Question

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.

Answer 1

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.

Answer 2

See the paper

H. F\"ollmer and Ph. Protter, On Itô's formula for multidimensional Brownian motion. Probab. Theory Relat. Fields 116, No.1, 1-20 (2000)

and its Zentralblatt review with further references:

http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0955.60077&format=complete