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The Malliavin calculus

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Abstract

The paper presents a review of the calculus of functional derivatives introduced by Malliaving and the Malliavin technique for establishing the existence of a density for the probability law of Wiener functionals. The approach of Malliavin, Stroock and Shigekawa is compared with that of Bismut.

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References

  1. Bichteler, K. and Jacod, J.: ‘Calcul de Malliavin pour la Diffusion avec Sauts: Existence d'une Density dans la Cas Unidimensionel’,Seminaire de Probabilites XVII, Lecture Notes in Mathematics, Vol. 986, Springer-Verlag, Berlin, 1983, pp. 132–157.

    Google Scholar 

  2. Bismut, J. M.: ‘Martingales, the Malliavin Calculus and Hypoellipticity Under Goneral Hörmander's Conditions’,Z. Wahrsch. 56 (1981), 469–505.

    Google Scholar 

  3. Bismut, J. M.: ‘Calcul de Variations Stochastique et Processus de Sauts’,Z. Wahr. Geb. 63 (1983), 147–235.

    Google Scholar 

  4. Bismut, J. M.: ‘The Calculus of Boundary Processes’, Publication 82 T 19, Univ. de Paris-Sud, Dept. de Mathematique, 1982.

  5. Bismut, J. M.:Large Deviations and the Malliavin Calculus, Birkhauser, Basle, 1984.

    Google Scholar 

  6. Bismut, J. M. and Michel, D.: ‘Diffusions Conditionelles’,J. Funet. Anal. 44 (1981), 174–211;45 (1982), 274–292.

    Google Scholar 

  7. Cameron, R. H.: ‘The First Variation of an Indefinite Wiener Integral’,Proc. Am. Math. Soc. 2 (1951), 914–924.

    Google Scholar 

  8. Geman, D. and Horowitz, J.: ‘Occupation Times’,Ann. Prob. 8 (1980), 1–67.

    Google Scholar 

  9. Holley, R. A. and Stroock, D.: ‘Diffusions on an Infinite Dimensional Torus’,J. Funet. Anal. 42 (1981), 29–63.

    Google Scholar 

  10. Ikeda, N. and Watanabe, S.:Stochastic Differential Equations and Diffusion Processes, North Holland/Kodansha, Amsterdam, 1981.

    Google Scholar 

  11. Kazamaki, N. and Sekiguchi, T.: ‘On the Transformation of Some Classes of Martingales by a Change of Law’,Tokohu Math. J. 31 (1979), 261–279.

    Google Scholar 

  12. Kunita, H.: ‘On the Decomposition of Solutions of Stochastic Differential Equations’, D.Williams (ed.), inStochastic Integrais, Lecture Notes in Mathematics, vol. 851, Springer-Verlag, Berlin, 1981.

    Google Scholar 

  13. Kuo, H. H.:Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin, 1975.

    Google Scholar 

  14. Malliavin, P.: ‘Stochastic Calculus of Variations and Hypoelliptic Operators’,Proceedings of International Symposium on Stochastic Differential Equations, Kyoto 1976, K. Itô (ed.), Kinokuniya-Wiley, 1978, pp. 195–263.

  15. Malliavin, P.: ‘C k Hypoellipticity with Degeneracy’, A.Friedman and M.Pinsky (eds.), inStochastic Analysis, Academic Press, New York, 1978, pp. 192–214.

    Google Scholar 

  16. Meyer, P. A.: ‘Flot d'une Equation Differentielle Stochastique’,Seminaire de Probabilites, No. XV, Lecture Notes in Mathematics, vol. 850, Springer-Verlag, Berlin, 1981, pp. 103–117.

    Google Scholar 

  17. Meyer, P. A.: ‘Note Sur les Processus d'Ornstein-Uhlenbeck’,Seminaire de Probabilites No. XVI, Lecture Notes in Mathematics, vol. 920, Springer-Verlag, Berlin, 1982, pp. 95–133.

    Google Scholar 

  18. Michel, D.: ‘Regularité de Lois conditionelles en théorie du filtrage non Linaire et calcul de variations stochastiques’,J. Funct. Anal. 41 (1981), 8–36.

    Google Scholar 

  19. Shigekawa, I.: ‘Derivatives of Wiener Functionals and Absolute Continuity of Induced Measures’,J. Math. Kyoto Univ. 20 (1980), 263–289.

    Google Scholar 

  20. Stroock, D.: ‘The Malliavin Calculus and its Applications to Second Order Parabolle Differential Equations’,Math. Systems Theory 14 (1981), 25–56;14 (1981), 141–171.

    Google Scholar 

  21. Stroock, D.: ‘The Malliavin Calculus, A Functional Analytical Approach’,J. Funct. Anal. 44 (1981), 212–257.

    Google Scholar 

  22. Stroock, D.: ‘Some Applications of Stochastic Calculus to Partial Differential Equations’,Ecole de Probabilites de Sainte Flour, Lecture Notes in Mathematics, Vol. 976, Springer-Verlag, Berlin, 1983, pp. 267–382.

    Google Scholar 

  23. Williams, D.:To Begin at the Beginning, Lecture Notes in Mathematics, vol. 851, Springer-Verlag, Berlin, 1981, pp. 1–55.

    Google Scholar 

  24. Zakai, M.: ‘Malliavin Derivatives and Derivatives of Functionals of the Wiener Process with Respect to a Scale Parameter’, to appear inAnn. Prob.

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Authors and Affiliations

  1. Department of Electrical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Moshe Zakai

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  1. Moshe Zakai
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The research was supported by the fund for the promotion of research at the Technion

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Zakai, M. The Malliavin calculus. Acta Applicandae Mathematicae 3, 175–207 (1985). https://doi.org/10.1007/BF00580703

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  • DOI: https://doi.org/10.1007/BF00580703

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