Let $M$ be a continuous local martingale with $M_0 = 0$ and $\langle M \rangle_\infty = \infty$ almost surely. Define the time change $\tau_s = \inf\{t \ge 0 : \langle M \rangle_t > s\}$ for $s \ge 0$. Then the process $W_s = M_{\tau_s}$ is a standard Brownian motion (with respect to the filtration $(\mathcal{F}_{\tau_s})_{s \ge 0}$), and \begin{align*} M_t = W_{\langle M \rangle_t} \quad \text{almost surely for all } t \ge 0. \end{align*}