= Inverse-clock proof of the Dambis-Dubins-Schwarz theorem
{title2=$B_u=M_{T_u},\quad M_t=B_{V_t}$}
For a continuous strictly increasing unbounded bracket $V$, the inverse times $T_u=\inf\{t:V_t>u\}$ satisfy $V_{T_u}=u$ and $T_{V_t}=t$. Optional sampling of the original martingale stopped at bounded bracket levels makes $M_{T_u}$ a continuous local martingale in $\mathcal F_{T_u}$. Time-changing the square-minus-bracket martingale gives its bracket $u$; the <Lévy characterization of Brownian motion> identifies it as Brownian.
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