A guide to the stochastic calculus of variations

  • Daniel L. Ocone
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1316)


Vector Field Brownian Motion Stochastic Differential Equation Differential Calculus Stochastic Integral 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R.F. Bass and M. Cranston, The Malliavin calculus for pure jump processes and applications to local time, Ann. Prob. 14 (1986), 490–532.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    G. Ben Arous, S. Kusuoka and D.W. Stroock, The Poisson kernel for certain degenerate elliptic operators, J. Functional. Anal. 56 (1984), 171–209.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    K. Bichteler and D. Fonken, A simple version of the Malliavin calculus in dimension N, in Cinlar, etal, ed, Seminar on Stochastic Processes, 1982, Birkhauser, Boston (1983).Google Scholar
  4. 4.
    K. Bichteler and D. Fonken, A simple version of the Malliavin calculus in dimension one, in Martingale Theory on Harmonic Analysis and Banach Spaces, Lecture Notes in Mathematics, 939 (1982), Springer, New York.Google Scholar
  5. 5.
    K. Bichteler and J. Jacod, Calcul de Malliavin pour les diffusions avec sauts; existence d'une densite dans le cas unidimensions, In Seminaire de Probabilites XVII (J. Azema and M. Yos, eds.), Lecture Notes in Math (1983) 132–157, Springer, New York.Google Scholar
  6. 6.
    J. M. Bismut, Martingales, the Malliavin calculus and hypoellipticity under general Hormander's conditions, Z. Wahrscheinlichkeits theorie verw. Gebiete 56 (1981), 469–505.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. M. Bismut, Calcul des variations stochastique et processus de sauts, Z. Wahrsch. verw. Gebiete 63 (1983), 147–235.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. M. Bismut, The calculus of boundary processes, Ann. Sci. Ecole Norm. Sup 17 (1984) 507–622.MathSciNetzbMATHGoogle Scholar
  9. 9.
    J. M. Bismut, Large Deviations and the Malliavin Calculus, Progress in Math. 45, Brikhauser, Boston (1984).zbMATHGoogle Scholar
  10. 10.
    J. M. Bismut, The Atiyah-Singer theorems: a probabilistic approach, 1. The index theorem, 2. The Lefschetz fixed point formulas, J. Funct. Anal. 57 (1984), 56–99 and 329–348.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J. M. Bismut, D. Michel, Diffusions Conditionelles, J. Funct. Anal. 44 (1981), 174–211; 45 (1982), 274–282.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    N. Bouleau and F. Hirsch, Proprietes d'absolue continuite dans l'espace de Dirichlet et Application aux equations differentielles stochastiques, In Seminaire de Probabilites xx, Lecture Notes in Math, 1204 (1986) Springer, New York.Google Scholar
  13. 13.
    N. Bouleau and F. Hirsch, Formes de Dirichlet generales et densite des variable aleatoire reeles l'espace de Wiener, preprint.Google Scholar
  14. 14.
    P. Cattiaux, Hypoellipticite et hypoellipticite partielle poure les diffusions avec une condition frontiere, Ann. Inst. Henri Poincare, 22 (1986) 67–112.MathSciNetzbMATHGoogle Scholar
  15. 15.
    M. Chaleyat-Maurel, Robustesse en theorie du filtrage non lineaire et calcul des variations stochastique, C.R. Acad. Sc. Paris 297 (1983), 541–544.MathSciNetzbMATHGoogle Scholar
  16. 16.
    M. Chaleyat-Maurel and D. Michel, Hypoellipticity theorems and conditional laws, Z. Wahrsch. verw. Gebiete 65 (1984), 573–597.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    J. M. C. Clark, The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Stat. 41 (1970) 1281–1295, 42 (1971) 1778-.MathSciNetCrossRefGoogle Scholar
  18. 18.
    A. B. Cruzeiro, "Diffusions sur l'espace de Wiener", C. R. Acad. Sc. Paris 302 (1986) 295–298.MathSciNetzbMATHGoogle Scholar
  19. 19.
