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Does anybody know a translation of Föllmer: Calcul d'Ito sans probabilités in English or German?

It seems to be a very interesting text - Abstract: "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)"

See also: Convergence and non-convergence of left-point and mid-point Riemann sums and first answer.

Other references along the same lines would also be appreciated - Thank you!

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Time to learn some French. –  Guntram May 23 '10 at 14:24
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3 Answers

up vote 5 down vote accepted

Föllmer's approach was mainly adopted by specialists in Mathematical Finance.

Have a look at Introduction to Stochastic Calculus for Finance by D. Sondermann. This is an intro lecture course based on the work of Föllmer. And the book contains an English translation of the original article by Föllmer in the Appendix.

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Hi vonjd You might be interested in the following article : arxiv.org/abs/1005.4357 –  The Bridge May 25 '10 at 14:50
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By the way I forgot to mention Rough Path Theory There is a book by Victoir and Fritz which seems to be quite complete account on the subject and that treats Itô stochastic Integration Theory


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By the way, there is a recent preprint in arXiv (Nutz2010.Pathwise construction of stochastic integrals) where the author develops a general construction for Stochastic integrals without probability. The construction includes the integrals that can be constructed via Follmer's approach.

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