Common-clock time change of orthogonal local martingales ($[M^i,M^j{]}_t={\delta }_{ij}{A}_t⟹(M_{{\tau }_r}^i-M_0^i{)}_i\text{ is Brownian}$)

If a continuous vector local martingale has bracket matrix $A_{t}I$ with $A_{\infty }=\infty$, inverse-clock time change gives bracket matrix $rI$. The Lévy characterization of multidimensional Brownian motion then gives Brownian motion in the common time-changed filtration. A continuous local martingale is constant on flat intervals of its bracket, so inverses using $A_t\ge r$ and $A_t>r$ give the same martingale values. The filtration at the earlier inverse is smaller; adaptation and independence from the larger past transfer the Brownian property to it. Nonzero initial values are retained by adding the initial vector back.