I am wondering if there is some characterization of the $n$ dimensional Brownian motion, with some more general covariance matrix $\Sigma(t)$ (rather than just $tI$. Specifically, I am interested in a Brownina process with $\Sigma=tA$ for some arbitrary scalar valued positive definite matrix $A$.
If we denote the process by $X_t$, can we write it for instance as $X_t=\int_0^t B(s)dB_s$ for some matrix valued function $B(s)$? Or does it hold, as it holds for classical $n$ dimensional Brownian motions that $\limsup\limits_{t \to \infty} B_t = \infty$ almost surely? If none of this holds, do you know some other characterization? Thank you!