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Ito formula for \(C^ 1\)-functions of semimartingales. (English) Zbl 0838.60045
We establish an Itô formula for \(C^1\) functions of processes whose time reversals are semimartingales and for \(C^1\) functions whose first derivatives are Hölder continuous of any parameter and the process comes out from a stochastic flow of homeomorphism.
Reviewer: F.Russo

MSC:
60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65 Brownian motion
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[1] [B] Bertoin, J.: Les processus de Dirichlet en tant qu’espaces de Banach. Stochastics18, 155-168 (1968) · Zbl 0602.60069
[2] [CJPS] Cinlar, E., Jacod, J., Protter, P., Sharpe, M.J.: Semimartingales and Markov processes. Z. Wahrscheinlicht, Verw. Geb.54, 161-219 (1980) · Zbl 0443.60074
[3] [D] Dozzi, M.: Processes with a multidimensional parameter. Pitman Research Notes Math. vol. 194 1989 · Zbl 0663.60039
[4] [DS] Dunford, N., Schwarz, J.T.: Linear operators. Part I. General theory. New York: Publishers Inc., 1967
[5] [Fo] Föllmer, H.: Calcul d’Itô sans probabilités. Séminaire de Probabilités XV, 1979/1980 (Leet. Notes Math., vol. 850, pp. 143-150) Berlin: Springer
[6] [FPS] Föllmer, H., Protter, P., Shiryaev, P.: Preprint. J. Bernoulli Soc., to appear in Bernoullli
[7] [Fu] Fukushima, M.: Dirichlet forms and Markov processes. Amsterdam:, North Holland 1980 · Zbl 0422.31007
[8] [HP] Haussmann, U., Pardoux, E.: Time reversal of diffusions. Ann. Probab.14, 1188-1205 (1983) · Zbl 0607.60065
[9] [J] Jeulin, T.: Grossissement de filtration (Lect. Notes Math., vol. 833) Berlin: Springer 1980
[10] [K] Kunita, H.: Lectures on stochastic flows and applications. Bombay: Tata Institute of Fundamental Research (1986) · Zbl 0625.60073
[11] [LZ] Lyons, T.J., Zhang, T.S.: Decomposition of Dirichlet processes and its applications. Ann. Probab.22, 494-524 (1994) · Zbl 0804.60044
[12] [MNS] Millet, A., Nualart, D., Sanz, M.: Integration by parts and time reversal for diffusion processes. Ann. Probab.17, 208-238 (1989) · Zbl 0681.60077
[13] [P] Pardoux, E.: Grossissement d’une filtration et retournement du temps d’une diffusion. Séminaire de Probabilités XX (Lect. Notes Math. 1204) Berlin: Springer 1986 · Zbl 0607.60042
[14] [Pr] Protter, P.: Stochastic integration and differential equations. A new approach. Berlin: Springer 1990 · Zbl 0694.60047
[15] [RV1] Russo, F., Vallois, P.: Forward, backward and symmetric stochastic integration. Probab. Theory Rel. Fields97, 403-421 (1993) · Zbl 0792.60046
[16] [RV2] Russo, F., Vallois, P.: The generalized covariation process and Itô formula. Stochastic Processes Appl., to appear (1995) · Zbl 0840.60052
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