    M. H. A. Davis, Functionals of diffusion processes as stochastic integrals, Math. Proc. Camb. Phil. Soc. 87 (1980), 157–166.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    R. Fefferman, R. Gundy, M. Silverstein, and E.M. Stein, Inequalities for ratios of functionals of harmonic functions, Proc. Nat. Acad. Sci. USA; 79 (1982) 7958–7960.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    G. S Ferreyra, Smoothness of the unnormalized conditional measures of stochastic nonlinear filtering, preprint.Google Scholar
  22. 22.
    B. Gaveau and P. Trauber, L'Integrale stochastique comme operateur de divergence dans l'espace fonctionnel, J. Funct. Anal. (46) (1982), 230–238.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    J. B. Gravereuax, J. Jacod, Operateur de Malliavin sur l'espace de Wiener-Poisson, C. R. Acad. Sc. Paris, t. 300, Ino. 31985 81.Google Scholar
  24. 24.
    U. Haussmann, On the integral representation of functionals of Ito processes, Stochastics, 3 (1979), 17–28.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    U. Haussmann, Functionals of Ito processes as stochastic integrals, SIAM J. Control and Opt. 16 (1978), 252–269.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    T. Hida, Stationary Stochastic Processes, Princeton University Press (197 ), Princeton.Google Scholar
  27. 27.
    T. Hida, Brownian Motion, Application of Math, Vol II (1980), Springer-Verlag.Google Scholar
  28. 28.
    T. Hida, Generalized multiple Wiener integrals, Proc. Japan Academy, Ser. A 54 (1978), 55–58.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    R. Holley and D. W. Stroock, Diffusions on an infinite dimensional torus, J. Funct. Anal. 42, 29–63 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland/Kodansha, Amsterdam/Tokyo (1981).zbMATHGoogle Scholar
  31. 31.
    A. El Kharroubi, Lois de probabilite martingales de la solution de certaines equations differentielles stochastique, C. R. Acad. Sc. Paris 296 (1983), 1013–1016.zbMATHGoogle Scholar
  32. 32.
    P. Kree, Solutions faibles d'equations aux derivees fonctionnelles I, in Seminair Pierre Lelong (analyse) 1972–1973, Lecture Notes in Math 410, Springer-Verlag (1974).Google Scholar
  33. 33.
    M. Kree, Propriete de trace en dimension infinie d'espaces du type Sobolev, Bull. Soc. Math. France 105 (1977), 141–163.MathSciNetzbMATHGoogle Scholar
  34. 34.
    M. Kree and P. Kree, Continuite de la divergence daus les espaces de Sobolev relatifs a l'espace de Wiener, C.R.A.S. Paris 296 (1983), 833.MathSciNetzbMATHGoogle Scholar
  35. 35.
    P. Kree, Regularite C des lois conditionelles par rapport a certaines variables aleatoires, C.R. Acad. Paris 296 (1983), 223–225.MathSciNetzbMATHGoogle Scholar
  36. 36.
    I. Kubo, Ito formula for generalized Brownian functionals, in Lecture Notes in Control and Inf. Sci. 49 (1983), Springer-Verlag.Google Scholar
  37. 37.
    H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463 (1975), Springer-Verlag.Google Scholar
  38. 38.
    H. H. Kuo, Donsker's delta function as a generalized Brownian functional and its application, in Lecture Notes in Control and Inf. Sci. 49, (1983) Springer-Verlag.Google Scholar
  39. 39.
    S. Kusuoka and D. W. Stroock, Applications of the Malliavin calculus, part I. Taniguchi Symp. in Katata (1982), ed. by K. Ito, Kinokuniya, Tokyo (1984) 277–306-part II, J. Fac. Sci., Univ. of Tokyo, sec 1A, 32, (1985), 1–76.Google Scholar
  40. 40.
    S. Kusuoka and D. W. Stroock, The Partial Malliavin Calculus and its Application to Nonlinear Filtering, Stochastics 12 (1984), 83–142.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    B. Lascar, Proprietes locale d'espaces de type Sobolev en dimension infinite, Comm. in P.D.E 1 (1976), 561–584.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    R. Leandre, Estimation en temps petit de la densite d'une diffusion hypoelliptique, C. R. Acad. Sc. Paris, 301 (1985) 801–804.MathSciNetzbMATHGoogle Scholar
  43. 43.
    P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, Proc. Intern. Symp. S.D.E. Kyoto, ed. by K. Ito, Kinokuniya, Tokyo (1978).Google Scholar
  44. 44.
