= Brownian motion transform by three times its running average
{c}
{title2=$\widehat W_t=W_t-(3/t)\int_0^tW_s\,ds$}
The displayed continuous centered <Gaussian process>, with value zero at time zero, has the <Brownian covariance kernel>. Hence it is a <Brownian motion> in its own natural filtration. More generally, replacing three by $c$ gives covariance $s+c(c-3)(s/2-s^2/(6t))$ for $0<s\leq t$, so three is the only nonzero valid coefficient. Its conditional future increment in the original Brownian filtration is $3(t-s)t^{-1}(s^{-1}\int_0^sW_u\,du-W_s)$, and is generally nonzero.
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