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A guide to the stochastic calculus of variations

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References

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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. J. M. Bismut, Martingales, the Malliavin calculus and hypoellipticity under general Hormander's conditions, Z. Wahrscheinlichkeits theorie verw. Gebiete 56 (1981), 469–505.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. M. Bismut, Calcul des variations stochastique et processus de sauts, Z. Wahrsch. verw. Gebiete 63 (1983), 147–235.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. M. Bismut, The calculus of boundary processes, Ann. Sci. Ecole Norm. Sup 17 (1984) 507–622.

    MathSciNet  MATH  Google Scholar 

  9. J. M. Bismut, Large Deviations and the Malliavin Calculus, Progress in Math. 45, Brikhauser, Boston (1984).

    MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. M. Bismut, D. Michel, Diffusions Conditionelles, J. Funct. Anal. 44 (1981), 174–211; 45 (1982), 274–282.

    Article  MathSciNet  MATH  Google Scholar 

  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. N. Bouleau and F. Hirsch, Formes de Dirichlet generales et densite des variable aleatoire reeles l'espace de Wiener, preprint.

    Google Scholar 

  14. P. Cattiaux, Hypoellipticite et hypoellipticite partielle poure les diffusions avec une condition frontiere, Ann. Inst. Henri Poincare, 22 (1986) 67–112.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  16. M. Chaleyat-Maurel and D. Michel, Hypoellipticity theorems and conditional laws, Z. Wahrsch. verw. Gebiete 65 (1984), 573–597.

    Article  MathSciNet  MATH  Google Scholar 

  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-.

    Article  MathSciNet  Google Scholar 

  18. A. B. Cruzeiro, "Diffusions sur l'espace de Wiener", C. R. Acad. Sc. Paris 302 (1986) 295–298.

    MathSciNet  MATH  Google Scholar 

  19. M. H. A. Davis, Functionals of diffusion processes as stochastic integrals, Math. Proc. Camb. Phil. Soc. 87 (1980), 157–166.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  21. G. S Ferreyra, Smoothness of the unnormalized conditional measures of stochastic nonlinear filtering, preprint.

    Google Scholar 

  22. B. Gaveau and P. Trauber, L'Integrale stochastique comme operateur de divergence dans l'espace fonctionnel, J. Funct. Anal. (46) (1982), 230–238.

    Article  MathSciNet  MATH  Google Scholar 

  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. U. Haussmann, On the integral representation of functionals of Ito processes, Stochastics, 3 (1979), 17–28.

    Article  MathSciNet  MATH  Google Scholar 

  25. U. Haussmann, Functionals of Ito processes as stochastic integrals, SIAM J. Control and Opt. 16 (1978), 252–269.

    Article  MathSciNet  MATH  Google Scholar 

  26. T. Hida, Stationary Stochastic Processes, Princeton University Press (197 ), Princeton.

    Google Scholar 

  27. T. Hida, Brownian Motion, Application of Math, Vol II (1980), Springer-Verlag.

    Google Scholar 

  28. T. Hida, Generalized multiple Wiener integrals, Proc. Japan Academy, Ser. A 54 (1978), 55–58.

    Article  MathSciNet  MATH  Google Scholar 

  29. R. Holley and D. W. Stroock, Diffusions on an infinite dimensional torus, J. Funct. Anal. 42, 29–63 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  30. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland/Kodansha, Amsterdam/Tokyo (1981).

    MATH  Google Scholar 

  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.

    MATH  Google Scholar 

  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. M. Kree, Propriete de trace en dimension infinie d'espaces du type Sobolev, Bull. Soc. Math. France 105 (1977), 141–163.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  35. P. Kree, Regularite C des lois conditionelles par rapport a certaines variables aleatoires, C.R. Acad. Paris 296 (1983), 223–225.

    MathSciNet  MATH  Google Scholar 

  36. I. Kubo, Ito formula for generalized Brownian functionals, in Lecture Notes in Control and Inf. Sci. 49 (1983), Springer-Verlag.

    Google Scholar 

  37. H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463 (1975), Springer-Verlag.

    Google Scholar 

  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. 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. S. Kusuoka and D. W. Stroock, The Partial Malliavin Calculus and its Application to Nonlinear Filtering, Stochastics 12 (1984), 83–142.

    Article  MathSciNet  MATH  Google Scholar 

  41. B. Lascar, Proprietes locale d'espaces de type Sobolev en dimension infinite, Comm. in P.D.E 1 (1976), 561–584.

    Article  MathSciNet  MATH  Google Scholar 

  42. R. Leandre, Estimation en temps petit de la densite d'une diffusion hypoelliptique, C. R. Acad. Sc. Paris, 301 (1985) 801–804.

