Skip to main content
SpringerLink
Log in

Estimate of transition kernel for Euler-Maruyama scheme for SDEs driven by \(\alpha \)-stable noise and applications

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

Abstract

In this paper, the discrete parametrix method is adopted to investigate the estimation of transition kernel for Euler-Maruyama scheme SDEs driven by \(\alpha \)-stable noise, which implies Krylov’s estimate and Khasminskii’s estimate. As an application, the convergence rate of Euler-Maruyama scheme for a class of multidimensional SDEs with singular drift (by the use of Zvonkin’s transformation) is obtained.

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

Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Bao, J., Huang, X., Yuan, C.: Convergence rate of Euler-Maruyama Scheme for SDEs with Hölder-Dini continuous drifts. J. Theor. Probab. 32, 848–871 (2019)

    Article  MATH  Google Scholar 

  2. Bao, J., Huang, X., Zhang, S.: Convergence rate of EM algorithm for SDEs under integrability condition. J. Appl. Probab. 1–24 (2022) https://doi.org/10.1017/jpr.2021.56

  3. Blumenthal, R.M., Getoor, R.K.: Some theorems on stable processes. Trans. Am. Math. Soc. 95, 263–273 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, Z.-Q., Zhang, X.: Heat kernels and analyticity of non-symmetric jump diffusion semigroups. Probab. Theory Relat. Fields. 165, 267–312 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Z.-Q., Zhang, X.: Heat kernels for time-dependent non-symmetric stable-like operators. J. Math. Anal. Appl. 465, 1–21 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dareiotis, K., Kumar, C., Sabanis, S.: On tamed Euler approximations of SDEs driven by Lévy noise with applications to delay equations. SIAM J. Numer. Anal. 54, 1840–1872 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Flandoli, M., Gubinelli, M., Priola, E.: Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift. Bull. Sci. Math. 134, 405–422 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gottlich, S., Lux, K., Neuenkirch, A.: The Euler scheme for stochastic differential equations with discontinuous drift coefficient: a numerical study of the convergence rate. Adv. Differ. Equ. 429, 1–21 (2019)

  9. Gyöngy, I., Martinez, T.: On stochastic differential equations with locally unbounded drift. Czechoslovak Math. J. 51, 763–783 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Halidias, N., Kloeden, P.E.: A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient. BIT Numer. Math. 48, 51–59 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Higham, D.J., Kloeden, P.E.: Strong convergence rates for backward Euler on a class of nonlinear jump diffusion problems. J. Comput. Appl. Math. 205, 949–956 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang, X., Liao, Z.: The Euler-Maruyama method for S(F)DEs with Hölder drift and \(\alpha \)-stable noise. Stoch. Anal. Appl. 36, 28–39 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang, X., Wang, F.-Y.: Degenerate SDEs with singular drift and applications to Heisenberg groups. J. Differ. Equ. 265, 2745–2777 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Knopova, V., Kulik, A.: Parametrix construction of the transition probability density of the solution to an SDE driven by \(\alpha \)-stable noise. Ann. Inst. H. Poincaré Probab. Statist. 54, 100–140 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Konakov, V., Mammen, E.: Local limits theorems for transition densities of Markov chains converging to diffusions. Probab. Theory Relat. Fields. 117, 551–587 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Konakov, V., Menozzi, S.: Weak error for stable driven stochastic differential equations: expansion of densities. J. Theor. Probab. 24, 454–478 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kloden, P.E., Platen, E.: Numerical solutions of stochastic differential equations. Springer, Berlin-Heidelberg (1992)

    Book  Google Scholar 

  18. Krylov, N.V., Röckner, M.: Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields. 131, 154–196 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kühn, F., Schilling, R.-L.: Strong convergence of the Euler-Maruyama approximation for a class of Lévy-driven SDEs. Stoch. Process Their Appl. 129, 2654–2680 (2019)

    Article  MATH  Google Scholar 

  20. Lemaire, V., Menozzi, S.: On some non asymptotic bounds for the Euler scheme. Electron. J. Probab. 15, 1645–1681 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Leobacher, G., Szölgyenyi, M.: A numerical method for SDEs with discontinuous drift. BIT Numer. Math. 56, 151–162 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Leobacher, G., Szölgyenyi, M.: A strong order 1/2 method for multidimensional SDEs with discontinuous drift. Ann. Appl. Probab. 27, 2383–2418 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Leobacher, G., Szölgyenyi, M.: Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient. Numer. Math. 138, 219–239 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Pamen, O.M., Taguchi, D.: Strong rate of convergence for the Euler-Maruyama approximation of SDE with Hölder continuous drift coefficient. Stoch. Process Their Appl. 127, 2542–2559 (2017)

    Article  MATH  Google Scholar 

  26. Triebel, H.: Interpolation theory, Functional Spaces. North-Holland Publishing company, Differential Operators (1978)

    MATH  Google Scholar 

  27. Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton University Press (1970)

  28. Wang, F.-Y., Wang, J.: Harnack inequalities for stochastic equations driven by Lévy noise. J. Math. Anal. Appl. 410, 513–523 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Xie, L., Zhang, X.: Ergodicity of stochastic differential equations with jumps and singular coefficients. Ann. Inst. H. Poincaré Probab. Statist. 56, 175–229 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang, X.: Strong solutions of SDEs with singural drift and Sobolev diffusion coefficients. Stoch. Process Their Appl. 115, 1805–1818 (2005)

    Article  MATH  Google Scholar 

  31. Zhang, X.: Stochastic differential equations with Sobolev drifts and driven by \(\alpha \)-stable processes. Ann. Inst. H. Poincaré Probab. Statist. 49, 1057–1079 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, X.: Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients. Electron. J. Probab. 16, 1096–1116 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, X.: Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients. Rev. Mat. Iberoam. 29, 25–52 (2013)

  34. Zvonkin, A.K.: A transformation of the phase space of a diffusion process that removes the drift. Math. USSR Sb. 93, 129–149 (1974)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

No other acknowledgement is presented.

Author information

Authors and Affiliations

  1. Center for Applied Mathematics, Tianjin University, 300072, Tianjin, China

    Xing Huang

  2. School of Mathematics, Nanjing University of Aeronautics and Astronautics, 211106, Nanjing, China

    Yongqiang Suo

  3. Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, Swansea, UK

    Chenggui Yuan

Authors
  1. Xing Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongqiang Suo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chenggui Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors reviewed the manuscript.

Corresponding author

Correspondence to Yongqiang Suo.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Ethical Approval and Consent to participate

The manuscript is original and have not been published elsewhere in any form or language (partially or in fully).

Consent for Publication

All of the authors are agree for publication.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, X., Suo, Y. & Yuan, C. Estimate of transition kernel for Euler-Maruyama scheme for SDEs driven by \(\alpha \)-stable noise and applications. Numer Algor 94, 1381–1402 (2023). https://doi.org/10.1007/s11075-023-01539-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-023-01539-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation