Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Convergence, Non-negativity and Stability of a New Tamed Euler–Maruyama Scheme for Stochastic Differential Equations with Hölder Continuous Diffusion Coefficient

  • 31 Accesses

Abstract

We propose and analyze a new tamed Euler–Maruyama approximation scheme for stochastic differential equations with Hölder continuous diffusion. This new scheme preserves the stability and non-negativity of the exact solution.

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

Fig. 1
Fig. 2

References

  1. 1.

    Bao, J., Yuan, C.: Convergence rate of EM scheme for SDDEs. Proc. Am. Math. Soc. 141, 3231–3243 (2013)

  2. 2.

    Gyöngy, I., Rásonyi, M.: A note on Euler approximations for SDEs with hölder continuous diffusion coefficients. Stoch. Proc. Appl. 121, 2189–2200 (2011)

  3. 3.

    Higham, D.J.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38, 753–769 (2000)

  4. 4.

    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)

  5. 5.

    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, 592–609 (2007)

  6. 6.

    Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Probab. 22, 1611–1641 (2012)

  7. 7.

    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)

  8. 8.

    Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, vol. 113. Springer, New York (1998)

  9. 9.

    Khasminskii, R.: Stochastic Stability of Differential Equations, 2nd edn. Stochastic Modelling and Applied Probability, vol. 66. Springer, Berlin (2012)

  10. 10.

    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Stochastic Modelling and Applied Probability, vol. 23. Springer, Berlin (1992)

  11. 11.

    Mao, X.: Stochastic Differential Equations and Their Applications. Horwood Publishing Series in Mathematics & Applications. Horwood Publishing Limited, Chichester (1997)

  12. 12.

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

  13. 13.

    Mao, X., Szpruch, L.: Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients. J. Comput. Appl. Math. 238, 14–28 (2013)

  14. 14.

    Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Scientific Computation. Springer, Berlin (2004)

  15. 15.

    Ngo, H.-L., Luong, D.-T.: Strong rate of tamed Euler–Maruyama approximation for stochastic differential equations with hölder continuous diffusion coefficient. Braz. J. Probab. Stat. 31, 24–40 (2017)

  16. 16.

    Ngo, H.-L., Taguchi, D.: On the Euler–Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients. IMA J. Numer. Anal. 37, 1864–1883 (2017)

  17. 17.

    Ngo, H.-L., Taguchi, D.: Strong rate of convergence for the Euler–Maruyama approximation of stochastic differential equations with irregular coefficients. Math. Comp. 85, 1793–1819 (2016)

  18. 18.

    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Grundlehren Der Mathematischen Wissenschaften, vol. 293. Springer, Berlin (1999)

  19. 19.

    Sabanis, S.: A note on tamed Euler approximations. Electron. Commun. Probab. 18(47), 10 (2013)

  20. 20.

    Sabanis, S.: Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients. Ann. Appl. Probab. 26, 2083–2105 (2016)

  21. 21.

    Saito, Y., Mitsui, T.: Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 33, 2254–2267 (1996)

  22. 22.

    Szpruch, Ł., Zhang, X.: V-integrability, asymptotic stability and comparison property of explicit numerical schemes for non-linear SDEs. Math. Comp. 87, 755–783 (2018)

  23. 23.

    Yamada, T., Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11, 155–167 (1971)

  24. 24.

    Zong, X., Wu, F., Huang, C.: Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients. Appl. Math. Comput. 228, 240–250 (2014)

Download references

Acknowledgments

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03-2017.316. The paper was completed during a scientific stay of the third author at the Vietnam Institute for Advanced Study in Mathematics (VIASM), whose hospitality is gratefully appreciated.

The authors thank the anonymous referees and Dai Taguchi for their valuable suggestions and comments.

Author information

Correspondence to Hoang-Long Ngo.

Additional information

Publisher’s Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kieu, T., Luong, D., Ngo, H. et al. Convergence, Non-negativity and Stability of a New Tamed Euler–Maruyama Scheme for Stochastic Differential Equations with Hölder Continuous Diffusion Coefficient. Vietnam J. Math. 48, 107–124 (2020). https://doi.org/10.1007/s10013-019-00373-3

Download citation

Keywords

  • Exponential stability
  • Hölder continuous diffusion
  • Non-negativity
  • Stochastic differential equation
  • Tamed Euler–Maruyama approximation

Mathematics Subject Classification (2010)

  • 65C30
  • 65L20
  • 60H10