= Solution
The <Dambis-Dubins-Schwarz theorem> states: let $M$ be a <continuous local martingale> with $M_0=0$ and $[M]_\infty=\infty$ almost surely. Set
$$
\tau_u=\inf\{t\ge0:[M]_t>u\},\qquad
\mathcal G_u=\mathcal F_{\tau_u}.
$$
Then $W_u=M_{\tau_u}$ is standard <Brownian motion> relative to $(\mathcal G_u)$ and
$$
\boxed{M_t=W_{[M]_t}.}
$$
Strict increase of the bracket is not required in the general theorem: $M$ is constant on intervals on which its bracket is constant. Continuity of $M$ is required. A nonzero initial value is handled by applying the theorem to $M-M_0$ and adding it back. If the terminal bracket is finite, the <finite-lifetime extension of the Dambis-Dubins-Schwarz theorem> supplies the <Brownian motion> up to that bracket time and, if necessary, an independent continuation on an enlarged probability space.
Back to article page
