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Let $(X_t)$ be a continuous one dimensional semimartingale and $f: [0,1] \times \mathbb R \rightarrow \mathbb R$ a continuous function such that $\forall x \in \mathbb R$, $f(x,\cdot)$ is $C^2$ and $\forall y \in \mathbb R$, $f(\cdot,y)$ is $C^2$ almost everywhere (and the number of points where $C^2$ fails is finite).

I would like to apply Itô's lemma on $f(t,X_t)$. The problem is that the "usual" statement of the lemma requires that $f$ is $C^2$, which is not the case here. Can I "cut" the integration in Itô's lemma at all points of failure ? For example, suppose that the only point where $C^2$ fail is $\frac 1 2$. Instead applying Itô's lemma on $f(t,X_t) - f(0,X_0)$, I would apply it to $f(t,X_t) - f(\frac 1 2,X_{\frac 1 2})$ and to $f(\frac 1 2,X_{\frac 1 2}) - f(0,X_0)$ and sum the results. Does that break anything ?

W. Volante
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  • If the number of points of failure is finite, you can indeed apply Ito lemma from one point to another and combine the results. – SBF Jun 14 '21 at 13:25
  • I believe that because you have finitely many discontinuities this is fine. – rubikscube09 Jun 14 '21 at 13:25
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    See here: https://mathoverflow.net/questions/341453/itos-formula-for-functions-that-are-c2-almost-everywhere for the more general case. In fact things hold if $f$ is a Sobolev function if one uses more advanced estimates from parabolic PDE theory (ABP estimates) – rubikscube09 Jun 14 '21 at 13:26
  • Does this https://math.stackexchange.com/questions/4965337/itos-formula-for-functions-that-are-c2-almost-everywhere/4965613#4965613 answer your question? – Thomas Kojar Sep 01 '24 at 04:40

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