Summary
We define three types of non causal stochastic integrals: forward, backward and symmetric. Our approach consists in approximating the integrator. Two optics are considered: the first one is based on traditional usual stochastic calculus and the second one on Wiener distributions.
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References
[AP] Asch, J., Potthoff, J.: Ito lemma without non-anticipatory conditions. Probab. Theory Relat. Fields88, 17–46 (1991)
[B] Balakrishnan, A.V.: Applied functional analysis. 2nd edn. Berlin Heidelberg New York: Springer 1981
[BY] Barlow, M., Yor, M.: Semi-martingale inequalities via Garsia-Rodemich Rumsey lemma. Application to local times. J. Funct. Anal.49, (2) (1982)
[BH] Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener space. Berlin New York: Walter de Gruyter 1991
[BM] Berger, M.A., Mizel, V.J.: An extension of the stochastic integral. Ann. Probab.10, (2) 435–450 (1982)
[DM] Dellacharie, C., Meyer, P.A.: Probabilités et Potentiel, Chapitres V à VIII, Théorie des martingales. Paris: Hermann 1975
[DS] Dunford, N., Schwartz, J.T.: Linear Operators. Part I, General Theory. New York: Wiley-Intersciene 1967
[HM] Hu, Y.Z., Meyer, P.A.: Sur l'approximation des intégrales multiples de Stratonovich. (Preprint)
[Ja] Jacod, J.: Calcul stochastique et problèmes de martingales (Lect. Notes Math., vol. 714) Berlin Heidelberg New York: Springer 1979
[Je] Jeulin, T.: Semi-martingales et grossissement d'une filtration. (Lect. Notes Math., vol. 833) Berlin Heidelberg New York: Springer 1980
[JK] Johnson, G.W., Kallianpur, G.: Some remarks on Hu and Meyer's paper and infinite dimensional calculus on finite additive canonical Hilbert space. Theory Probab. Appl. (SIAM)34, 679–689 (1989)
[K1] Kunita, H.: On backward stochastic differential equations. Stochastics6, 293–313 (1982)
[K2] Kunita, H.: Stochastic differential equations and stochastic flow of diffeomorphisms. Ecole d'été de Saint-Flour XII. (Lect. Notes Math., vol. 1097) Heidelberg New York: Springer 1982
[KR] Kuo, H.H., Russek, A.: White noise approach to stochastic integration J. Multivariate Anal.24, 218–236 (1988)
[N] Nualart, D.: Non causal stochastic integrals and calculus. Stochastic analysis and related topics (Proceedings Silivri 1986). Korzelioglu, H., Ustunel, A.S. (eds.) (Lect. Notes Math., vol. 1316, pp. 80–129) Berlin Heidelberg New York: Springer 1986
[NP] Nualart, D., Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Theory Relat. Fields78, 535–581 (1988)
[NZ] Nualart, D., Zakaï, M.: Generalized stochastic integrals and the Malliavin calculus. Probab. Theory Relat. Fields73, 255–280 (1986)
[O] Ogawa, S.: Une remarque sur l'approximation de l'intégrale stochastique du type noncausal par une suite d'intégrales de Stieltjes. Tohoku Math. J.36, 41–48 (1984)
[RY] Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Berlin Heidelberg New York: Springer 1991
[R] Rosinski, J.: On stochastic integration by series of Wiener integrals. Technical report no 112. Chapel Hill (1985)
[RV] Russo, F., Vallois, P.: Intégrales progressive, rétrograde et symétrique de processus non adaptés. Note C.R. Acad. Sci. Sér. I312, 615–618 (1991)
[S] Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton: Princeton University Press 1970
[SU] Solé, J.L., Utzet, F.: Stratonovich integral and trace. Stochastics29, 203–220 (1990)
[T] Thieullen, M.: Calcul stochastique non adaté pour des processus à deux paramètres: formules de changement de variables de type Stratonovitch et de type Skorohod. Probab. Theory Relat. Fields89, 457–485 (1991)
[W] Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus. Bombay: Tata Institute of Fundamental Research. Berlin Heidelberg New York: Springer 1984
[Z] Zakaï, M.: Stochastic integration, trace and skeleton of Wiener functionals. Stochastics33, 93–108 (1990)