The generalized Ito formula to the Sobolev spaces will be discussed here, which stems from [Kry80]. I would like to thank Prof Krylov on his confirmation of this result by email communication.

**Proposition 1** * Let be a diffusion on the filtered probability space , with generator , initial time , and initial state . Suppose for some . Define *

*
* Then, for any -stopping time , we have

* *

*Proof:*

For any -stopping time , we have inequality, by Theorem 2.10.2 of book [Kry80],

If we apply (2) to a function , then

which yields

From (2) and (3), we conclude equality holds for (2).

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