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In the conventional Ito's formula, it is required that $F$ is $C^2$ everywhere. However I've seen mentioning of a slightly weaker condition, where $F$ is $C^1$ everywhere but $C^2$ almost everywhere. Is there any reference for this? I couldn't find it in the textbooks by Oksendal or Protter.

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  • $\begingroup$ I did not look into it, but the title of [R. Aebi, Itô's Formula for Non-Smooth Functions, Publ. RIMS, Kyoto Univ. 28 (1992), 595-602] (link) suggests it may contain an answer. $\endgroup$ – Mateusz Kwaśnicki Sep 13 '19 at 8:11
  • $\begingroup$ Also, see this question at Math.SE. $\endgroup$ – Mateusz Kwaśnicki Sep 13 '19 at 8:13
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There is a comment to this effect in Revuz and Yor's Continuous Martingales and Brownian Motion after the proof for continuous semimartingale vector $X=(X^1,\dots,X^d)$ and $F\in C^2$ (pg 147):

Remark 1°) The differentiability properties of $F$ may be somewhat relaxed. For instance, if some of the $X^{i}$'s are of finite variation, $F$ needs only be of class $C^1$ in the corresponding coordinates; the proof goes through just the same.

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