Abstract
We define the Skorohod integral of an operator-valued process with respect to a cylindrical Hilbertian Wiener process. We study the resulting process, and establish a generalized Itô formula. We define also a Stratonovitch integral, and establish the corresponding chain rule. Our work is inspired by the finite-dimensional results in [10].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Références
A. Badrikian, Séminaire sur les Fonctions Aléatories Linéaires et les Mesures Cylindriques, Lecture Notes in Mathematics, Vol. 139, Springer-Verlag, Berlin (1970).
A. Bensoussan, Filtrage optimal des systémes linéaires, Dunod, Paris (1970).
D. L. Burkholder, A geometrical characterisation of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab., 9:6 (1981).
J. Gärtner, E, Pardoux, About Itô and Stratonovitch integrals with respect to a space-time Wiener process (en préparation).
A. Grorud, E. Pardoux, Calcul Stochastique Anticipatif dans les Espaces de Hilbert, C. R. Acad. Sci. Paris Ser. I Math., 307:1001–1004 (1988).
L. Gross, Measurable functions on Hilbert spaces, Trans. Amer. Math. Soc., 105:372-390.
B. Hajek, Stochastic equations of hyperbolic type and a two-parameter Stratonovitch calculus, Ann. Probab., 10(2):451–463 (1982).
H. Korezlioglu, Passage from two parameters to infinite dimension, in Stochastic Partial Differential Equations and Applications (Eds. G. DaPrato and L. Tubaro), pp. 131–153, Lecture Notes in Mathematics, Vol. 1236, Springer-Verlag, Berlin (1987).
M. Métivier, J. Pellaumail, Stochastic Integration, Academic Press, New York (1980).
D. Nualart, E. Pardoux, Stochastic calculus with anticipating integrands, Probab. Theory Rel. Fields 78:535–551 (1988).
D. Ocone, E. Pardoux,A Generalized, Itô-Ventsell formula. Application to a class of anticipating stochastic differential equations, Ann. Inst. H. Poincaré, 25(1):39–71 (1989).
E. Pardoux, Applications of anticipating stochastic calculus to stochastic differential equations, in Stochastic Analysis and Related Topics, II (Eds. H. Korezlioglu and A. S. Usternel), Lecture Notes in Mathematics, Vol. 1444, Springer-Verlag, Berlin (1990).
H. Sugita, Sobolev spaces of Wiener functionals and Malliavin's calculus, J. Math. Kyoto Univ., 25(1):31–48 (1985).
J. Walsh, An introduction to stochastic partial differential equations, in Ecole d'été de Probabilité de St-Flour XIV (Ed. P. L. Hennequin), pp. 265–439, Lectures Notes in Mathematics, Vol. 1180, Springer-Verlag, Berlin (1986).
Author information
Authors and Affiliations
Additional information
Communicated by D. Ocone
Rights and permissions
About this article
Cite this article
Grorud, A., Pardoux, E. Intégrales Hilbertiennes anticipantes par rapport à un processus de Wiener cylindrique et calcul stochastique associé. Appl Math Optim 25, 31–49 (1992). https://doi.org/10.1007/BF01184155
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01184155