Generalized Ito formula for the functions in Sobolev functions

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 {X} be a diffusion on the filtered probability space {(\Omega, \mathcal{F}, \mathbb{P}, \mathbb{F}= \{\mathcal{F}_t\})}, with generator {L}, initial time {s}, and initial state {x\in \mathbb{R}^d}. Suppose {v\in W_{d+1,loc}^{1,2}(Q)} for some {Q\subset \mathbb{R}^{d+1}}. Define

\displaystyle \tau_Q^{s,x} = \inf\{r>s: X_r \notin Q\}.

Then, for any {\mathbb{F}}-stopping time {\tau \le \tau_Q^{s,x}}, 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)

Proof:

For any {\mathbb{F}}-stopping time {\tau \le \tau_Q^{s,x}}, we have inequality, by Theorem 2.10.2 of book [Kry80],

\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 {v_1 = -v}, then

\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],

which yields

\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).

\Box

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