 573–592 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mahmoud, A., Xiao, Y., Tian, B.: Convergence and stability of two classes of theta methods with variable step size for a stochastic pantograph differential equations. J. Adv. Math. Appl. 5(2), 95–106 (2016)

    Article  Google Scholar 

  15. Mao, X.: Stochastic differential equations and applications. Horvood, Chichester (2007)

    MATH  Google Scholar 

  16. Mao, X.: Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions. Appl. Math. Comput. 217(12), 5512–5524 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Mao, X., Lam, J., Xu, S., et al.: Razumikhin method and exponential stability of hybrid stochastic delay interval systems. J. Math. Anal. Appl. 314(1), 45–66 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mao, X., Rassias, M.J.: Khasminskii-type theorems for stochastic differential delay equations. Stoch. Anal. Appl. 23(5), 1045–1069 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Milošević, M.: Existence, uniqueness, almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the Euler-Maruyama approximation. Appl. Math. Comput. 237, 672–685 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Pavlović, G., Janković, S.: Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay. J. Comput. Appl. Math. 236(7), 1679–1690 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Schurz, H.: Almost sure asymptotic stability and convergence of stochastic theta methods applied to systems of linear SDEs in \(R^{d}\). Random Operators and Stochastic Equations 19(2), 111–129 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Song, M., Lu, Y., Liu, M.: Stability of analytical solutions and convergence of numerical methods for non-linear stochastic pantograph differential equations. arXiv:1502.00061 (2015)

  23. Wu, F., Hu, S.: Razumikhin-type theorems on general decay stability and robustness for stochastic functional differential equations. Int. J. Robust Nonlinear Control 22(7), 763–777 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wu, F., Mao, X., Szpruch, L.: Almost sure exponential stability of numerical solutions for stochastic delay differential equations. Numer. Math. 115(4), 681–697 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Xiao, Y., Mahmoud, A., Tian, B.: Convergence and stability of split-step theta methods with variable step-size for stochastic pantograph differential equations. Int. J. Comput. Math. 95(5), 939–960 (2017)

    Article  MathSciNet  Google Scholar 

  26. Xiao, Y., Song, M., Liu, M.: Convergence and stability of the semi-implicit euler method with variable stepsize for a linear stochastic pantograph differential equation. Int. J. Numer. Anal. Model 8, 214–225 (2011)

    MathSciNet  MATH  Google Scholar 

  27. You, S., Mao, W., Mao, X., et al.: Analysis on exponential stability of hybrid pantograph stochastic differential equations with highly nonlinear coefficients. Appl. Math. Comput. 263, 73–83 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Zhang, H., Xiao, Y., Guo, F.: Convergence and stability of a numerical method for nonlinear stochastic pantograph equations. J. Frankl. Inst. 351(6), 3089–3103 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zong, X., Wu, F., Huang, C.: Preserving exponential mean square stability and decay rates in two classes of theta approximations of stochastic differential equations. J. Differ. Equ. Appl. 20(7), 1091–1111 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work was supported by the National Natural Science Foundation of China (Nos. 11471066 and 11572081) and the Fundamental Research Funds for the Central Universities (DUT17JC07, DUT17GJ102).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, Liaoning, China

    Ping Guo & Chong–Jun Li

Authors
  1. Ping Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chong–Jun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chong–Jun Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, P., Li, C. Almost sure stability with general decay rate of exact and numerical solutions for stochastic pantograph differential equations. Numer Algor 80, 1391–1411 (2019). https://doi.org/10.1007/s11075-018-0531-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-018-0531-1

Keywords

Mathematics Subject Classification (2010)

Search

Navigation