Consider classical statement of Ito's formula: Let $X$ be a continuous
semimartingale and $F \in C^2(\mathbb{R}^d, \mathbb{R})$; then $F(X)$
is a continuous semimartingale and
$$F(X\_t) = F(X\_0) + \sum\_i \int\_0^t {\partial\_i F} dX\_s^i + \frac 1 2
\sum\_{i,j} {\partial^2\_{ij} F} d \langle X^i, X^j \rangle\_s.$$
In the above Ito formula, how much does function $F$ extendable in a
Sobolev space? For example, is Ito formula true if $F\in W^{2,p}$ for
some $p>1$? Note that, if we use Ito-Tanaka formula, then there exists 
some extra term from local time, and we wish to find Sobolev regularity
to make sure this term being zero.



Generalized Ito's formula
One can also use the Alexandrov-Bakelman-Pucci-Krylov-Tso estimates from parabolic PDE to show that Ito's Lemma holds for functions in $W^{2,p}$ when $X$ is a diffusion with uniformly positive definite covariance and $p$ is large enough. This result be found, for example, in Krylov's "Controlled Diffusion Processes" Ch 2 Section 10. 


