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Usually Ito's lemma is stated for $C^{1,2}(\mathbb{R}^{d+1},\mathbb{R})$ functions.

My question is does Ito still hold if the domain is restricted. That is if the semi-martingale $Z_t$ is only considered on $[t_1,t_2]\times B$ where B is an open ball in $\mathbb{R}^d$ and $f$ is $C^{1,2}$ thereon then does the Ito formula still hold?

More generally,what i need precisely is to know if $f$ is a piecewise $C^{1}$ function in the time parameter and twice differentiable in the space parameter, then does Ito still hold? If not can it be modified, if so how?

References are always welcome.

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Ito for diffusion is local so it holds locally if conditions are local.

Let B a open ball of $\mathbb{R}^d $ Let I be a open time interval. Let f be $C^{1,2}(I,B) $. Let $A\subset B $ strictly in $B $ (you take margins wrt to the boundary). Let $t_1,t_2\in I $ with $t_1 <t_2$ and $x\in B $

Then define $\tau$ the exit time of $A $ starting from $t_1,x$

You can now write Ito lemma between $t_1$ and $\min (\tau ,t_2)$

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