Potential Analysis

, 31:375 | Cite as

Pathwise Numerical Approximations of SPDEs with Additive Noise under Non-global Lipschitz Coefficients

  • Arnulf JentzenEmail author


We consider the pathwise numerical approximation of nonlinear parabolic stochastic partial differential equations (SPDEs) driven by additive white noise under local assumptions on the coefficients only. We avoid the standard global Lipschitz assumption in the literature on the coefficients by first showing convergence under global Lipschitz coefficients but with a strong error criteria and then by applying a localization technique for one sample path on a bounded set.


Parabolic stochastic partial differential equation Higher order approximation Strong error criteria Global Lipschitz Pathwise approximation 

Mathematics Subject Classifications (2000)

60H15 65C30 35K90 


  1. 1.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  2. 2.
    Grecksch, W., Kloeden, P.E.: Time-discretised Galerkin approximation of parabolic stochastic PDEs. Bull. Aust. Math. Soc. 54(1), 79–85 (1996)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Gyöngy, I., Nualart, D.: Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise. Stoch. Process. their Appl. 58(1), 57–72 (1995)zbMATHCrossRefGoogle Scholar
  4. 4.
    Gyöngy, I., Nualart, D.: Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise. Potential Anal. 7(4), 725–757 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gyöngy, I.: A note on Euler’s approximations. Potential Anal. 8(3), 205–216 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gyöngy, I.: Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise II. Potential Anal. 11(1), 1–37 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gyöngy, I., Krylov, N.: On the splitting-up method and stochastic partial differential equations. Ann. Probab. 31(2), 564–591 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gyöngy, I., Millet, A.: On discretization schemes for stochastic evolution equations. Potential Anal. 23(2), 99–134 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gyöngy, I., Millet, A.: Rate of convergence of space time approximations for stochastic evolution equations. Potential Anal. 30(1), 29–64 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hausenblas, E.: Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comput. Appl. Math. 147(2), 485–516 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hausenblas, E.: Approximation for semilinear stochastic evolution equations. Potential Anal. 18(2), 141–186 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hutzenthaler, M., Jentzen, A.: Non-globally Lipschitz Counterexamples for the stochastic Euler scheme. arXiv:0905.0273v1 (2009)
  13. 13.
    Jentzen, A., Kloeden, P.E., Neuenkirch, A.: Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients. Numer. Math. 112(1), 41–64 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jentzen, A., Neuenkirch, A.: A random Euler scheme for Carathéodory differential equations. J. Comput. Appl. Math. 224(1), 346–359 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Jentzen, A., Kloeden, P.E.: Overcoming the order barrier in the numerical approximation of SPDEs with additive space-time noise. Proc. R. Soc. A 465(2102), 649–667 (2009)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Kloeden, P.E., Shott, S.: Linear-implicit strong schemes for Itô-Galerkin approximation of stochastic PDEs. J. Appl. Math. Stoch. Anal. 14(1), 47–53 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lord, G.J., Rougemont, J.: A numerical scheme for stochastic PDEs with Gevrey regularity. IMA J. Numer. Anal. 54(4), 587–604 (2004)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Lord, G.J., Shardlow, T.: Post processing for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 45(2), 870–889 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Mishura, Y.S., Shevchenko, G.M.: Approximation schemes for stochastic differential equations in a Hilbert space. Theory Probab. Appl. 51(3), 442–458 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Müller-Gronbach, T., Ritter, K., Wagner, T.: Optimal pointwise approximation of a linear stochastic heat equation with additive space-time noise. In: Monte Carlo and Quasi Monte Carlo Methods 2006, pp. 577–589 (2007)Google Scholar
  21. 21.
    Müller-Gronbach, T., Ritter, K.: Lower bounds and nonuniform time discretization for approximation of stochastic heat equations. Found. Comput. Math. 7(2), 135–181 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Müller-Gronbach, T., Ritter, K.: An implicit Euler scheme with non-uniform time discretization for heat equations with multiplicative noise. BIT 47(2), 393–418 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Müller-Gronbach, T., Ritter, K., Wagner, T.: Optimal pointwise approximation of infinite-dimensional Ornstein-Uhlenbeck processes. Stoch. Dyn. 8(3), 519–541 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Pettersson, R., Signahl, M.: Numerical approximation for a white noise driven SPDE with locally bounded drift. Potential Anal. 22(4), 375–393 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Prévot, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Springer, Berlin (2007)zbMATHGoogle Scholar
  26. 26.
    Roman, L.J.: On numerical solutions of stochastic differential equations. PhD Thesis, University of Minnesota, Minnesota (2009)Google Scholar
  27. 27.
    Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Springer, New York (2002)zbMATHGoogle Scholar
  28. 28.
    Shardlow, T.: Numerical methods for stochastic parabolic PDEs. Numer. Funct. Anal. Optim. 20(1–2), 121–145 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (1997)zbMATHGoogle Scholar
  30. 30.
    Yan, Y.: Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43(4), 1363–1384 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Yoo, H.: Semi-discretization of stochastic partial differential equations on ℝ1 by a finite-difference method. Math. Comput. 69(230), 653–666 (1999)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Institute of MathematicsJohann Wolfgang Goethe-UniversityFrankfurt am MainGermany

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