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In "On the Gap Between Deterministic and Stochastic Ordinary Differential Equations," The Annals of Probability, Vol. 6, No. 1 (Feb., 1978), pp. 19-41, Hector J. Sussmann showed that a stochastic differential equation can be solved by simply solving, for each sample path of the process, the corresponding non stochastic ordinary differential equation, and that for the particular case of a Wiener process, the solution obtained turns out to be the solution in the sense of Stratonovich.

The paper observes that the results are not valid for equations with several stochastic inputs, because of commutativity issues. My question is: has the problem been solved for the multi-input case? Do comparable results exist?

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    $\begingroup$ In some multidimensional cases, the results remain true see e.g. Doss, H., Liens entre équations différentielles stochastiques et ordinaires Ann. Inst. H. Poincaré Sect. B (N.S.) 13 (1977), 99–125. But there is a real obstruction in general, and the ``modern'' approach to treat this is through the rough paths approach of Terry Lyons. See Lyons, T; Caruana, M; Lévy, T Differential equations driven by rough paths. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004. Lecture Notes in Mathematics, 1908. Springer, Berlin, 2007 for a review $\endgroup$ Dec 24, 2013 at 5:47
  • $\begingroup$ This was a very useful comment, I'm coming back to write thanks, ofer zeitouni! $\endgroup$
    – Pait
    Jan 9, 2014 at 21:11

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