    P. Malliavin, Ck-hypoellipticity with degeneracy, Stochastic Analysis, Academic Press (1978), 199–214, 327–340.Google Scholar
  45. 45.
    P. Malliavin, Calcul des variations, integrales stochastique et complexe de Rham sur l'espace de Wiener, C.R.A.S. Paris 299 (1984) 347–350.MathSciNetzbMATHGoogle Scholar
  46. 46.
    P. A. Meyer, Demonstration de certaines inequalites de Littlewood-Paley, Seminaire de Probabilities x 1976, Lecture Notes in Mathematics 511, Springer-Verlag (1977).Google Scholar
  47. 47.
    P. A. Meyer, Transformations de Riesz pour les lois Gaussiennes, Seminaire de Probabilities XVIII, 1982/83, Lecture Notes in Mathematics 1059, Springer-Verlag (1984).Google Scholar
  48. 48.
    P. A. Meyer, Quelques resultats analytique sur le semi-groupe d'Ornstein-Uhlenbeck en dimension infinie, in Theory and Application of Random Fields, Lecture Notes in Control and Information Sciences 49, Springer-Verlag (1983).Google Scholar
  49. 49.
    P. A. Meyer, Note sur les processes d'Ornstein-Uhlenbeck, Seminaire de Probabilities, XVI, 1980/81, Lecture Notes in Mathematics 920, Springer-Verlag (1982).Google Scholar
  50. 50.
    P. A. Meyer, Retour sur la theorie de Littlewood-Paley, Seminaire de Probabilities XV 1979/80, Lecture Notes in Mathematics 850, Springer-Verlag (1981).Google Scholar
  51. 51.
    D. Michel, Regularite des lois conditionelles en theorie du filtrage non-lineaire et calcul des variations stochastique, J. Funct. Anal. 41 (1981), 8–36.MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    D. Michel, Conditional laws and Hormander's condition, in Stochastic Analysis, Proceedings of the Taniguchi International Symposium on Stochastic Analysis, 1982, K. Ito, ed., North-Holland, Amsterdam (1984).Google Scholar
  53. 53.
    J. M. Moulinier, Fonctionelles oscillantes stochastique et hypoellipticite, Bull. Sc. Math 109 (1985) 37–60.MathSciNetzbMATHGoogle Scholar
  54. 54.
    D. Nulart and M. Zakai, Generalized stochastic integrals and the Malliavin Calculus, Probability Theory and Related Fields, 73 (1986), 255–280.MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    D. Nualart and E. Pardoux, Stochastic Calculus with anticipating integrands, preprint.Google Scholar
  56. 56.
    D. Ocone, Malliavin's Calculus and stochastic integral representation of functionals of diffusion processes, Stochastics, 12 (1984), 161–185.MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    D. Ocone, Application of Wiener space analysis to nonlinear filtering, in Theory and Application of Nonlinear Control Systems, C. I. Byrnes and A. Lindquist (editors), North-Holland Elsevier Publishers (1986).Google Scholar
  58. 58.
    D. Ocone, Stochastic calculus of variations for stochastic partial differential equations, preprint.Google Scholar
  59. 59.
    D. Ocone, Probability distributions of solutions to some stochastic partial differential equations, Proceedings of the Trento Conference on Stochastic Partial Differential Equations, to appear in Lecture Notes in Mathematics, Springer-Verlag.Google Scholar
  60. 60.
    D. Ocone, Existence of densities for statistics in the cubic sensor problem, to appear in the Proceedings of the Workshop on Stochastic Control and Filtering, Institute of Mathematics and its Applications, Minneapolis, June 1986.Google Scholar
  61. 61.
    E. Pardoux and P. Protter, Two-sided stochastic integrals and calculus, preprint.Google Scholar
  62. 62.
    I. Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ. 20 (1980) 263–269.MathSciNetzbMATHGoogle Scholar
  63. 63.
    A. V. Skorohod, On a generalization of a stochastic integral, Theory of Prob. and Appl. XX (1975), 279–233.MathSciNetGoogle Scholar
  64. 64.
