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.
In the post Other versions of a weak Ito formula?, they also give a proof in the case of Sobolev function $F\in W_{\text{loc}}^{2,p}(\mathbb{R}^{n})$.
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 $F$.
- Some other extensions (eg. convex $F$) are listed here Can we apply Ito formula to quadratic variation of $C^1$ function of semimartingale?.
