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I am currently reading about Föllmer's construction of a stochastic Integral that is defined in a pathwise sense. But I am not sure what exactly the purpose of such a construction is. The main applications seem to be in the area of Finance. But in my studies in that area I have not run into situations where the fact that the Itö integral is defined probabalistic causes any problems.

Could someone maybe provide an intuition as to why such a construction is useful or maybe give an example of when it is?

Jose Avilez
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El Duderino
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  • Not sure this is the same construction but you can try to look up "applications of rough path theory" to get some ideas. – Ian Jan 08 '21 at 16:44
  • @Ian there is a connection between rough paths theory and Föllmer's integral. For the details, see section 5.3 of "A course on Rough Paths" by Friz and Hairer. – Jose Avilez Aug 06 '21 at 21:08

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Föllmer's pathwise stochastic calculus often shows up in robust finance, as it allows us to replicate many classical results in a model-free manner. In particular, the theory focuses on $C[0,1]$ paths which admit quadratic variation along a refining sequence of partitions. The advantage of model-free finance lies in mitigating so-called "Knightian risk", which refers to the appearance of unknown unknowns when misspecifying a probability model.

One neat example of its use is shown in this paper which recovers the Black-Scholes PDE in a pathwise manner. For an exhaustive list of examples, you may see the list of papers that cite Föllmer's Calcul d'Itô sans probabilités.

As an aside, Föllmer's pathwise calculus is far more elementary than semi-martingale calculus, which makes it a good tool for teaching mathematical finance to undergraduate students.

Jose Avilez
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