I am studying stochastic calculus on my own so this should be a basic question. I saw in https://link.springer.com/article/10.1007/BF01259552 the statement that for $B$ an brownian matrix (with independent brownian motions as components), $\ln(\det(B^TB))$ is a local martingale.
I understand that I could use Itos formula to rewrite it as a integral over brownian motion since ln is twice differentiable, but I have trouble formulating the exact properties satisfied so I can conclude that $\ln(\det(B^TB))$ is a local martingale. I am mostly confused because in the determinant I have squares of brownian motions. I feel like this should be pretty basic and since I am not able, I would appreciate every detail.
This could be related to Logarithm of Brownian Motion - local martingale but not martingale.
Edit: I have just realized, that I have not given the whole statement, namely the author of the paper looks at $t_0 = \inf (t: det(B^TB) = 0)$ and says that $\ln(\det(B^TB))$ is a local martingale on the interval $[0,t_0)$. I have not seen that processes are local martingales on some interval of a stopping time. Is then the conclusion that it is a local martingale on the interval trivial because on this interval the logarithm is twice differentiable and therefore itos formula is applicable?