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Stochastic Equations for Complex Systems pp 1–36Cite as

An Introduction to the Malliavin Calculus and Its Applications

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Abstract

The purpose of these notes is to provide an introduction to the Malliavin calculus and its recent application to quantitative results in normal approximations, in combination with Stein’s method. The basic differential operators of the Malliavin calculus and their main properties are presented. We explain the connection of these operators with the Wiener chaos expansion and the Ornstein-Uhlenbeck semigroup. We survey several applications of the Malliavin calculus including stochastic integral representation, density formulas, smoothness of densities and normal approximations.

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References

  1. Bally V, Caramellino L (2011) Riesz transform and integration by parts formulas for random variables. Stoch Process Appl 121:1332–1355

    Article  MATH  MathSciNet  Google Scholar 

  2. Bally V, Caramellino L (2013) Positivity and lower bounds for the density of Wiener functionals. Potential Anal 39:141–168

    Article  MATH  MathSciNet  Google Scholar 

  3. Bismut JM (1981) Martingales, the Malliavin calculus and hypoellipticity under general Hörmander’s conditions. Z Wahrsch Verw Gebiete 56:469–505

    Article  MATH  MathSciNet  Google Scholar 

  4. Bismut JM (1984) Large deviations and the Malliavin calculus. Progress in mathematics, vol 45. Birkhäuser, Boston

    Google Scholar 

  5. Bouleau N, Hirsch F (1991) Dirichlet forms and analysis on Wiener space. Walter de Gruyter & Co., Berlin

    Book  MATH  Google Scholar 

  6. Breuer P, Major P (1983) Central limit theorems for nonlinear functionals of Gaussian fields. J Multivar Anal 13:425–441

    Article  MATH  MathSciNet  Google Scholar 

  7. Carbery A, Wright J (2001) Distributional and \(L^q\) norm inequalities for polynomials over convex bodies in \(\mathbb{R}^n\). Math Res Lett 8:233–248

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen L, Goldstein L, Shao QM (2011) Normal approximation by Stein’s method. Springer, Berlin

    Book  MATH  Google Scholar 

  9. Dawson DA, Li Z, Wang H (2001) Superprocesses with dependent spatial motion and general branching densities. Electron J Probab 6:1–33

    Article  MathSciNet  Google Scholar 

  10. Gaveau B, Trauber P (1982) L’intégrale stochastique comme opérateur de divergence dans l’espace fonctionnel. J Funct Anal 46:230–238

    Article  MATH  MathSciNet  Google Scholar 

  11. Hu Y, Lu F, Nualart D (2013) Hölder continuity of the solutions for a class of nonlinear SPDE’s arising from one dimensional superprocesses. Probab Theory Relat Fields 156:27–49

    Article  MATH  MathSciNet  Google Scholar 

  12. Hu Y, Lu F, Nualart D (2014) Convergence of densities for some functionals of Gaussian processes. J Funct Anal 266:814–875

    Article  MATH  MathSciNet  Google Scholar 

  13. Hu Y, Nualart D, Tindel S, Xu F, Density convergence in the Breuer-major theorem for Gaussian stationary sequences. Bernoulli (to appear)

    Google Scholar 

  14. Kusuoka S, Stroock D (1984) Applications of the Malliavin calculus. I. Stochastic analysis (Katata/Kyoto, 1982). North-Holland, Amsterdam

    Google Scholar 

  15. Kusuoka S, Stroock D (1985) Applications of the Malliavin calculus. II. J Fac Sci Univ Tokyo Sect IA Math 32:1–76

    MATH  MathSciNet  Google Scholar 

  16. Kusuoka S, Stroock D (1987) Applications of the Malliavin calculus. III. J Fac Sci Univ Tokyo Sect IA Math 34:391–442

    MATH  MathSciNet  Google Scholar 

  17. Li Z, Wang H, Xiong J, Zhou X (2011) Joint continuity for the solutions to a class of nonlinear SPDE. Probab Theory Relat Fields 153(3–4):441–469

    MathSciNet  Google Scholar 

  18. Malliavin P (1978) Stochastic calculus of variation and hypoelliptic operators. In: Proceedings of the international symposium on stochastic differential equations, Kyoto, 1976. Wiley, New York-Chichester-Brisbane, pp 195-263

