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
From (2) and (3), we conclude equality holds for (2).
Nice. Do you know any reference on this generalized Dynkin’s Formula? I’m curious what Prof. Krylov responded.