Merzbach, Ely; Zakai, Moshe Bimeasures and measures induced by planar stochastic integrators. (English) Zbl 0615.60034 J. Multivariate Anal. 19, 67-87 (1986). The class of all one parameter processes which are stochastic integrators with respect to predictable processes was characterized by Dellacherie [see C. Dellacherie and P.-A. Meyer, Probabilités et potentiel. Chapitres V à VIII: Théorie des martingales. (1980; Zbl 0464.60001)] and K. Bichteler [Ann. Probab. 9, 49-89 (1981; Zbl 0458.60057)]. The two-parameter case is considerably more difficult. For example, a weak martingale need not be a stochastic integrator. The author characterizes two-parameter stochastic processes that are \(L^ 2\) stochastic integrators in terms of measures and bimeasures defined on the product spaces. It is shown that X is an \(L^ 2\) stochastic integrator iff there exists a bimeasure \(\beta\) (d\(\omega\times dz\), \(d\omega\) ’\(\times dz')\) on \({\tilde \Omega}\times {\tilde \Omega}\) where \({\tilde \Omega}=(\Omega \times R_{z_ 0})\), such that \[ E\int_{R_{z_ 0}}\phi dX\int_{R_{z_ 0}}\psi dX=\int_{{\tilde \Omega}}\int_{{\tilde \Omega}}\phi_{\zeta}(\omega)\psi_{\zeta}(\omega ')\beta (d\omega \times dz,\quad d\omega '\times dz'). \] This condition is further specialized to processes \(M^ 1_{s,t}\) that are 1-martingales. Then the author considers a special class of \(L^ 2\) integrators that is named the class of ”measure inducing integrators” because their ”double Doleans function” can be extended to a finite measure on some \(\sigma\)- field. It is proved that if \(X_ z\) is a ”measure inducing integrator” and the conditional independence assumption (F-4) is satisfied then \(X^ 2_ z\) admits a Doob-Meyer-Cairoli decomposition. Reviewer: Yu.S.Mishura Cited in 6 Documents MSC: 60G05 Foundations of stochastic processes 60H05 Stochastic integrals Keywords:stochastic integrators; weak martingale; bimeasures; Doob-Meyer-Cairoli decomposition Citations:Zbl 0464.60001; Zbl 0458.60057 PDFBibTeX XMLCite \textit{E. Merzbach} and \textit{M. Zakai}, J. Multivariate Anal. 19, 67--87 (1986; Zbl 0615.60034) Full Text: DOI References: [1] Bakry, D., Semimartingales à Deux indices, (Sem. de Proba. XVI. Sem. de Proba. XVI, Lecture Notes in Math., Vol. 920 (1982), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York), 355-369 · Zbl 0473.60050 [2] Bichteler, K., Stochastic integration and \(L_p\)-theory of semimartingales, Ann. Probab., 9, 49-89 (1981) · Zbl 0458.60057 [3] Brennan, M. D., Planar semimartingales, J. Multivariate Anal., 9, 465-486 (1979) · Zbl 0456.60049 [4] Cairoli, R.; Wlash, J. B., Stochastic integrals in the plane, Acta Math., 134, 11-183 (1975) [5] Cairoli, R.; Walsh, J. B., Régions d’Arrêt, localisations et prolongement de Martingales, Z. Wahrsch. Verw. Geb., 44, 276-306 (1978) · Zbl 0369.60043 [6] Clarkson, J. A.; Adams, C. R., On definitions of bounded variation for functions of two variables, Trans. Amer. Math. Soc., 35, 824-854 (1933) · Zbl 0008.00602 [7] Dellacherie, C.; Meyer, P. A., (Probabilités et Potentiel, Vol. I and Vol. II (1980), Hermann: Hermann Paris) [8] Dunford, N.; Schwartz, J. T., (Linear Operators, Part I, General Theory (1963), Interscience: Interscience New York) [9] Follmer, H., Quasimartingales à Deux Indices, C. R. Acad. Sci. Paris Ser. I. Math., 288, 61-64 (1979) · Zbl 0397.60044 [10] Frechet, M., Sur les fonctionnelles bilinéaires, Trans. Ann. Math. Soc., 16, 215-234 (1915) · JFM 45.0546.01 [11] Hidebrandt, T. H., (Introduction to the Theory of Integration (1963), Academic Press: Academic Press New York) [12] Horowitz, J., Une Remarque sur les Bimesures, (Sem. de Proba. XI. Sem. de Proba. XI, Lecture Notes in Math., Vol. 581 (1977), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York), 59-64 · Zbl 0367.28007 [13] Hurzeler, H. E., Quasimartingale und Stochastische Integratoren Mit Halbgeordnaten Indexmengen, Dissertation ETH Zurich (1982) [14] Levy, P., Sur les fonctionnelles bilinéaires, C. R. Acad. Sci. Paris Ser. I. Math., 222, 125-127 (1946) · Zbl 0063.03532 [15] Loomis, L. H., (An Introduction to Abstract Harmonic Analysis (1953), Van Nostrand: Van Nostrand New York) · Zbl 0052.11701 [16] Merzbach, E., (Processus Stochastiques à Indices Partiellement Ordonnés, Vol. 55 (1979), Ecole Polyt: Ecole Polyt Palaiseau), Rap. Int. [17] Merzbach, E.; Zakai, M., Predictable and dual predictable projections of two-parameter stochastic processes, Z. Wahrsch. Verw. Gebiete, 53, 263-269 (1980) · Zbl 0437.60040 [18] Meyer, P. A., Théorie Elémentaire des Processus à Deux Indices, (Lecture Notes in Math., Vol. 863 (1981), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York), 1-39 · Zbl 0461.60072 [19] Morse, M.; Transue, W., \(C\)-Bimeasures ▵ and their integral extensions, Ann. of Math., 64, 480-504 (1956) · Zbl 0073.27302 [20] Parthasarathy, K. R., (Probability Measures on Metric Spaces (1967), Academic Press: Academic Press New York) · Zbl 0153.19101 [21] Stoica, L., On two-parameter semimartingales, Z. Wahrsch. Verw. Gebiete., 45, 257-268 (1978) · Zbl 0373.60057 [22] Wong, E.; Zakai, M., Weak martingales and stochastic integrals in the plane, Ann. Probab., 4, 570-586 (1976) · Zbl 0359.60053 [23] Zakai, M., Some remarks on integration with respect to weak martingales, (Lecture Notes in Math., 863 (1981), Springer Verlag: Springer Verlag Berlin-Heidelberg-New York), 149-161 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.