I have another question on Ito's formula, but maybe a rather strange one. In the book of Protter (Theorem 33, pg. 81) we can see that for a general $n$ dimensional semimartingale $(X_t)_{t \geq 0}$ and a $C^2$ function $f$ we can write:
Suppose I have a process $(X_t)_{t \geq 0}$ defined on $[0,T)$. In the interval $[0,T_1)$ it behaves like a semimartingale $(X^1_t)_{t \geq 0}$ and on the interval $[T_1,T)$ it has some other dynamics, given by the semimartingale $(X^2_t)_{t \geq 0}$. How could I write $f(X_{T_1}) - f(X_0)$? If the function $f$ is $C^2$ as we in fact assume then my guess is just it would be just the same as in the picture replaceing $t$ by $T_1$. Is this correct?
Edit: Here $T_1$ is not a fixed point but a stopping time, but by construction it is smaller than $T$.
