Brownian motion produced by the Dambis-Dubins-Schwarz theorem need not be independent of its clock
= Brownian motion produced by the Dambis-Dubins-Schwarz theorem need not be independent of its clock
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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, $M_t=(W_t^2-t)/2=\int_0^tW_s\,dW_s$ has clock $\int_0^tW_s^2ds$, and their dependence can be detected by a nonzero mixed moment.