zbMATH — the first resource for mathematics

Forward, backward and symmetric stochastic integration. (English) Zbl 0792.60046
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.
Reviewer: F.Russo

60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J65 Brownian motion
Full Text: DOI
[1] [AP] Asch, J., Potthoff, J.: Ito lemma without non-anticipatory conditions. Probab. Theory Relat. Fields88, 17-46 (1991) · Zbl 0695.60054 · doi:10.1007/BF01193581
[2] [B] Balakrishnan, A.V.: Applied functional analysis. 2nd edn. Berlin Heidelberg New York: Springer 1981 · Zbl 0459.46014
[3] [BY] Barlow, M., Yor, M.: Semi-martingale inequalities via Garsia-Rodemich Rumsey lemma. Application to local times. J. Funct. Anal.49, (2) (1982) · Zbl 0505.60054
[4] [BH] Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener space. Berlin New York: Walter de Gruyter 1991 · Zbl 0748.60046
[5] [BM] Berger, M.A., Mizel, V.J.: An extension of the stochastic integral. Ann. Probab.10, (2) 435-450 (1982) · Zbl 0499.60066 · doi:10.1214/aop/1176993868
[6] [DM] Dellacharie, C., Meyer, P.A.: Probabilités et Potentiel, Chapitres V à VIII, Théorie des martingales. Paris: Hermann 1975
[7] [DS] Dunford, N., Schwartz, J.T.: Linear Operators. Part I, General Theory. New York: Wiley-Intersciene 1967
[8] [HM] Hu, Y.Z., Meyer, P.A.: Sur l’approximation des intégrales multiples de Stratonovich. (Preprint)
[9] [Ja] Jacod, J.: Calcul stochastique et problèmes de martingales (Lect. Notes Math., vol. 714) Berlin Heidelberg New York: Springer 1979 · Zbl 0414.60053
[10] [Je] Jeulin, T.: Semi-martingales et grossissement d’une filtration. (Lect. Notes Math., vol. 833) Berlin Heidelberg New York: Springer 1980 · Zbl 0444.60002
[11] [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) · Zbl 0766.60061 · doi:10.1137/1134084
[12] [K1] Kunita, H.: On backward stochastic differential equations. Stochastics6, 293-313 (1982) · Zbl 0533.60073
[13] [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
[14] [KR] Kuo, H.H., Russek, A.: White noise approach to stochastic integration J. Multivariate Anal.24, 218-236 (1988) · Zbl 0636.60053 · doi:10.1016/0047-259X(88)90037-1
[15] [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
[16] [NP] Nualart, D., Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Theory Relat. Fields78, 535-581 (1988) · Zbl 0629.60061 · doi:10.1007/BF00353876
[17] [NZ] Nualart, D., Zakaï, M.: Generalized stochastic integrals and the Malliavin calculus. Probab. Theory Relat. Fields73, 255-280 (1986) · Zbl 0601.60053 · doi:10.1007/BF00339940
[18] [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) · Zbl 0551.60058 · doi:10.2748/tmj/1178228902
[19] [RY] Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Berlin Heidelberg New York: Springer 1991 · Zbl 0731.60002
[20] [R] Rosinski, J.: On stochastic integration by series of Wiener integrals. Technical report no 112. Chapel Hill (1985) · Zbl 0661.60065
[21] [RV] Russo, F., Vallois, P.: Intégrales progressive, rétrograde et symétrique de processus non adaptés. Note C.R. Acad. Sci. Sér. I312, 615-618 (1991)
[22] [S] Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton: Princeton University Press 1970 · Zbl 0207.13501
[23] [SU] Solé, J.L., Utzet, F.: Stratonovich integral and trace. Stochastics29, 203-220 (1990) · Zbl 0706.60056
[24] [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) · Zbl 0725.60053 · doi:10.1007/BF01199789
[25] [W] Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus. Bombay: Tata Institute of Fundamental Research. Berlin Heidelberg New York: Springer 1984
[26] [Z] Zakaï, M.: Stochastic integration, trace and skeleton of Wiener functionals. Stochastics33, 93-108 (1990) · Zbl 0722.60049
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.