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Stochastic Anticipating Calculus

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Probability Towards 2000

Part of the book series: Lecture Notes in Statistics ((LNS,volume 128))

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Abstract

The purpose of the stochastic anticipating calculus is to develop a differential and integral calculus involving stochastic processes which are not necessarily adapter to the Brownian motion {W t ,t ≥0}. This stochastic calculus is mainly used to formulate and solve stochastic differential equations of the form.

$$\left\{ \begin{gathered} {\text{ }}d{X_t} = \sigma (t,{X_t})d{W_t} + b(t,{X_t})dt, \hfill \\ {\text{ }}{X_0} \in {\mathbb{R}^m}, \hfill \\ \end{gathered} \right.$$

where the coefficients σ(t,x), b(t,x) or the initial condition X 0 depend on the whole trajectory of the Brownian motion W.

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References

  1. J. Asch and J. Potthoff: Itô’s lemma without nonanticipatory conditions. Probab. Theory Rel. Fields 88(1991) 17–46.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. A. Berger: A Malliavin-type anticipative stochastic calculus. Ann. Probab. 16(1988) 231–245.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. A. Berger and V. J. Mizel: An extension of the stochastic integral. Ann. Probab. 10(1982) 435–450.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. M. Bismut: Martingales, the Malliavin Calculus and hypoellipticity under general Hormander’s condition. Z. für Wahrscheinlichkeitstheorie verw. Gebiete 56(1981) 469–505.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Buckdahn: Quasilinear partial stochastic differential equations without nonanticipation requirement. Preprint 176, Humboldt Universität, Berlin, 1988.

    Google Scholar 

  6. R. Buckdahn: Anticipative Girsanov transformations. Probab. Theory Rel. Fields 89(1991) 211–238.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Buckdahn: Linear Skorohod stochastic differential equations. Probab. Theory Rel. Fields 90(1991) 223–240.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Buckdahn: Anticipative Girsanov transformations and Skorohod stochastic differential equations. Memoires of the AMS 533, 1994.

    Google Scholar 

  9. R. Buckdahn: Skorohod stochastic differential equations of diffusion type. Probab. Theory Rel. Fields 92(1993) 297–324.

    Google Scholar 

  10. R. Buckdahn and D. Nualart: Linear stochastic differential equations and Wick products. Probab. Theory Rel Fields. 99(1994) 501–526.

    Google Scholar 

  11. M.E. Caballero, B. Fernandez, and D. Nualart: Smoothness of distributions for solutions of anticipating stochastic differential equations. Stochastics and Stochastics Reports 52(1995) 303–322.

    MathSciNet  MATH  Google Scholar 

  12. R. H. Cameron and W. T. Martin: Transformation of Wiener integrals by nonlinear transformations. Trans. Amer. Math. Soc. 66(1949) 253–283.

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Donati-Martin: Equations différentielles stochastiques dans R avec conditions au bord. Stochastics and Stochastics Reports 35(1991) 143–173.

    MathSciNet  MATH  Google Scholar 

  14. C. Donati-Martin and D. Nualart: Markov property for elliptic stochastic partial differential equations. Stochastics and Stochastics Reports 46(1994) 107–115.

    MathSciNet  MATH  Google Scholar 

  15. J. Gamier: Stochastic invariant imbedding: Application to stochastic differential equations with boundary conditions. Probab. Theory Rel Fields 102(1995) 249–271.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Grorud, D. Nualart, and M. Sanz: Hilbert-valued anticipating stochastic differential equations. Ann. Inst. Henri Poincaré 30(1994) 133–161.

    MATH  Google Scholar 

  18. T. Hida, H. H. Kuo, J. Potthoff and L. Streit: White Noise: An Infinite Dimensional Calculus. Kluwer, 1993.

    Google Scholar 

  19. P. Imkeller and D. Nualart: Integration by parts on Wiener space and the existence of occupation densities. Ann. Probab. 22(1994) 469–493.

