Skip to main content
SpringerLink
Log in

Wong–Zakai Approximation for Stochastic Differential Equations Driven by G-Brownian Motion

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

In this paper, we build the Wong–Zakai approximation for Stratonovich-type stochastic differential equations driven by G-Brownian motion and obtain the quasi-sure convergence rate under Hölder norm by a rough path argument. As a corollary, we obtain the quasi-continuity of solutions of random rough differential equations driven by lifted martingales under a sequence of singular measures.

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. Bayer, C., Friz, P., Riedel, S., Schoenmakers, J.: From rough path estimates to multilevel monte carlo. SIAM J. Numer. Anal. 54(3), 1449–1483 (2016)

    Article  MathSciNet  Google Scholar 

  2. Denis, L., Hu, M., Peng, S.: Function spaces and capacity related to a sublinear expectation: application to \(G\)-Brownian motion paths. Potential Anal. 34, 139–161 (2011)

    Article  MathSciNet  Google Scholar 

  3. Deya, A., Neuenkirch, A., Tindel, S.: A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 48(2), 518–550 (2012)

    Article  MathSciNet  Google Scholar 

  4. Epstein, L.G., Ji, S.: Ambiguous volatility and asset pricing in continuous time. Rev. Financial Stud. 26(7), 1740–1786 (2013)

    Article  Google Scholar 

  5. Friz, P., Hairer, M.: A course on rough paths. With an Introduction Regularity Structures. Universitext, Springer (2014)

    Book  Google Scholar 

  6. Friz, P., Victoir, N.: Multidimensional Stochastic Processes as Rough Paths. Cambridge Studies in Advanced Mathematics, vol. 120. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  7. Hairer, M., Pardoux, É.: A Wong-Zakai theorem for stochastic PDEs. J. Math. Soc. Japan 67(4), 1551–1604 (2015)

    Article  MathSciNet  Google Scholar 

  8. Hu, M., Ji, S., Peng, S., Song, Y.: Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by \(G\)-Brownian motion. Stoch. Process Appl. 124(2), 1170–1195 (2014)

    Article  MathSciNet  Google Scholar 

  9. Gubinelli, M.: Controlling rough paths. J. Function. Anal. 216, 86–140 (2004)

    Article  MathSciNet  Google Scholar 

  10. Gubinelli, M.: Ramification of rough paths. J. Differ. Equ. 248(4), 693–721 (2010)

    Article  MathSciNet  Google Scholar 

  11. Geng, X., Qian, Z., Yang, D.: \(G\)-Brownian motion as rough paths and differential equations driven by \(G\)-Brownian motion. Séminaire de Probabilités XLVI, Lecture Notes in Mathematics 2123, (2014). https://doi.org/10.1007/978-3-319-11970-0_6. Springer, New York, London

  12. Kelly, D., Melbourne, I.: Smooth approximation of stochastic differential equations. Ann. Probab. 44(1), 479–520 (2016)

    Article  MathSciNet  Google Scholar 

  13. Lyons, T.: Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14(2), 215–310 (1998)

    Article  MathSciNet  Google Scholar 

  14. Lyons, T., Qian, Z., et al.: System Control and Rough Paths. Oxford University Press, Oxford (2002)

    Book  Google Scholar 

  15. Peng, S.: \(G\)-expectation, \(G\)-Brownian motion and related stochastic calculus of Itô type. In: Stochastic Analysis and Applications, Abel Symp. Springer, Berlin 2, 541–567 (2007)

  16. Peng, S.: Multi-dimensional \(G\)-Brownian motion and related stochastic calculus under \(G\)-expectation. Stochast. Process. Appl. 118(12), 2223–2253 (2008)

    Article  MathSciNet  Google Scholar 

  17. Peng, S.: Nonlinear expectations and stochastic calculus under uncertainty. With Robust CLT and \(G\)-Brownian Motion. Probability Theory and Stochastic Modelling, Springer (2019)

  18. Jin, H., Peng, S.: Optimal Unbiased Estimation for Maximal Distribution. Available at arXiv 1611, 07994 (2016)

  19. Peng, S., Zhou, Q.: A hypothesis-testing perspective on the \(G\)-normal distribution theory. Stat. Prob. Lett. 156, 108623 (2020)

    Article  MathSciNet  Google Scholar 

  20. Peng, S., Zhang, H.: Stochastic calculus with respect to \(G\)-Brownian motion viewed through rough paths. Sci. China Math. 60, 1–20 (2016)

    Article  MathSciNet  Google Scholar 

  21. Stroock, D. W. and Varadhan, S. R. S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), vol. III: Probability Theory, Univ. California Press, Berkeley, CA, pp. 333–359 (1972)

  22. Vorbrink, J.: Financial markets with volatility uncertainty. J. Math. Econ. 53, 64–78 (2014)

    Article  MathSciNet  Google Scholar 

  23. Wong, E., Zakai, M.: On the relationship between ordinary and stochastic differential equations. Int. J. Eng. Sci. 3, 213–29 (1965)

    Article  Google Scholar 

  24. Wong, E., Zakai, M.: On the convergence of oridinary integrals to stochastic integrals. Ann. Math. Stat. 36, 1560–1564 (1965)

    Article  Google Scholar 

Download references

Acknowledgements

H.Z. is supported by the Youth Program of National Natural Science Foundation No. 11901104 and Postdoctoral Science Foundation of China.

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, China

    Shige Peng

  2. School of Mathematics, Fudan University, Shanghai, China

    Huilin Zhang

Authors
  1. Shige Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huilin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huilin Zhang.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peng, S., Zhang, H. Wong–Zakai Approximation for Stochastic Differential Equations Driven by G-Brownian Motion . J Theor Probab 35, 410–425 (2022). https://doi.org/10.1007/s10959-020-01058-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-020-01058-1

Keywords

Mathematics Subject Classification (2020)

Search

Navigation