Abstract
In this paper, we prove that the process of product variation of a two–parameter smooth martingale admits an ∞ modification, which can be constructed as the quasi–sure limit of sum of the corresponding product variation.
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References
Cairoli, R., Walsh, J. B.: Stochastic integrals in the plane. Acta Math., 134, 111–183 (1975)
Merzbach, E., Zakai, M.: Predictable and dual predictable projections for two-parameter stochastic processes. Z. Wahrsch. Verw. Gebiete, 53, 263–269 (1980)
Chevalier, L., Martingales continuous à deux paramèter. Bull. Sci. Math., 106, 19–62 (1982)
Nualart, D.: On the quadratic variation of two-parameter continuous martingales. Ann. Probab., 12, 445–456 (1984)
Malliavin, P., Nualurt, D.: Quasi sure analysis of stochastic flows and Banach space valued smooth functionals on the Wiener space. J. Funct. Anal., 112, 287–317 (1993)
Ren, J.: Analysis quasi sûre des équation différentielles stochastiques. Bull Sci. Math, 114(2), 187–213 (1990)
Liang, Z.: Quasi sure quadratic variations of two-parameter smooth Martingales on the Wiener space. J. Math. Kyoto Univ., 36(3), 619–640 (1996)
Hajek, B.: Stochastic equations of hyperbolic type and a two-parameter stratonovich calculus. Ann. Probab., 10, 451–463 (1982)
Norris, J. R.: Twisted sheets. J. Funct. Anal., 132, 273–334 (1995)
Nualart, D.: Une formule d’Itô pour les martingales continues à deux indices et quelques applications. Ann. Inst. Henri Poincaré, 20, 251–275 (1984)
Sanz, M.: r-variations for two-parameter continuous martingales and Itô’s formula. Stochastic Processes Appl., 32, 69–92 (1989)
Guyon, X., Prum, B.: Differents types de variations-produit pour une semi-martingale représentable a deux paramèteres. Ann. Inst. Fourier, Grenoble, 29(3), 295–317 (1979)
Guyon, X., Prum, B.: Variations-produit et formule de Itô pour les semi-Martingales représentables à deux paramèter. Z. Wahrsch. Verw. Gebiete, 56, 361–397 (1981)
Liu, J.: Two-parameter quasi-sure analysis and stochastic differential equations, Thesis of Huazhong University of Science and Technology, 2003
Huang, Z., Yan, J.: Introduction to Infinite Dimensional Stochastic Analysis, Sicence Press/Kluwer Academic Publishers, Beijing/New York, 2000
Malliavin, P.: Implicit functions in finite corank on the Wiener space, Taniguchi Intern. Symp. SA. Katata, 369–386, 1982, Kinokunia, Tokyo, 1983
Nualart, D., Sanz, M.: Malliavin calculus for two-parameter Wiener functionals. Z. Wahrsch. Verw. Gebiete, 70, 573–590 (1985)
Nualart, D., Sanz, M.: Stochastic differential equations in the plane: smoothness the solution. J. Multi. Anal., 31, 1–29 (1989)
Wong, E., Zakai, M.: Differentiation formula for stochastic integrals in the plane. Stochastic Processes Appl., 6, 339–349 (1978)
Liu, J., Xie, P.: Quasi sure approximation of stochastic integral, Preprint
Ren, J.: Quasi sure quadratic variation of smooth martingales. J. Math. Kyoto Univ., 34(1), 191–206 (1994)
Meyer, P. A.: Théorie élémentaire des process à deux indices. Lecture Notes in Math., 863, 1–39 (1981)
Nualart, D.: Variation quadratiques et inégalités pour les martingales à deux indices. Stochastics, 15, 51–63 (1985)
Kusuoka, S., Stroock, D.: Application of the Malliavin calculus, Part I, Taniguchi Intern. Symp. SA. Katata, 271–306, 1982
Liu, J., Ren, J.: Regularities of local times of two parameter martingales. Journal of Mathematics of Kyoto University, 44(3), 501–521 (2004)
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Supported by Project 973 and NSF (No. 10526020) of China
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Liu, J.C., Ren, J.G. Quasi–sure Product Variation of Two-parameter Smooth Martingales on the Wiener Space. Acta Math Sinica 22, 1103–1114 (2006). https://doi.org/10.1007/s10114-005-0661-y
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DOI: https://doi.org/10.1007/s10114-005-0661-y