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.


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.

  • $\begingroup$ It seems that in Krylov’s book, the Generalized Ito formula is shown for $W^{2,2}$ function before the process exits a bounded region. May you clarify why we need $W^{2,p}$? Is it for extending the formula without the exiting time? $\endgroup$ – John Oct 30 '20 at 1:51

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:



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