I need help solving the following exercise out of chapter 1 of Karatzas, Shreve:
5.20 Exercise. Suppose $X\in\mathscr{M}_2$ has stationary, independent increments, and $(\mathscr{F}_t)_t$ is the filtration generated by $X$. Show that $\langle X\rangle_t=t\mathbb{E}[X_1^2]$ for all $t\geq 0$.
My approach: I only need to show that the process $(X^2-t\mathbb{E}[X_1^2])_t$ is a martingale. Integrability and adaptedness are clear. Thus only the martingale-property is left: Let $0\leq s\leq t$, then $$ \mathbb{E}[X_t^2-t\mathbb{E}[X_1^2]|\mathscr{F}_s] = \mathbb{E}[X_t^2|\mathscr{F}_s]-t\mathbb{E}[X_1^2]=\mathbb{E}[X_t^2-X_s^2|\mathscr{F}_s]+X_s^2-t\mathbb{E}[X_1^2]. $$ I have that $X_t-X_s$ is independent of $\mathscr{F}_s$ by independence of increments. I would like to do the following $$ \tag{a}\mathbb{E}[X_t^2-X_s^2|\mathscr{F}_s]=\mathbb{E}[(X_t-X_s)(X_t+X_s)|\mathscr{F}_s]=\mathbb{E}[X_t^2-X_s^2]. $$ Question 1. Is it true that $(X_t-X_s)(X_t+X_s)$ is independent of $\mathscr{F}_s$? Can I argue in this way?
Then I would like to show that for any $r\geq 0$ the following holds: $$ \tag{b}\mathbb{E}[X_r^2]=r\mathbb{E}[X_1^2]. $$ As $\mathbb{E}[X_r]=\mathbb{E}[X_0]=0$ it is enough to show that $\frac{1}{\sqrt{a}}X_a\sim X_1$ (equal in distribution). But I think that this only holds for a Brownian motion but not for a general $L^2$ martingale. I would like to exploit the fact that $X$ has stationary increments, but comparing $X_r$ and $X_1$ does not yield any good.
Question 2. Is (b) true? If yes, how to continue?
Probably this argument is wrong and one needs to be more precise since I can't figure out, where to exploit the filtration.
Question 3. Where do we use that the filtration is generated by $X$?