zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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

MSC:
60H05Stochastic integrals
60H07Stochastic calculus of variations and the Malliavin calculus
60H30Applications of stochastic analysis
60J65Brownian motion
References:
[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
[3][BY] Barlow, M., Yor, M.: Semi-martingale inequalities via Garsia-Rodemich Rumsey lemma. Application to local times. J. Funct. Anal.49, (2) (1982)
[4][BH] Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener space. Berlin New York: Walter de Gruyter 1991
[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
[10][Je] Jeulin, T.: Semi-martingales et grossissement d’une filtration. (Lect. Notes Math., vol. 833) Berlin Heidelberg New York: Springer 1980
[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) · doi:10.1137/1134084
[12][K1] Kunita, H.: On backward stochastic differential equations. Stochastics6, 293-313 (1982)
[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
[20][R] Rosinski, J.: On stochastic integration by series of Wiener integrals. Technical report no 112. Chapel Hill (1985)
[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
[23][SU] Solé, J.L., Utzet, F.: Stratonovich integral and trace. Stochastics29, 203-220 (1990)
[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)