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A Short Review of the Malliavin Calculus in Hilbert Spaces

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2093)

Abstract

So far, the Malliavin calculus has been rarely used in articles covering the numerical analysis of stochastic processes, in particular for SPDEs, and we find it appropriate to provide a gentle and self-contained introduction into this theory.

Keywords

  • Divergence Operator
  • Wiener Process
  • Derivative Operator
  • Part Formula
  • Stochastic Evolution Equation

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. C. Carstensen, An adaptive mesh-refining algorithm allowing for an H 1 stable L 2 projection onto Courant finite element spaces. Constr. Approx. 20(4), 549–564 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. D.L. Cohn, Measure Theory (Birkhäuser, Boston [u.a.], 1993)

    Google Scholar 

  3. G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44 (Cambridge University Press, Cambridge, 1992)

    Google Scholar 

  4. A. Grorud, É. Pardoux, Intégrales Hilbertiennes anticipantes par rapport à un processus de Wiener cylindrique et calcul stochastique associé. Appl. Math. Optim. 25(1), 31–49 (1992)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. J.A. León, D. Nualart, Stochastic evolution equations with random generators. Ann. Probab. 26(1), 149–186 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, in Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) (Wiley, New York, 1978), pp. 195–263

    Google Scholar 

  7. D. Nualart, The Malliavin Calculus and Related Topics, 2nd edn. Probability and Its Applications (New York) (Springer, Berlin, 2006)

    MATH  Google Scholar 

  8. D. Nualart, Application of Malliavin calculus to stochastic partial differential equations, in A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1962 (Springer, Berlin, 2009), pp. 73–109

    Google Scholar 

  9. M. Veraar, The stochastic Fubini theorem revisited. Stochastics 84(4), 543–551 (2012)

    MathSciNet  MATH  Google Scholar 

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Kruse, R. (2014). A Short Review of the Malliavin Calculus in Hilbert Spaces. In: Strong and Weak Approximation of Semilinear Stochastic Evolution Equations. Lecture Notes in Mathematics, vol 2093. Springer, Cham. https://doi.org/10.1007/978-3-319-02231-4_4

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