could anyone help me with this one line proof?
I couldn't understand the proof in Chapter 1, 4.50 proposition, part(c)，'only if' direction (page53) in Jean Jacod and AN. Shiryear's Limit Theorems for Stochastic Processes.
The line I got confused is the equation on top of page 54: how to see the following relation holds:
$\sup_t E[X_t^2] = \sup_t\lim_{n\rightarrow\infty}\uparrow E[X_{(t\wedge\tau_n)}^2]$.
Here $X$ is a local martingale, and $\tau_n$ is the localizing sequence for the local martingale $X^2 - X_0^2 - [X,X]$. For the 'only if' direction, it is also assumed that $[X,X]\in\mathcal{A}$, i.e $E([X,X]_\infty) < \infty$ and $X_0\in L^2$.
I was trying to convince myself $E[X_t^2] = \lim_{n\rightarrow\infty}\uparrow E[X_{(t\wedge\tau_n)^2}]$. I do know $\lim_{n\rightarrow\infty} X^2_{t\wedge\tau_n} = X_t^2$ a.s, and $\{Y_n:= X_{t\wedge\tau_n}, n\in\mathbb{N}\}$ is a martingale, hence $Y_n^2 = X^2_{t\wedge\tau_n}$ is a submartingale, and this makes righthand of the above equation as the limit of an increasing sequence. But I don't see the monotonicity of the sequence $X^2_{t\wedge\tau_n}$, nor a dominating random variable for it. So why the switch order of the limit and the expectation hold? I suppose this is some martingale inequality that I don't know?
Thank you so much!
Since $\mathbb{E}[X,X]_{\infty}<\infty$ and $[X,X]_t$ is non-decreasing (in $t$), we have $\mathbb{E}[X,X]_t < \infty$ for each $t \geq 0$. Applying Doob's maximal inequality we find that
$$\mathbb{E}\left( \sup_{s \leq t \wedge \tau_n} X_s^2 \right) \leq 4 \mathbb{E}(X_{t \wedge \tau_n}^2) = 4 (\mathbb{E}(X_0^2)+\mathbb{E}[X,X]_{t \wedge \tau_n}).$$
By monotonicity, $\mathbb{E}[X,X]_{t \wedge \tau_n} \leq \mathbb{E}[X,X]_t<\infty$, and so
$$\mathbb{E}\left( \sup_{s \leq t \wedge \tau_n} X_s^2 \right) \leq 4 (\mathbb{E}(X_0^2)+\mathbb{E}(X_t^2)).$$
It follows from the monotone convergence theorem that
$$\mathbb{E}\left( \sup_{s \leq t} X_s^2 \right) \leq 4 (\mathbb{E}(X_0^2)+\mathbb{E}(X_t^2))< \infty.$$
In particular, $\sup_{s \leq t} X_s^2$ is an integrable dominating function for $X_{t \wedge \tau_n}^2$. Since $X_{t \wedge \tau_n} \to X_t$ almost surely, the dominated convergence theorem gives
$$\mathbb{E}(X_t^2) = \lim_{n \to \infty} \mathbb{E}(X_{t \wedge \tau_n}^2).$$