= Brownian motion produced by the Dambis-Dubins-Schwarz theorem need not be independent of its clock
{c}
The <Brownian motion> $B$ in the <Dambis-Dubins-Schwarz theorem> representation $M_t=B_{[M]_t}$ need not be independent of the bracket clock. Let $W$ be standard <Brownian motion> and take
$$
M_t=\frac{W_t^2-t}{2}=\int_0^tW_s\,dW_s,\qquad A_t=[M]_t=\int_0^tW_s^2ds.
$$
The <Gaussian fourth moment> and the <Brownian covariance kernel> give $\mathbb E[(W_t^2-t)W_s^2]=2s^2$ for $s\leq t$, and hence
$$
\mathbb E[M_tA_t]=\int_0^ts^2ds=\frac{t^3}{3}>0\qquad(t>0).
$$
If the entire driving <Brownian motion> $B$ were independent of the clock $A$, conditioning on $A_t$ would give $\mathbb E[B_{A_t}A_t]=0$, contradicting this identity. Gaussian increments in bracket time therefore do not justify an independent random-clock model.
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