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XV: 06, 103-117, LNM 850 (1981)
MEYER, Paul-André
Flot d'une équation différentielle stochastique (Stochastic calculus)
Malliavin showed very neatly how an (Ito) stochastic differential equation on $R^n$ with $C^{\infty}$ coefficients, driven by Brownian motion, generates a flow of diffeomorphisms. This consists of three results: smoothness of the solution as a function of its initial point, showing that the mapping is 1--1, and showing that it is onto. The last point is the most delicate. Here the results are extended to stochastic differential equations on $R^n$ driven by continuous semimartingales, and only partially to the case of semimartingales with jumps. The essential argument is borrowed from Kunita and Varadhan (see Kunita's talk in the Proceedings of the Durham Symposium on SDE's, LN 851)
Comment: The results on semimartingales with jumps have been proved independently by Uppman. Some dust has been swept under the rugs about the non-explosion of the solution, and the results should be considered valid only in the globally Lipschitz case. See also Uppman 1624 and Léandre 1922
Keywords: Stochastic differential equations, Flow of a s.d.e.
Nature: Exposition, Original additions
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