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


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.

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  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). 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 

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

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Correspondence to Huilin Zhang.

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Peng, S., Zhang, H. Wong–Zakai Approximation for Stochastic Differential Equations Driven by G-Brownian Motion . J Theor Probab 35, 410–425 (2022).

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