Exercise in stochastic analysis

Question

The problem comes from Karatzas's book 'Brownian motion and stochastic analysis'. Exercise 5.20.

Suppose $X$ is in the space of square integrable martingales with stationary, independent increments. Then $\langle X \rangle_t = t(EX_1^2), T \ge 0$.

Answer 1

You have to see $E(X_{1}^2)$ like some kinda variance $\sigma^2$ for process. Standard brownian motion has variance 1, so its bracket process is 1*t.

That being said, depending on the tools you have there is two or three ways to solve this (One being really straightforward). Edit : Yes ! It is $t \cdot \sigma^2$ .