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Generalized stochastic integrals and the malliavin calculus
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  • Published: September 1986

Generalized stochastic integrals and the malliavin calculus

  • David Nualart1,2 &
  • Moshe Zakai1,2 

Probability Theory and Related Fields volume 73, pages 255–280 (1986)Cite this article

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Summary

The paper first reviews the Skorohod generalized stochastic integral with respect to the Wiener process over some general parameter space T and it's relation to the Malliavin calculus as the adjoint of the Malliavin derivative. Some new results are derived and it is shown that every sufficiently smooth process {ut, t∈T} can be decomposed into the sum of a Malliavin derivative of a Wiener functional, and a process whose generalized integral over T vanishes. Using the results on the generalized integral, the Bismut approach to the Malliavin calculus is generalized by allowing non adapted variations of the Wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions.

Let e i be a non-random complete orthonormal system on T, the Ogawa integral ∫u \(\mathop \delta \limits^{{\text{ }}o} \) W is defined as ∑ i (e i u) ∫ e i dW where the integrals are Wiener integrals. Conditions are given for the existence of an intrinsic Ogawa integral i.e. independent of the choice of the orthonormal system and results on it's relation to the Skorohod integral are derived.

The transformation of measures induced by (W + ∫ u d μu non adapted is discussed and a Girsanov-type theorem under certain regularity conditions is derived.

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References

  1. Bismut, J.M.: Martingales, the Malliavin calculus and hypoellipticity under general Hörmander's conditions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 469–505 (1981)

    Google Scholar 

  2. Cameron, R.H., Martin, W.T.: The transformation of Wiener integrals by non-linear transformations. Trans. Am. Math. Soc. 66, 253–283 (1949)

    Google Scholar 

  3. Federer, H.: Geometric measure theory. Berlin-Heidelberg-New York: Springer 1969

    Google Scholar 

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

    Google Scholar 

  5. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion proceses. Amsterdam-Oxford-New York: North Holland/Kodanska 1981

    Google Scholar 

  6. Ito, K.: Multiple Wiener integral. J. Math. Soc. Japan. 3, 157–169 (1951)

    Google Scholar 

  7. Kunita, H.: On backward stochastic differential equations. Stochastic 6, 293–313 (1982)

    Google Scholar 

  8. Kuo, H.H.: Gaussian measures in Banach spaces. Lecture Notes Math. 463. Berlin-Heidelberg-New York: Springer 1975

    Google Scholar 

  9. Kusuoka, S.: The non-linear transformation of Gaussian measure on Banach space and its absolute continuity (I). J. Fac. Sci. Univ. Tokyo Univ. Sec. IA 567–597 (1982)

    Google Scholar 

  10. Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. In: Ito, K. (ed.). Proc. of Int. Symp. Stoch. D. Eqs. Kyoto 1976, pp. 195–263. Tokyo: Kmokuniya-Wiley (1978)

    Google Scholar 

  11. Malliavin, P.: Calcul des variations, intégrales stochastiques et complexes de Rham sur l'espace de Wiener. C.R. Acad. Sci. Paris, t. 299, Serie I, 8, 347–350 (1984)

    Google Scholar 

  12. Ogawa, S.: Sur le produit direct du bruit blanc par luimême. C.R. Acad. Sc. Paris, t. 288 (Serie A), 359–362 (1979)

    Google Scholar 

  13. Ogawa, S.: Une remarque sur l'approximation de l'intégrale stochastique du type noncausal par une suite des intégrales de Stieltjes. Tohoku Math. Journ. 36, 41–48 (1984)

    Google Scholar 

  14. Ramer, R.: On non-linear transformations of Gaussian measures. J. Funct. Anal. 15, 166–187 (1974)

    Google Scholar 

  15. Shigekawa, I.: Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ. 20, 263–289 (1980)

    Google Scholar 

  16. Shigekawa, I.: de Rham-Hodge-Kodaira's decomposition on abstract Wiener space. (preprint)

  17. Skorohod, A.V.: On a generalization of a stochastic integral. Theory Probab. Appl. XX, 219–233 (1975)

    Google Scholar 

  18. Stroock, D.: The Malliavin calculus. A. functional analytical approach. J. Funct. Anal. 44, 212–257 (1981)

    Google Scholar 

  19. Zakai, M.: The Malliavin calculus. Acta Applicandae Mat. 3, 175–207 (1985)

    Google Scholar 

  20. Zakai, M.: Malliavin derivatives and derivatives of functionals of the Wiener process with respect to a scale parameter. Ann. Probab. 13, 609–615 (1985)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Facultat de Matematiques, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    David Nualart & Moshe Zakai

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

    David Nualart & Moshe Zakai

Authors
  1. David Nualart
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  2. Moshe Zakai
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Additional information

The work of M.Z. was supported by the Fund for Promotion of Research at the Technion

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Nualart, D., Zakai, M. Generalized stochastic integrals and the malliavin calculus. Probab. Th. Rel. Fields 73, 255–280 (1986). https://doi.org/10.1007/BF00339940

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  • Received: 24 May 1985

  • Issue Date: September 1986

  • DOI: https://doi.org/10.1007/BF00339940

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Keywords

  • Parameter Space
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Regularity Condition
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