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XV: 09, 143-150, LNM 850 (1981)
Calcul d'Ito sans probabilités (Stochastic calculus)
It is shown that if a deterministic continuous curve has a ``quadratic variation'' in a suitable sense (which however depends explicitly on a nested sequence of time subdivisions, for example the standard dyadic one), then it satisfies a deterministic ``Ito formula'' when composed with a twice differentiable function. Thus the only place where probability really appears in the derivation of Ito's formula is in the fact that, given any sequence of subdivisions, almost every path of a semimartingale admits a quadratic variation relative to this sequence (though no path may exist which has a quadratic variation relative to all sequences)
Comment: This subject is developed by T. Lyons' work on differential equations driven by non-smooth functions (in Rev. Math. Iberoamericana 14, 1998)
Keywords: Stochastic integrals, Change of variable formula, Quadratic variation
Nature: Original
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