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
![\displaystyle \mathbb{E}[v(\tau, X_\tau)] = v(s,x) + \mathbb{E} \Big[ \int_s^\tau L v(r,X_r) dr\Big]. \ \ \ \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+++%5Cmathbb%7BE%7D%5Bv%28%5Ctau%2C+X_%5Ctau%29%5D+%3D+v%28s%2Cx%29+%2B+%5Cmathbb%7BE%7D+%5CBig%5B+%5Cint_s%5E%5Ctau+L+v%28r%2CX_r%29+dr%5CBig%5D.+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0 "\displaystyle \mathbb{E}[v(\tau, X_\tau)] = v(s,x) + \mathbb{E} \Big[ \int_s^\tau L v(r,X_r) dr\Big]. \ \ \ \ \ (1)")
Proof:
For any
![\displaystyle \mathbb{E}[v(\tau, X_\tau)] \le v(s,x) + \mathbb{E} \Big[ \int_s^\tau L v(r,X_r) dr\Big]. \ \ \ \ \ (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+++%5Cmathbb%7BE%7D%5Bv%28%5Ctau%2C+X_%5Ctau%29%5D+%5Cle+v%28s%2Cx%29+%2B+%5Cmathbb%7BE%7D+%5CBig%5B+%5Cint_s%5E%5Ctau+L+v%28r%2CX_r%29+dr%5CBig%5D.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0 "\displaystyle \mathbb{E}[v(\tau, X_\tau)] \le v(s,x) + \mathbb{E} \Big[ \int_s^\tau L v(r,X_r) dr\Big]. \ \ \ \ \ (2)")
If we apply (2) to a function 
![\displaystyle \mathbb{E}[v_1(\tau, X_\tau)] \le v_1(s,x) + \mathbb{E} \Big[ \int_s^\tau L v_1(r,X_r) dr\Big],](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathbb%7BE%7D%5Bv_1%28%5Ctau%2C+X_%5Ctau%29%5D+%5Cle+v_1%28s%2Cx%29+%2B+%5Cmathbb%7BE%7D+%5CBig%5B+%5Cint_s%5E%5Ctau+L+v_1%28r%2CX_r%29+dr%5CBig%5D%2C+&bg=ffffff&fg=000000&s=0 "\displaystyle \mathbb{E}[v_1(\tau, X_\tau)] \le v_1(s,x) + \mathbb{E} \Big[ \int_s^\tau L v_1(r,X_r) dr\Big],")
![\displaystyle \mathbb{E}[v(\tau, X_\tau)] \ge v(s,x) + \mathbb{E} \Big[ \int_s^\tau L v(r,X_r) dr\Big]. \ \ \ \ \ (3)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+++%5Cmathbb%7BE%7D%5Bv%28%5Ctau%2C+X_%5Ctau%29%5D+%5Cge+v%28s%2Cx%29+%2B+%5Cmathbb%7BE%7D+%5CBig%5B+%5Cint_s%5E%5Ctau+L+v%28r%2CX_r%29+dr%5CBig%5D.+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0 "\displaystyle \mathbb{E}[v(\tau, X_\tau)] \ge v(s,x) + \mathbb{E} \Big[ \int_s^\tau L v(r,X_r) dr\Big]. \ \ \ \ \ (3)")
From (2) and (3), we conclude equality holds for (2).
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