  In this article we consider a Brownian motion with drift of the form
\[dS_t=\mu_t dt+dB_t\qquadfor t\ge0,\] with a specific nontrivial
$(\mu_t)_{t\geq0}$, predictable with respect to $\mathbb{F}^B$, the natural
filtration of the Brownian motion $B=(B_t)_{t\ge0}$. We construct a process
$H=(H_t)_{t\ge0}$, also predictable with respect to $\mathbb{F}^B$, such that
$((H\cdot S)_t)_{t\ge 0}$ is a Brownian motion in its own filtration.
Furthermore, for any $\delta>0$, we refine this construction such that the
drift $(\mu_t)_{t\ge0}$ only takes values in $]\mu-\delta,\mu+\delta[$, for
fixed $\mu>0$.