    D. W. Stroock, The Malliavin calculus and its applications to second order parabolic differential operators, I. II, Math. Systems Theory 14 (1981), 25–65, 141–171.MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    D. W. Stroock, The Malliavin calculus, a functional analytic approach, J. Funct. Anal. 44 (1981), 212–257.MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    D. W. Stroock, Some Applications of Stochastic Calculus to Partial Differential Equations, in Ecole d'Ete de Probabilites de Saint-Fleur XI-1981, Springer-Verlag Notes 976, Springer-Verlag, Berlin (1983).Google Scholar
  67. 67.
    D. W. Stroock, Lecture Notes on the Malliavin Calculus and applications, Institute for Mathematics and its Applications, Minneapolis, Preprint #218, (1986).Google Scholar
  68. 68.
    I. Segal, Tensor Algebras over Hilbert spaces, Trans. Amer. Math. Soc. 81, 1956, 106–134.MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    H. Sugita, Sobolev spaces of Wiener functionals and Malliavin's calculus, J. Math. Kyoto Univ. 25 (1985), 31–48.MathSciNetzbMATHGoogle Scholar
  70. 70.
    H. Sugita, On a characterization of the Sobolev spaces over an abstract Wiener space, J. Math. Kyoto Univ. 25 (1985) 717–725.MathSciNetzbMATHGoogle Scholar
  71. 71.
    S. Taniguchi, Malliavin's Stochastic Calculus of Variations for Manifold-Valued Wiener functionals, Zeit. fur. Wahr. 65 (1984).Google Scholar
  72. 72.
    A. S. Ustunel, Une extension du calcul d'Ito via le calcul stochastique des variations, C.R.A.S. Paris 300 (1985) 277–279.MathSciNetzbMATHGoogle Scholar
  73. 73.
    A. S. Ustunel, Extension of Ito's Calculus via Malliavin's Calculus, preprint.Google Scholar
  74. 74.
    A. S. Ustunel, Representation of the distributions on Wiener space and stochastic calculus of variations, preprint.Google Scholar
  75. 75.
    A. S. Ustunel, La formule de changement de variable pour l'integrale anticipante de Skorohod, C.R.A.S. to appear.Google Scholar
  76. 76.
    C. Varsan, On the regularity of the probabilities associated with diffusions, preprint, Dept. Math, Nat. Institute for Scientific and Technical Creation, Bd. Pacii 220, 79622, Bucharest, Romania.Google Scholar
  77. 77.
    S. Watanabe, Stochastic Differential Equations and Malliavin Calculus, Tata Institute of Fundamental Research, Springer-Verlag, 1984.Google Scholar
  78. 78.
    S. Watanabe, Analysis of Wiener functionals (Malliavin calculus) and its application to heat kernels, Ann. Prob. (15) (1987), 1–39.CrossRefzbMATHGoogle Scholar
  79. 79.
    M. Zakai, The Malliavin Calculus, Acta Appl. Math. 3 (1985), 175–207.MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    M. Reed and B. Simon, Methods of Mathematical Physics I: Functional Analysis, Academic Press (1972).Google Scholar
  81. 81.
    D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, New York (1979).zbMATHGoogle Scholar
  82. 82.
    G. Kallianpur, The role of reproducing kernel Hilbert spaces in the study of Gaussian processes, in Advances in Probability and Related Topics, ed. by P. Ney, Marcel-Dekker (1970), 49–83.Google Scholar
  83. 83.
    J. Neveu, Processus aleatoires gaussiens, Presses de l'Universite de Montreal, (1968).Google Scholar
  84. 84.
    H. J. Sussmann, Product expansions of exponential Lie series and the discretization of stochastic differential equations, Institute for Mathematics and its Applications preprint, to appear.Google Scholar
  85. 85.
    H.J. Sussmann, Nonexistence of finite dimensional filters for the cubic sensor problem, preprint.Google Scholar
  86. 86.
    J. Norris, Simplified Malliavin calculus, in Seminaire de Probabilities XX, Lecture Notes in Mathematics 1204. Springer-Verlag, New York, 1986.Google Scholar
  87. 87.
    H. Federer, Geometric Measure Theory, Springer-Verlag, New York (1969).zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Daniel L. Ocone
    • 1
  1. 1.Mathematics DepartmentRutgers UniversityNew BrunswickUSA

Personalised recommendations