Summary
We study the stochastic integral defined by Skorohod in [24] of a possibly anticipating integrand, as a function of its upper limit, and establish an extended Itô formula. We also introduce an extension of Stratonovich's integral, and establish the associated chain rule. In all the results, the adaptedness of the integrand is replaced by a certain smoothness requirement.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Berger, M., Mizel, V.: An extension of the stochastic integral. Ann. Probab. 10, 435–450 (1982)
Bismut, J.M.: Martingales, the Malliavin calculus and hypoellipticity under general Hörmander's conditions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 469–505 (1981)
Föllmer, H.: Calcul d'Itô sans probabilités. Séminaire de Probabilités XV (Lect. Notes Math., vol. 850, pp. 143–150) Berlin Heidelberg New York: Springer 1981
Gaveau, B., Trauber, P.: L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel. J. Funct. Anal. 46, 230–238 (1982)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Tokyo: North Holland/Kodanska (1981)
Ikeda, N., Watanabe, S.: An introduction to Malliavin's Calculus. Proceedings of the Taniguchy International Symposium on Stochastic Analysis. Katata and Kyoto, 1982, pp. 1–52. Tokyo: Kinokuniya 1984
Ito, K.: Multiple Wiener integral. J. Math. Soc. Japan 3, 157–169 (1951)
Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. Ecole d'Eté de Probabilités de Saint-Flour XII 1982. (Lect. Notes Math. vol. 1097, pp. 144–303) Berlin Heidelberg New York Tokyo: Springer 1984
Kunita, H.: On backward stochastic differential equations. Stochastics 6, 293–313 (1982)
Kuo, H.H., Russek, A.: Stochastic integrals in terms of white noise. Preprint Louisiana State Univ., Baton Rouge LA, USA
Kree, M.: Propriété de trace en dimension infinie, d'espaces du type Sobolev. Bull. Soc. Math. France 105, 141–163 (1977)
Kree, M., Kree, P.: Continuité de la divergence dans les espaces de Sobolev relatifs à l'espace de Wiener. Note C.R.A.S. t. 296, 833–836 (1983)
Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. Proceedings of the International Symposium on Stochastic Differential Equations. Kyoto 1976, pp. 195–263. Tokyo: Kinokuniya-Wiley 1978
Meyer, P.A.: Transformations de Riesz pour les lois Gaussiennes. Séminaire de Probabilités XVIII (Lect. Notes Math. vol. 1059, pp. 179–193) Berlin Heidelberg New York Tokyo: Springer 1984
Nualart, D., Pardoux, E.: Stochastic calculus associated with Skorohod's integral. Stochastic Differential Systems, Proc. 5th IFIP Workshop on Stochastic Differential System, Eisenach, eedings, (Lect. Notes Control Inform. Sci. vol. 96, pp. 363–372) Berlin Heidelberg New York Tokyo: Springer 1987
Nualart, D., Zakai, M.: Generalized stochastic integrals and the Malliavin Calculus. Probab. Theor. Rel. Fields 73, 255–280 (1986)
Ocone, D.: Malliavin's calculus and stochastic integral representation of functionals of diffusion processes. Stochastic 12, 161–185 (1984)
Ogawa, S.: Quelques propriétés de l'intégrale stochastique du type noncausal. Japan J. Appl. Math. 1, 405–416 (1984)
Pardoux, E., Protter, Ph.: Two-sided stochastic integral and calculus. Probab. Theor. Rel. Fields 76, 15–50 (1987)
Rosinski, J.: On stochastic integration by series of Wiener integrals. Preprint Univ. North Carolina, Chapell Hill, NC, USA
Sekiguchi, T., Shiota, Y.: L 2-theory of noncausal stochastic integrals. Math. Rep. Toyama Univ. 8, 119–195 (1985)
Sevljakov, A. Ju.: The Itô formula for the extended stochastic integral. Theor. Probab. Math. Statist. 22, 163–174 (1981)
Shigekawa, I.: Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ. 20-2, 263–289 (1980)
Skorohod, A.V.: On a generalization of a stochastic integral. Theor. Prob. Appl. 20, 219–233 (1975)
Ustunel, A.S.: La formule de changement de variable pour l'intégrale anticipante de Skorohod. C.R. Acad. Sci., Paris, Ser. I 303, 329–331 (1986)
Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus. Tata Institute of Fundamental Research. Berlin Heidelberg New York Tokyo: Springer 1984
Yor, M.: Sur quelques approximations d'intégrales stochastiques. Séminaire de Probabilités XI (Lect. Notes Math. vol. 581, pp. 518–528) Berlin Heidelberg New York Tokyo: Springer 1977
Zakai, M.: The Malliavin calculus. Acta Appl. Math. 3-2, 175–207 (1985)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nualart, D., Pardoux, E. Stochastic calculus with anticipating integrands. Probab. Th. Rel. Fields 78, 535–581 (1988). https://doi.org/10.1007/BF00353876
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00353876