I am reading the book: "P. Protter, Stochastic Integration and Differential Equations. Second edition, Springer-Verlag, (2004)", where the following are mentioned:
(1) Local martingales are not always locally square integrable.
(2) A martingale $M$ is square integrable, namely, $\,\mathbb{E}(M_t^2)<\infty$ for all $t\in[0,\infty)$, if and anly if $\,\,\mathbb{E}[M]_t<\infty\,$ for all $t\in[0,\infty)$.
(3) Any local martingale is a special semimartingale.
On top of that, in the exercise part of Chap IV, it says as:
Exercise 1. A semimartingale $X$ is special if and only if the increasing process $[X]$ is locally integrable, that is, there exists a sequence of stopping times $\,T_n\underset{n}{\nearrow}\infty$ a.s. such that $\,\mathbb{E}[X]_{T_n}<\infty$ for each $n\in\mathbb{N}$.
I could not understand the assertion in Exercise 1. If the assertion is correct, given any local martingale $M$, which is a special semimartingale by (3), we can choose a sequence of stopping times $S_n\le T_n$ which tends to $\infty$ a.s. and which makes each $M_{\cdot\wedge S_n}$ martingale, and hence $M_{\cdot\wedge S_n}$ becomes square integrable martingale by (2). This contradicts to (1).
The assertion in Exercise 1 above implies that local martingales $M$ always have locally integrable quadratic variation (increasing process) $[M]$.
I would realy appreciate if you tell us what is correct and what is wrong in this argument.