Let $W_t$ be a $d$-dimensional standard Brownian motion with independent components, and let $\mathcal F_t$ be the natural filtration induced by $W_t$. Moreover, let $G_t$ be a $\mathcal F_t$- adapted process with values in $\mathbb R^{d\times d}$, and define the $d$-dimensional Itô integral $$ M_t := \int_0^t G_s\, dW_s. $$ It is known that $M_t$ is a $\mathbb R^d$-valued martingale and that its quadratic variation is given by $$ \langle M \rangle_t = \int_0^t G_sG_s^\top \, ds. $$ In the one-dimensional case $d = 1$, one can show that there exists a Brownian motion $\hat W_t$ such that it holds $$ M_t = \hat W_{\langle M\rangle_t}, \quad a.s., $$ which is known as a time-change formula for martingales. For dimension $d > 1$, is it possible (and under which conditions) to show that $M_t$ is actually equal to a time-changed rescaled multi-dimensional Brownian motion?
I know there exists Knight's theorem (e.g. [Chapter V, Theorem (1.9) in Revuz & Yor, Continuous Martingales and Brownian Motion]), but under specific conditions on $M_t$. Are there other results, more specific for multi-dimensional Itô integrals?