Dambis-Dubins-Schwarz theorem

Let $M$ be a continuous local martingale with $M_0=0$ and $[M{]}_{\infty }=\infty$. Forthe process $B_s=M_{{\tau }_s}$ is Brownian motion andWhen $[M{]}_{\infty }<\infty$, one can enlarge the probability space and continue $B$ independently beyond that time.

The Brownian motion $B$ in $M_t=B_{[M{]}_t}$ is generally constructed from $M$, so it need not be independent of the quadratic variation $[M]$. For example, $M_t=(W_t^2-t)/2={\int }_0^t{W}_{s}d{W}_s$ has clock ${\int }_0^t{W}_s^{2}ds$, and their dependence can be detected by a nonzero mixed moment.