×

Stochastic calculus with anticipating integrands. (English) Zbl 0629.60061

We study the stochastic integral defined by A. V. Skorohod in Teor. Veroyatn. Primen. 20, 223-238 (1975; Zbl 0333.60060) 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.

MSC:

60H05 Stochastic integrals

Citations:

Zbl 0333.60060
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Berger, M., Mizel, V.: An extension of the stochastic integral. Ann. Probab. 10, 435-450 (1982) · Zbl 0499.60066 · doi:10.1214/aop/1176993868
[2] Bismut, J.M.: Martingales, the Malliavin calculus and hypoellipticity under general H?rmander’s conditions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 469-505 (1981) · Zbl 0445.60049 · doi:10.1007/BF00531428
[3] F?llmer, H.: Calcul d’It? sans probabilit?s. S?minaire de Probabilit?s XV (Lect. Notes Math., vol. 850, pp. 143-150) Berlin Heidelberg New York: Springer 1981
[4] Gaveau, B., Trauber, P.: L’int?grale stochastique comme op?rateur de divergence dans l’espace fonctionnel. J. Funct. Anal. 46, 230-238 (1982) · Zbl 0488.60068 · doi:10.1016/0022-1236(82)90036-2
[5] Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Tokyo: North Holland/Kodanska (1981) · Zbl 0495.60005
[6] 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 · Zbl 0546.60055
[7] Ito, K.: Multiple Wiener integral. J. Math. Soc. Japan 3, 157-169 (1951) · Zbl 0044.12202 · doi:10.2969/jmsj/00310157
[8] Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. Ecole d’Et? de Probabilit?s de Saint-Flour XII 1982. (Lect. Notes Math. vol. 1097, pp. 144-303) Berlin Heidelberg New York Tokyo: Springer 1984
[9] Kunita, H.: On backward stochastic differential equations. Stochastics 6, 293-313 (1982) · Zbl 0533.60073
[10] Kuo, H.H., Russek, A.: Stochastic integrals in terms of white noise. Preprint Louisiana State Univ., Baton Rouge LA, USA · Zbl 0636.60053
[11] Kree, M.: Propri?t? de trace en dimension infinie, d’espaces du type Sobolev. Bull. Soc. Math. France 105, 141-163 (1977) · Zbl 0377.46021
[12] Kree, M., Kree, P.: Continuit? de la divergence dans les espaces de Sobolev relatifs ? l’espace de Wiener. Note C.R.A.S. t. 296, 833-836 (1983)
[13] 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
[14] Meyer, P.A.: Transformations de Riesz pour les lois Gaussiennes. S?minaire de Probabilit?s XVIII (Lect. Notes Math. vol. 1059, pp. 179-193) Berlin Heidelberg New York Tokyo: Springer 1984
[15] 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 · Zbl 0633.60075
[16] Nualart, D., Zakai, M.: Generalized stochastic integrals and the Malliavin Calculus. Probab. Theor. Rel. Fields 73, 255-280 (1986) · Zbl 0601.60053 · doi:10.1007/BF00339940
[17] Ocone, D.: Malliavin’s calculus and stochastic integral representation of functionals of diffusion processes. Stochastic 12, 161-185 (1984) · Zbl 0542.60055
[18] Ogawa, S.: Quelques propri?t?s de l’int?grale stochastique du type noncausal. Japan J. Appl. Math. 1, 405-416 (1984) · Zbl 0637.60070 · doi:10.1007/BF03167066
[19] Pardoux, E., Protter, Ph.: Two-sided stochastic integral and calculus. Probab. Theor. Rel. Fields 76, 15-50 (1987) · Zbl 0608.60058 · doi:10.1007/BF00390274
[20] Rosinski, J.: On stochastic integration by series of Wiener integrals. Preprint Univ. North Carolina, Chapell Hill, NC, USA · Zbl 0661.60065
[21] Sekiguchi, T., Shiota, Y.: L 2-theory of noncausal stochastic integrals. Math. Rep. Toyama Univ. 8, 119-195 (1985) · Zbl 0583.60050
[22] Sevljakov, A. Ju.: The It? formula for the extended stochastic integral. Theor. Probab. Math. Statist. 22, 163-174 (1981)
[23] Shigekawa, I.: Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ. 20-2, 263-289 (1980) · Zbl 0476.28008
[24] Skorohod, A.V.: On a generalization of a stochastic integral. Theor. Prob. Appl. 20, 219-233 (1975) · Zbl 0333.60060 · doi:10.1137/1120030
[25] Ustunel, A.S.: La formule de changement de variable pour l’int?grale anticipante de Skorohod. C.R. Acad. Sci., Paris, Ser. I 303, 329-331 (1986)
[26] Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus. Tata Institute of Fundamental Research. Berlin Heidelberg New York Tokyo: Springer 1984
[27] Yor, M.: Sur quelques approximations d’int?grales stochastiques. S?minaire de Probabilit?s XI (Lect. Notes Math. vol. 581, pp. 518-528) Berlin Heidelberg New York Tokyo: Springer 1977
[28] Zakai, M.: The Malliavin calculus. Acta Appl. Math. 3-2, 175-207 (1985) · Zbl 0553.60053 · doi:10.1007/BF00580703
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.