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Potential Analysis

, Volume 30, Issue 1, pp 29–64 | Cite as

Rate of Convergence of Space Time Approximations for Stochastic Evolution Equations

Article

Abstract

Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators driven by a finite dimensional Brownian motion are considered. Under some regularity condition assumed for the solution, the rates of convergence of various numerical approximations are estimated under strong monotonicity and Lipschitz conditions. The abstract setting involves general consistency conditions and is then applied to a class of quasilinear stochastic PDEs of parabolic type.

Keywords

Stochastic evolution equations Monotone operators Coercivity Space time approximations Galerkin method Wavelets Finite elements 

Mathematics Subject Classifications (2000)

Primary 60H15 Secondary 65M60 

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References

  1. 1.
    Ciarlet, P.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, vol. 4. North Holland, Amsterdam (1978)MATHCrossRefGoogle Scholar
  2. 2.
    Cohen, A.: Numerical Analysis of Wavelet Methods. Studies in Mathematics and its Applications, vol. 32. Elsevier, Amsterdam (2003)MATHGoogle Scholar
  3. 3.
    Gyöngy, I., Krylov, N.V.: On stochastic equations with respect to semi-martingales II, Ito formula in Banach spaces. Stochastics 6, 153–173 (1982)MATHGoogle Scholar
  4. 4.
    Gyöngy, I.: On stochastic equations with respect to semimartingales III. Stochastics 7, 231–254 (1982)MathSciNetGoogle Scholar
  5. 5.
    Gyöngy, I.: Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise II. Potential Anal. 11, 1–37 (1999)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gyöngy, I., Martinez, T.: Solutions of partial differential equations as extremals of convex functionals. Acta Math. Hungar. 109, 127–145 (2005)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gyöngy, I., Millet, A.: On discretization schemes for stochastic evolution equations. Potential Anal. 23, 99–134 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gyöngy, I., Millet, A.: Rate of convergence of implicit approximations for stochastic evolution equations. In: Baxendale, P., Lototsky, S. (eds.) Stochastic Differential Equations: Theory and Applications (A volume in honor of Boris L. Rosovskii), vol. 2, pp. 281–310. World Scientific Interdisciplinary Mathematical Sciences. World Scientific, Singapore (2007)Google Scholar
  9. 9.
    Krylov, N.V., Rosovskii, B.L.: On Cauchy problem for linear stochastic partial differential equations. Math. USSR Izvestija 11(4), 1267–1284 (1977)MATHCrossRefGoogle Scholar
  10. 10.
    Krylov, N.V., Rosovskii, B.L.: Stochastic evolution equations. J. Sov. Math. 16, 1233–1277 (1981)MATHCrossRefGoogle Scholar
  11. 11.
    Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Études Mathématiques. Dunod Gauthiers-Villars (1969)Google Scholar
  12. 12.
    Mueller-Gronbach, T., Ritter, K.: An Implicit Euler Scheme with Non-uniform Time Discretization for Heat Equations with Multiplicative Noise (2006). arXiv math.PR/0604600
  13. 13.
    Pardoux, E.: Équations aux dérivées partielles stochastiques nonlinéares monotones. Étude de solutions fortes de type Itô, Thèse Doct. Sci. Math. Univ. Paris Sud (1975)Google Scholar
  14. 14.
    Pardoux, E.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3–2, 127–167 (1979)MathSciNetGoogle Scholar
  15. 15.
    Pardoux, E.: Filtrage non linéaire et équations aux derivées partielles stochastiques associées. In: École d’été de Probabilités de Saint-Flour 1989. Lecture Notes in Math., vol. 1464, pp. 67–163. Springer, New York (1981)Google Scholar
  16. 16.
    Rozovskii, B.: Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering. Kluwer, Dordrecht (1990)Google Scholar
  17. 17.
    Yan, Y.B.: Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise. Numer. Math. 44, 829–847 (2004)MATHCrossRefGoogle Scholar
  18. 18.
    Yoo, H.: An analytic approach to stochastic partial differential equations and its applications. Thesis, University of Minnesota (1998)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.School of MathematicsUniversity of EdinburghEdinburghUK
  2. 2.Maxwell Institute for Mathematical SciencesUniversity of EdinburghEdinburghUK
  3. 3.Laboratoire de Probabilités et Modèles AléatoiresUniversités Paris 6-Paris 7Paris Cedex 05France
  4. 4.SAMOS-MATISSE, Centre d’Économie de la SorbonneUniversité Paris 1 Panthéon SorbonneParis Cedex 13France

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