    Article  MathSciNet  MATH  Google Scholar 

  20. K. Itô: Multiple Wiener integral. J. Math. Soc. Japan 3(1951) 157–169.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Jacod: Grossissement initial, hypothèse (H’), et théorème de Girsanov. Lecture Notes in Math. 1118(1985) 15–35.

    Article  Google Scholar 

  22. T. Jeulin: Semimartingales et grossissement d’une filtration. Lecture Notes in Math. 833, Springer-Verlag, 1980.

    Google Scholar 

  23. A. Kohatsu-Higa and J. León: Anticipating stochastic differential equations of Stratonovich type. Preprint.

    Google Scholar 

  24. A. Kohatsu-Higa, J. Lesn and D. Nualart: Stochastic differential equations with random coefficients. Preprint.

    Google Scholar 

  25. H. Kunita: First order stochastic partial differential equations. In: Stochastic Analysis, Proc. Taniguchi Inter. Symp., Katata and Kyoto 1982, North-Holland, 1984, 249–270.

    Google Scholar 

  26. H. H. Kuo and A. Russek: White noise approach to stochastic integration. Journal Multivariate Analysis 24(1988) 218–236.

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Kusuoka: The nonlinear transformation of Gaussian measure on Banach space and its absolute continuity (I). J. Fac. Sci. Univ. Tokyo IA29 (1982) 567–597.

    MathSciNet  MATH  Google Scholar 

  28. J. Ma, Ph. Protter and M. San Martin: Anticipating integrals for a class of martingales. Preprint.

    Google Scholar 

  29. P. Malliavin: Stochastic calculus of variations and hypoelliptic operators. In: Proc. Inter. Symp. on Stoch. Diff. Equations, Kyoto 1976, Wiley 1978, 195–263.

    Google Scholar 

  30. T. Masuda: Absolute continuity of distributions of solutions of anticipating stochastic differential equations. J. Functional Anal. 95(1991) 414–432.

    Article  MathSciNet  MATH  Google Scholar 

  31. A. Millet and D. Nualart: Support theorems for a class of anticipating stochastic differential equations. Stochastics and Stochastics Reports 39(1992) 1–24.

    MathSciNet  MATH  Google Scholar 

  32. A. Millet, D. Nualart and M. Sanz: Large deviations for a class of anticipating stochastic differential equations. Annals of Probability 20(1992) 1902–1931.

    Article  MathSciNet  MATH  Google Scholar 

  33. D. Nualart: Noncausal stochastic integrals and calculus. In: Stochastic Analysis and Related Topics, eds.: H. Korezlioglu and A. S. ‘st’nel, Lecture Notes in Math. 1316 (1988) 80–129.

    Google Scholar 

  34. D. Nualart: The Malliavin Calculus and Related Topics. Springer-Verlag, 1995.

    Google Scholar 

  35. D. Nualart: Analysis on Wiener space and anticipating stochastic calculus. In: École d’Été de Probabilités de Saint Flour 1995. Lecture Notes in Math. To appear.

    Google Scholar 

  36. D. Nualart and E. Pardoux: Stochastic calculus with anticipating integrands. Probab. Theory Rel Fields 78(1988) 535–581.

    Article  MathSciNet  MATH  Google Scholar 

  37. D. Nualart and E. Pardoux: Boundary value problems for stochastic differential equations. Ann. Probab. 19(1991) 1118–1144.

    Article  MathSciNet  MATH  Google Scholar 

  38. D. Nualart and E. Pardoux: Second order stochastic differential equations with Dirichlet boundary conditions. Stochastic Processes and Their Applications 39(1991) 1–24.

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Nualart and M. Zakai: Generalized stochastic integrals and the Malliavin calculus. Probab. Theory Rel. Fields 73(1986) 255–280.

    Article  MathSciNet  MATH  Google Scholar 

  40. D. Nualart and M. Zakai: Generalized multiple stochastic integrals and the representation of Wiener functionals. Stochastics 23(1988) 311–330.

    MathSciNet  MATH  Google Scholar 

  41. D. Ocone and E. Pardoux: A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations. Ann. Inst. Henri Poincaré 25(1989) 39–71.

