Summary
This work is devoted to derive Itô-type formulae for anticipative stochastic processes with nonmonotonous time using the Malliavin Calculus techniques and the fundamental theorem of the differential calculus. The same method is applied also to give an Itô-Ventcell type formula in the anticipative case.
Article PDF
Similar content being viewed by others
References
Bismut, J.-M.: Mécanique aléatoire. (Lect. Notes Math., vol. 866) Berlin Heidelberg New York: Springer 1981
Dellacherie, C., Meyer, P.A.: Probabilités et potentiel, Chap. V–VIII. Paris: Hermann 1980
Gaveau, B., Trauber, P.: L'Integrale stochastique comme opérateur de divergence dans l'espace fonctionnel. J. Funct. Anal. 46, 230–238 (1982)
Hitsuda, M.: Formula for Brownian partial derivatives. Faculty of Integrated Arts and Sciences, Hiroshima University, vol. 3, pp. 1–15 (1979)
Krée, P.: Continuité de la divergence dans les espaces de Sobolev relatifs à l'espace de Wiener. C.R. Acad. Sci. Paris, Ser. I 296, 833 (1983)
Kuo, H.-H.: Gaussian measures on Banach spaces. (Lect. Notes Math., vol. 463) Berlin Heidelberg New York: Springer 1975
Malliavin, P.: Implicit functions in finite corank on the Wiener space. In: Ito, K., Watanabe, S. (eds.) Tanaguchi Supposium, Katata 1982. Amsterdam: North Holland 1984
Meyer, P.A.: Flot d'une équation differentielle stochastique. In: Azema, J. Yor M. (eds.) Séminaire de probabilités XV 1979/80. (Lect. Notes Math., vol. 850, pp. 103–117) Berlin Heidelberg New York: Springer 1981
Nualart, D., Pardoux, E.: Stochastic calculus with anticipating integrands. Preprint (1986)
Nualart, D., Zakai, M.: Generalized stochastic integrals and the Malliavin calculus. Preprint (1985)
Pardoux, E., Protter, P.: A two-sided stochastic integral and its calculus. Preprint (1985)
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. Theory Proba. Math. Stat. 22, 163–174 (1981)
Schaefer, H.H.: Topological vector spaces. (Graduate Texts in Mathematics, vol. 3) Berlin Heidelberg New York: Springer 1971
Skorohod, A.V.: On a generalization of a stochastic integral. Theory Probab. Appl. 20, 219–233 (1975)
Ustunel, A.S.: Une extension du calcul d'Itô via le calcul des variations stochastiques. C.R. Acad. Sci., Paris, Ser. I 300, 277–279 (1985)
Ustunel, A.S.: Extension of Itô's calculus via Malliavin's calculus. Stochastics 23, 353–375 (1988)
Ustunel, A.S.: Representation of the distributions on Wiener space and stochastic calculus of variations. J. Funct. Anal. 70, 126–139 (1987)
Ustunel, A.S.: La formule de changement de variable pour l'intégrale anticipante de Skorohod. C.R. Acad. Sci., Paris, Serie I, 303, 329–331 (1986)
Ustunel, A.S.: Some comments on the filtering of diffusions and the Malliavin Calculus. In: Korezliogli, H., Ustunel, A.S. (eds.) Proceedings, Sihvri. (Lect. Notes Math. vol. 1316, pp. 247–266). Berlin Heidelberg New York: Springer 1988
Watanabe, S.: Stochastic differential equations and Malliavin calculus. Bombay: Tata Institute of Fundamental Research 1984
Zheng, Wei-An, Meyer, P.A.: Intégrales stochastiques non monotones. In: Azema, J., Yor, M. (eds.) Séminaire de probabilités XVIII, 1982/83. Proceedings, 1984. (Lect. Notes Math., vol. 1059, pp. 154–171) Berlin Heidelberg New York: Springer 1984
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ustunel, A.S. The Itô formula for anticipative processes with nonmonotonous time scale via the Malliavin Calculus. Probab. Th. Rel. Fields 79, 249–269 (1988). https://doi.org/10.1007/BF00320921
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00320921