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

Question

In the conventional Itô'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.

Answer 1

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.

Answer 2

1. As mentioned by Mateusz Kwaśnicki in the comments, in "Itô's Formula for Non-Smooth Functions", Aebi proves 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.

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

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

• Gerard referenced Krylov's "Controlled Diffusion Processes" Ch. 2 Section 10.
• Anatoly Kochubei referenced H. Föllmer and Ph. Protter, On Itô's formula for multidimensional Brownian motion. Probab. Theory Relat. Fields 116, No.1, 1–20 (2000). Consider a $d$-dimensional Brownian motion $X = (X_1,\dotsc,X_d )$ and a function $F$ which belongs locally to the Sobolev space $W^{1,2}$. We prove an extension of Itô’s formula where the usual second order terms are replaced by the quadratic covariations $[f_k (X), X_k ]$ involving the weak first partial derivatives $f_k$ of $F$.

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