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In the classical Ito's formula, it is required that $F$ is $C^2(R^n)$. However I was curious if it holds under a weaker condition, where $F$ is $C^1$, $C^2$ almost everywhere. Is there any reference for this?

In thi case, probably one also needs that the process $y(t)$ has an absolutely continuous density wrt to the Lebesgue measure, so that $y(t)\not \in A$ a.s., where $A\subset R^n$ with Lebesgue measure zero is the set where $F$ is not $C^1,C^2$.

The question is also related to the one in https://mathoverflow.net/questions/341453/itos-formula-for-functions-that-are-c2-almost-everywhere which does not have a definite answer.

carlos85
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  • it works for $\mathcal C^1$ and piecewise $\mathcal C^2$. The proof is more or less the same than the normal Itô formula. – Surb Aug 31 '24 at 10:13
  • Piecewise I understand of course. But here $A$ can be very nasty, e.g. $A=\mathbb Q$ – carlos85 Aug 31 '24 at 10:22
  • a function that is not continuous on $\mathbb Q$ is nowhere continuous... – Surb Aug 31 '24 at 10:42
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    How things diverge from Ito's formula even for such simple almost everywhere smooth functions such as $|x|$ is demonstrated by Tanaka's formula which you find in almost every reference on Ito calculus. – Kurt G. Aug 31 '24 at 11:52
  • @Surb, no consider the Thomae’s Function https://en.wikipedia.org/wiki/Thomae%27s_function, which is continuous at every irrational point and discontinuous at very rational point – carlos85 Aug 31 '24 at 13:38
  • @KurtG. does Tanaka's method works also when $A$ is not a finite set (like $\mathbb Q$)? – carlos85 Aug 31 '24 at 13:40
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    Sounds a bit crazy. I won't elaborate further as it is a standard result you should study first. – Kurt G. Aug 31 '24 at 15:52

1 Answers1

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  1. As mentioned in the comments of Itô's Formula for functions that are C2 almost everywhere

in "Ito's Formula for Non-Smooth Functions", they prove a version of Itô formula when $D_{x_{i}}D_{x_{j}}F\in L^{1}$, where the derivatives are understood in the sense of distributions.

  1. In the post Other versions of a weak Ito formula?, they also give a proof in the case of Sobolev function $F\in W_{\text{loc}}^{2,p}(\mathbb{R}^{n})$.

  2. In the post Generalized Ito's formula, they further give the following two references:

  1. Some other extensions (eg. convex F) are listed here Can we apply Ito formula to quadratic variation of $C^1$ function of semimartingale?.
Thomas Kojar
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