    Google Scholar 

  19. Malliavin P (1997) Stochastic analysis. Grundlehren der Mathematischen Wissenschaften, vol 313. Springer, Berlin

    Google Scholar 

  20. Meyer PA (1984) Transformations de Riesz pour les lois gaussiennes. Seminaire de Probabilités XVIII. Lecture notes in mathematics, vol 1059. Springer, Berlin, pp 179–193

    Google Scholar 

  21. Nourdin I, Nualart D, Fisher information and the fourth moment problem. Annals Inst Henri Poincaré (to appear)

    Google Scholar 

  22. Nourdin I, Peccati G (2012) Normal approximations with Malliavin calculus. From Stein’s method to universality. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  23. Nourdin I, Viens F (2009) Density formula and concentration inequalities with Malliavin calculus. Electron J Probab 14:2287–2309

    Article  MATH  MathSciNet  Google Scholar 

  24. Nualart D (1998) Analysis on Wiener space and anticipating stochastic calculus. Lectures on probability theory and statistics (Saint-Flour, 1995). Lecture notes in mathematics, vol 1690. Springer, Berlin, pp 123–227

    Google Scholar 

  25. Nualart D (2006) The Malliavin calculus and related topics, 2nd edn. Springer, New York

    MATH  Google Scholar 

  26. Nualart D (2009) Malliavin calculus and its applications. CBMS regional conference series in mathematics, vol 110. AMS & NSF

    Google Scholar 

  27. Nualart D, Ortiz-Latorre S (2007) Central limit theorems for multiple stochastic integrals. Stoch Process Appl 118:614–628

    Article  MathSciNet  Google Scholar 

  28. Nualart D, Pardoux E (1988) Stochastic calculus with anticipating integrands. Probab Theory Relat Fields 78:535–581

    Article  MATH  MathSciNet  Google Scholar 

  29. Nualart D, Peccati G (2005) Central limit theorems for sequences of multiple stochastic integrals. Ann Probab 33:177–193

    Article  MATH  MathSciNet  Google Scholar 

  30. Ocone D (1984) Malliavin calculus and stochastic integral representation of diffusion processes. Stochastics 12:161–185

    Article  MATH  MathSciNet  Google Scholar 

  31. Peccati G, Tudor C (2005) Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités XXXVIII. Lecture notes in mathematics, vol 1857. Springer, Berlin, pp 247–262

    Google Scholar 

  32. Pisier G (1988) Riesz transforms. A simple analytic proof of P.A. Meyer’s inequality. Seminaire de Probabilités XXIII. Lecture notes in mathematics, vol 1321. Springer, Berlin, pp 485–501

    Google Scholar 

  33. Skorohod AV (1975) On a generalization of a stochastic integral. Theory Probab Appl 20:219–233

    Article  Google Scholar 

  34. Stroock DW (1983) Some applications of stochastic calculus to partial differential equations. In: Eleventh Saint Flour probability summer school 1981 (Saint Flour, 1981). Lecture notes in mathematics, vol 976. Springer, Berlin, pp 267–382

    Google Scholar 

  35. Stroock DW (1987) Homogeneous chaos revisited. Seminaire de Probabilités XXI. Lecture notes in mathematics, vol 1247, Springer, Berlin, pp 1–8

    Google Scholar 

  36. Üstünel AS (1995) An introduction to analysis on Wiener space. Lecture notes in mathematics, vol 1610. Springer, Berlin

    Google Scholar 

  37. Watanabe S (1984) Lectures on stochastic differential equations and Malliavin calculus. Tata Institute of Fundamental Research. Springer, Berlin

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The University of Kansas, Lawrence, KS, 66045, USA

    David Nualart

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  1. David Nualart
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Correspondence to David Nualart .

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Editors and Affiliations

  1. University of Wyoming, Laramie, Wyoming, USA

    Stefan Heinz

  2. University of Wyoming, Laramie, Wyoming, USA

    Hakima Bessaih

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Nualart, D. (2015). An Introduction to the Malliavin Calculus and Its Applications. In: Heinz, S., Bessaih, H. (eds) Stochastic Equations for Complex Systems. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-18206-3_1

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  • DOI: https://doi.org/10.1007/978-3-319-18206-3_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-18205-6

  • Online ISBN: 978-3-319-18206-3

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