    MathSciNet  MATH  Google Scholar 

  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. P. Malliavin, Ck-hypoellipticity with degeneracy, Stochastic Analysis, Academic Press (1978), 199–214, 327–340.

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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. 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. 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. 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. 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. D. Michel, Regularite des lois conditionelles en theorie du filtrage non-lineaire et calcul des variations stochastique, J. Funct. Anal. 41 (1981), 8–36.

    Article  MathSciNet  MATH  Google Scholar 

  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. J. M. Moulinier, Fonctionelles oscillantes stochastique et hypoellipticite, Bull. Sc. Math 109 (1985) 37–60.

    MathSciNet  MATH  Google Scholar 

  54. D. Nulart and M. Zakai, Generalized stochastic integrals and the Malliavin Calculus, Probability Theory and Related Fields, 73 (1986), 255–280.

    Article  MathSciNet  MATH  Google Scholar 

  55. D. Nualart and E. Pardoux, Stochastic Calculus with anticipating integrands, preprint.

    Google Scholar 

  56. D. Ocone, Malliavin's Calculus and stochastic integral representation of functionals of diffusion processes, Stochastics, 12 (1984), 161–185.

    Article  MathSciNet  MATH  Google Scholar 

  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. D. Ocone, Stochastic calculus of variations for stochastic partial differential equations, preprint.

    Google Scholar 

  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. 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. E. Pardoux and P. Protter, Two-sided stochastic integrals and calculus, preprint.

    Google Scholar 

  62. I. Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ. 20 (1980) 263–269.

    MathSciNet  MATH  Google Scholar 

  63. A. V. Skorohod, On a generalization of a stochastic integral, Theory of Prob. and Appl. XX (1975), 279–233.

    MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  65. D. W. Stroock, The Malliavin calculus, a functional analytic approach, J. Funct. Anal. 44 (1981), 212–257.

    Article  MathSciNet  MATH  Google Scholar 

  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. D. W. Stroock, Lecture Notes on the Malliavin Calculus and applications, Institute for Mathematics and its Applications, Minneapolis, Preprint #218, (1986).

    Google Scholar 

  68. I. Segal, Tensor Algebras over Hilbert spaces, Trans. Amer. Math. Soc. 81, 1956, 106–134.

    Article  MathSciNet  MATH  Google Scholar 

  69. H. Sugita, Sobolev spaces of Wiener functionals and Malliavin's calculus, J. Math. Kyoto Univ. 25 (1985), 31–48.

    MathSciNet  MATH  Google Scholar 

  70. H. Sugita, On a characterization of the Sobolev spaces over an abstract Wiener space, J. Math. Kyoto Univ. 25 (1985) 717–725.

    MathSciNet  MATH  Google Scholar 

  71. S. Taniguchi, Malliavin's Stochastic Calculus of Variations for Manifold-Valued Wiener functionals, Zeit. fur. Wahr. 65 (1984).

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  73. A. S. Ustunel, Extension of Ito's Calculus via Malliavin's Calculus, preprint.

    Google Scholar 

  74. A. S. Ustunel, Representation of the distributions on Wiener space and stochastic calculus of variations, preprint.

    Google Scholar 

  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. 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. S. Watanabe, Stochastic Differential Equations and Malliavin Calculus, Tata Institute of Fundamental Research, Springer-Verlag, 1984.

    Google Scholar 

  78. S. Watanabe, Analysis of Wiener functionals (Malliavin calculus) and its application to heat kernels, Ann. Prob. (15) (1987), 1–39.

    Article  MATH  Google Scholar 

  79. M. Zakai, The Malliavin Calculus, Acta Appl. Math. 3 (1985), 175–207.

    Article  MathSciNet  MATH  Google Scholar 

  80. M. Reed and B. Simon, Methods of Mathematical Physics I: Functional Analysis, Academic Press (1972).

    Google Scholar 

  81. D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, New York (1979).

    MATH  Google Scholar 

  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. J. Neveu, Processus aleatoires gaussiens, Presses de l'Universite de Montreal, (1968).

    Google Scholar 

  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. H.J. Sussmann, Nonexistence of finite dimensional filters for the cubic sensor problem, preprint.

    Google Scholar 

  86. J. Norris, Simplified Malliavin calculus, in Seminaire de Probabilities XX, Lecture Notes in Mathematics 1204. Springer-Verlag, New York, 1986.

    Google Scholar 

  87. H. Federer, Geometric Measure Theory, Springer-Verlag, New York (1969).

    MATH  Google Scholar 

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Hayri Korezlioglu Ali Süleyman Ustunel

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Ocone, D.L. (1988). A guide to the stochastic calculus of variations. In: Korezlioglu, H., Ustunel, A.S. (eds) Stochastic Analysis and Related Topics. Lecture Notes in Mathematics, vol 1316. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0081929

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