I want to show that $B_t^3 - 3tB_t$ is a martingale. I have shown it is a local martingale via its Ito decomposition:
$B_t^3 - 3tB_t = \int_{0}^{t} 3B_t^2 - 3t dB_t$
I want to argue that since it has finite quadratic variation, it is a martingale. In terms of showing that the expectation of the quadratic variation is finite, I think it might be easiest to show that:
$ \mathbb{E}[\langle \int_{0}^{t} 3B_t^2 - 3t dB_t \rangle] $
$= \mathbb{E}[\int_{0}^{t} (3B_t^2 - 3t)^2 d\langle B_t \rangle] $
$= \mathbb{E}[\int_{0}^{t} (3B_t^2 - 3t)^2 dt]$
is finite.
How can I show this is finite? I am stuck at this point.