Rate of Convergence for Wong–Zakai-Type Approximations of Itô Stochastic Differential Equations

  • Bilel Kacem Ben Ammou
  • Alberto Lanconelli


We consider a class of stochastic differential equations driven by a one-dimensional Brownian motion, and we investigate the rate of convergence for Wong–Zakai-type approximated solutions. We first consider the Stratonovich case, obtained through the pointwise multiplication between the diffusion coefficient and a smoothed version of the noise; then, we consider Itô equations where the diffusion coefficient is Wick-multiplied by the regularized noise. We discover that in both cases the speed of convergence to the exact solution coincides with the speed of convergence of the smoothed noise toward the original Brownian motion. We also prove, in analogy with a well-known property for exact solutions, that the solutions of approximated Itô equations solve approximated Stratonovich equations with a certain correction term in the drift.


Stochastic differential equations Wong–Zakai theorem Wick product 

Mathematics Subject Classification (2010)

60H10 60H30 60H05 



The author acknowledges the support of the Italian INDAM-GNAMPA.


  1. 1.
    Bogachev, V.I.: Gaussian Measures. American Mathematical Society, Providence (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Brezniak, Z., Flandoli, F.: Almost sure approximation of Wong–Zakai type for stochastic partial differential equations. Stoch. Proc. Appl. 55, 329–358 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Da Pelo, P., Lanconelli, A.: On a new probabilistic representation for the solution of the heat equation. Stochastics 84, 171–181 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Da Pelo, P., Lanconelli, A., Stan, A.I.: A Hölder–Young–Lieb inequality for norms of Gaussian Wick products. Inf. Dim. Anal. Quantum Prob. Relat. Top. 14, 375–407 (2011)CrossRefzbMATHGoogle Scholar
  5. 5.
    Da Pelo, P., Lanconelli, A., Stan, A.I.: An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems. Stoch. Proc. Appl. 123, 3183–3200 (2013)CrossRefzbMATHGoogle Scholar
  6. 6.
    Gyöngy, I., Michaletzky, G.: On Wong–Zakai approximations with \(\delta \)-martingales. Proc. Math. Phys. Eng. Sci. 460, 309–324 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gyöngy, I., Shmatkov, A.: Rate of convergence of Wong–Zakai approximations for stochastic partial differential equations. Appl. Math. Optim. 54, 315–341 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gyöngy, I., Stinga, P.R.: Rate of convergence of Wong–Zakai approximations for stochastic partial differential equations. In: Seminar on Stochastic Analysis, Random Fields and Applications VII, vol. 67. Birkhäuser, Basel (2013)Google Scholar
  9. 9.
    Hairer, M., Pardoux, E.: A Wong–Zakai theorem for stochastic PDEs. J. Math. Soc. Jpn. 67, 1551–1604 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Holden, H., Øksendal, B., Ubøe, J., Zhang, T.-S.: Stochastic Partial Differential Equations, II edn. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hu, Y.: Analysis on Gaussian Spaces. World Scientific Publishing Co Pte. Ltd., Hackensack (2017)zbMATHGoogle Scholar
  12. 12.
    Hu, Y., Kallianpur, G., Xiong, J.: An approximation for Zakai equation. Appl. Math. Optim. 45, 23–44 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hu, Y., Nualart, D.: Rough path analysis via fractional calculus. Trans. Am. Math. Soc. 361, 2689–2718 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hu, Y., Øksendal, B.: Wick approximation of quasilinear stochastic differential equations. In: Körezlioğlu, H., Øksendal, B., Üstünel, A.S. (eds.) Stochastic Analysis and Related Topics, vol. V, pp. 203–231, Birkhäuser, Basel (1996)Google Scholar
  15. 15.
    Janson, S.: Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics, vol. 129. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  16. 16.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991)zbMATHGoogle Scholar
  17. 17.
    Konecny, F.: A Wong–Zakai approximation of stochastic differential equations. J. Multivar. Anal. 13, 605–611 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lanconelli, A., Stan, A.I.: Some norm inequalities for Gaussian Wick products. Stoch. Anal. Appl. 28, 523–539 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Londono, J.A., Villegas, A.M.: Numerical performance of some Wong–Zakai type approximations for stochastic differential equations. Int. J. Pure Appl. Math. 107, 301–315 (2016)Google Scholar
  20. 20.
    Naganuma, N.: Exact convergence rate of the Wong–Zakai approximation to RDEs driven by Gaussian rough paths. Stochastics 7, 1041–1059 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nualart, D.: Malliavin Calculus and Related Topics, II edn. Springer, New York (2006)zbMATHGoogle Scholar
  22. 22.
    Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proceedings 6th Berkeley Symposium Mathematical Statistics and Probability, vol. 3, pp. 333–359. University of California Press, Berkeley (1972)Google Scholar
  23. 23.
    Tessitore, G., Zabczyk, G.J.: Wong–Zakai approximations of stochastic evolution equations. J. Evol. Equ. 6, 621–655 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wong, E., Zakai, M.: On the relation between ordinary and stochastic differential equations. Int. J. Eng. Sci. 3, 213–229 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wong, E., Zakai, M.: Riemann–Stieltjes approximations of stochastic integrals. Z. Wahrscheinlichkeitstheorie verw. Geb. 12, 87–97 (1969)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Tunis - El ManarNabeulTunisia
  2. 2.Dipartimento di MatematicaUniversitá degli Studi di Bari Aldo MoroBariItaly

Personalised recommendations