    MathSciNet  MATH  Google Scholar 

  42. D. Ocone and E. Pardoux: Linear stochastic differential equations with boundary conditions. Probab. Theory Rel. Fields 82(1989) 489–526.

    Article  MathSciNet  MATH  Google Scholar 

  43. S. Ogawa: Quelques propriétés de l’intégrale stochastique de type noncausal. Japan J. Appl. Math. 1(1984) 405–416.

    Article  MathSciNet  Google Scholar 

  44. S. Ogawa: The stochastic integral of noncausal type as an extension of the symmetric integrals. Japan J. Appl Math. 2(1984) 229–240.

    Article  Google Scholar 

  45. S. Ogawa: Une remarque sur l’approximation de l’intégrale stochastique du type noncausal par une suite des intégrates de Stieltjes. Tohoku Math. J. 36(1984) 41–48.

    Article  MathSciNet  MATH  Google Scholar 

  46. E. Pardoux: Applications of anticipating stochastic calculus to stochastic differential equations. In: Stochastic Analysis and Related Topics II, eds.: H. Korezlioglu and A. S. Üstünel, Lecture Notes in Math. 1444 (1990) 63–105.

    Google Scholar 

  47. E. Pardoux and P. Protter: Two-sided stochastic integrals and calculus. Probab. Theory Rel. Fields 76(1987) 15–50.

    Article  MathSciNet  MATH  Google Scholar 

  48. R. Ramer: On nonlinear transformations of Gaussian measures. J. Functional Anal 15(1974) 166–187.

    Article  MathSciNet  MATH  Google Scholar 

  49. J. Rosinski: On stochastic integration by series of Wiener integrals. Appl. Math. Optimization 19(1989) 137–155.

    Article  MathSciNet  MATH  Google Scholar 

  50. F. Russo and P. Vallois: Forward, backward and symmetric stochastic integration. Probab. Theory Rel. Fields 97(1993) 403–421.

    Article  MathSciNet  MATH  Google Scholar 

  51. F. Russo and P. Vallois: Product of two multiple stochastic integrals. Preprint.

    Google Scholar 

  52. T. Sekiguchi and Y. Shiota: L 2-theory of noncausal stochastic integrals. Math. Rep. Toyama Univ. 8(1985) 119–195.

    MathSciNet  MATH  Google Scholar 

  53. A. V. Skorohod: On a generalization of a stochastic integral. Theory Probab. Appl. 20(1975) 219–233.

    Article  Google Scholar 

  54. D. W. Stroock: Some applications of stochastic calculus to partial differential equations. In: École d’Été de Probabilités de Saint Flour, Lecture Notes in Math. 976 (1983) 267–382.

    Google Scholar 

  55. A. S. Üstünel: The Itô formula for anticipative processes with nonmonotonous time via the Malliavin calculus. Probab. Theory Rel. Fields 79(1988) 249–269.

    Article  Google Scholar 

  56. A. S. Üstünel and M. Zakai: Transformations of the Wiener measure under noninvertible shifts. Probab. Theory Rel. Fields 99(1994) 485–500.

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Departament d’Estadistica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007, Barcelona, Spain

    David Nualart (Facultat de Mathematiques)

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  1. David Nualart
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Editors and Affiliations

  1. Centro Vito Volterra, Universita degli Studi di Roma Tor Vergata, Via della Ricerca Scientifica, 00133, Roma, Italy

    L. Accardi

  2. Department of Statistics, Columbia University, 2990 Broadway Mail Code 4403, 10027, New York, NY, USA

    C. C. Heyde

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Nualart, D. (1998). Stochastic Anticipating Calculus. In: Accardi, L., Heyde, C.C. (eds) Probability Towards 2000. Lecture Notes in Statistics, vol 128. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2224-8_15

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  • DOI: https://doi.org/10.1007/978-1-4612-2224-8_15

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-98458-2

  • Online ISBN: 978-1-4612-2224-8

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