Skip to main content
SpringerLink
Log in

A numerical method for SDEs with discontinuous drift

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

In this paper we introduce a transformation technique, which can on the one hand be used to prove existence and uniqueness for a class of SDEs with discontinuous drift coefficient. One the other hand we present a numerical method based on transforming the Euler–Maruyama scheme for such a class of SDEs. We prove convergence of order \(1/2\). Finally, we present numerical examples.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Asmussen, S., Taksar, M.: Controlled diffusion models for optimal dividend pay-out. Insur. Math. Econ. 20(1), 1–15 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berkaoui, A.: Euler scheme for solutions of stochastic differential equations with non-Lipschitz coefficients. Port. Math. 61(4), 461–478 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Étoré, P., Martinez, M.: Exact simulation for solutions of one-dimensional stochastic differential equations involving a local time at zero of the unknown process. Monte Carlo Methods Appl. 19(1), 41–71 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Étoré, P., Martinez, M.: Exact simulation for solutions of one-dimensional stochastic differential equations with discontinuous drift. ESAIM: Probability and Statistics. (2014), accepted. arXiv:1301.3019

  5. Gyöngy, I.: A note on Euler’s approximation. Potential Anal. 8, 205–216 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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(1), 51–59 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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(4), 1611–1641 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. In: Graduate Texts in Mathematics, 2nd edn. Springer, New York (1991)

  9. Kloeden, P.E., Platen, E.: Numerical solutions of stochastic differential equations. In: Stochastic Modelling and Applied Probability. Springer, Berlin (1992)

  10. Kohatsu-Higa, A., Lejay, A., Yasuda, K.: Weak approximation errors for stochastic differential equations with non-regular drift. (2013), preprint. Inria, hal-00840211

  11. Leobacher, G., Szölgyenyi, M., Thonhauser, S.: On the existence of solutions of a class of SDEs with discontinuous drift and singular diffusion. Electron. Commun. Probab. 20(6), 1–14 (2015). arXiv:1311.6226

  12. Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Horwood Publishing Limited, New Delhi (2007)

  13. Veretennikov, A.Yu.: On strong solutions and explicit formulas for solutions of stochastic integral equations. Math. USSR Sb. 39(3), 387–403 (1981)

  14. Veretennikov, A.Yu.: On stochastic equations with degenerate diffusion with respect to some of the variables. Math. USSR Izv. 22(1), 173–180 (1984)

  15. Zvonkin, A.K.: A transformation of the phase space of a diffusion process that removes the drift. Math. USSR Sb. 22(129), 129–149 (1974)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors thank Evelyn Buckwar (Johannes Kepler University Linz) for valuable discussions.

Author information

Authors and Affiliations

  1. Department of Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz, 4040, Linz, Austria

    Gunther Leobacher & Michaela Szölgyenyi

Authors
  1. Gunther Leobacher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michaela Szölgyenyi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michaela Szölgyenyi.

Additional information

Communicated by David Cohen.

G. Leobacher and M. Szölgyenyi are supported by the Austrian Science Fund (FWF): Project F5508-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Leobacher, G., Szölgyenyi, M. A numerical method for SDEs with discontinuous drift. Bit Numer Math 56, 151–162 (2016). https://doi.org/10.1007/s10543-015-0549-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-015-0549-x

Keywords

Mathematics Subject Classification

Search

Navigation