# Ito's Formula for functions that are $C^2$ almost everywhere

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.

• 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. – Mateusz Kwaśnicki Sep 13 '19 at 8:11
• Also, see this question at Math.SE. – Mateusz Kwaśnicki Sep 13 '19 at 8:13

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.