Localisation in the proof of Ito's formula

Question

I am reading Karatza's and Schreve's book "Stochastic Calculus and Brownian Motion" and I don't understand a strange thing as follows:

Let $X=X_0 + A +M $ be a semimartingale, where $A$ is a continuous process of bounded variation and $M$ is a continuous local martingale. In the proof of Itó's formula, he defines a sequence of stopping times $(T_n)$ by

$$ T_n = \begin{array}{cc} \Bigg\{ & \begin{array}{cc} 0, & |X_0| > n \\ \inf\{ t \geq 0: |A_t| > n \text{ or } |M_t|>n \text{ or } \langle M \rangle_t >n\}, & |X_0| \leq n. \end{array}\end{array} $$

Then he claims that by considering the stopped process $X^{T_n}$, we can assume that $X_0$ is bounded. I don't understand this, as $X^{T_n}_0 = X_0$ is not necessarily bounded.

Answer 1

The thing is that the formula is trivially true at time $t=0$ so you only have to prove it for the case where $|X_0|<n$, which is why Karatzas and Shreve only consider this case making $|X_0|<n$ a property that $X$ respects.

Best regards

Answer 2

You are right, the stopped process is not bounded, if $X_0$ is not bounded:

For fixed $n$ there is a set $A$ with $\mathbb P(A)>0$, such that $|X_0| > n$ on $A$ and by $T_n$'s definition: $|X^{T_n}_t|=|X_0|>n$ on $A$.

However, the proof still works, because it only involves differences of the form $X^{T_n}_t-X^{T_n}_s$, which are combinations of the processes of $A^{T_n}$ and $M^{T_n}$ and thus bounded.