Divergence of the backward Euler method for ordinary stochastic differential equations


This paper is based on the analysis of the backward Euler method for stochastic differential equations. It is motivated by the paper (Hutzenthaler et al. Proc. R. Soc. A 467, 1563–1576, 2011), where authors studied the equations with superlinearly growing coefficients. The main goal of this paper is to reveal sufficient conditions of the strong and weak Lp-divergence of the backward Euler method at finite time, for all \(p\in (0,\infty )\). Theoretical results are supported by examples.

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


  1. 1.

    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations Applications of Mathematics (New York), vol. 23. Springer, Berlin (1992)

    Google Scholar 

  2. 2.

    Milstein, G.N.: Numerical Integration of Stochastic Differential Equations Mathematics and Its Applications, vol. 313. Kluwer Academic Publishers Group, Dordrecht (1995)

    Google Scholar 

  3. 3.

    Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006)

    Google Scholar 

  4. 4.

    Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. A 467, 1563–1576 (2011)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations. Ann. Appl. Probab. 23, 1913–1966 (2013)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Hutzenthaler, M., Jentzen, A.: Convergence of the stochastic Euler scheme for locally Lipschitz coefficients. Found. Comput. Math. 11, 657–706 (2011)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Hutzenthaler, M., Jentzen, A.: Numerical approximations of stochastic differential equations with nonglobally Lipschitz continuous coefficients. Mem. Amer. Math. Soc. 236, 1112 (2015)

    MATH  Google Scholar 

  8. 8.

    Hutzenthaler, M., Jentzen A.: On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients. arXiv:1401.0295v1 (2014)

  9. 9.

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

    MathSciNet  Article  Google Scholar 

  10. 10.

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

    MathSciNet  Article  Google Scholar 

  11. 11.

    Milošević, M.: Implicit numerical methods for highly nonlinear neutral stochastic differential equations with time-dependent delay. Appl. Math. Comput. 244, 741–760 (2014)

    MathSciNet  MATH  Google Scholar 

  12. 12.

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

    MathSciNet  Article  Google Scholar 

  13. 13.

    Mattingly, J.C., Stuart, A.M., Higham, D.J.: Ergodicity for SDEs and approximations, locally Lipschitz vector fields and degenerate noise. Stochastic Process. Appl. 101, 185–232 (2002)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Higham, D.J., Mao, X., Stuart, A.M.: Exponential mean-square stability of numerical solutions to stochastic differential equations. LMS J. Comput. Math. 6, 297–313 (2003)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Roberts, G.O., Tweedie, R.L.: Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2, 341–363 (1996)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Mao, X., Szpruch, L.: Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients. Stochastics An International Journal of Probability and Stochastic Processes 85, 144–171 (2013)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Zhou, S., Xie, S., Fang, Z.: Almost sure exponential stability of the backward Euler-Maruyama discretization for highly nonlinear stochastic functional differential equation. Appl. Math. Comput. 236, 150–160 (2014)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Mao, X.: Stochastic differential equations and their applications, Chichester, UK (1997)

  19. 19.

    Khasminskii, R.Z.: Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Groningen, the Netherlands, 1980. Russian Translation Nauka, Moscow (1969)

    Google Scholar 

Download references


The author is very thankful to the reviewers for their valuable suggestions which improved the paper.

Author information



Corresponding author

Correspondence to Marija Milošević.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author’s research is supported by Grant No. 174007 of the Ministry of Science, Republic of Serbia.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Milošević, M. Divergence of the backward Euler method for ordinary stochastic differential equations. Numer Algor 82, 1395–1407 (2019). https://doi.org/10.1007/s11075-019-00661-6

Download citation


  • Ordinary stochastic differential equations
  • Backward Euler method
  • Strong L p-divergence
  • Super-linear growth conditions
  • One-sided Lipschitz condition

Mathematics Subject Classification (2010)

  • 60H10