Hiding a drift

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 $\mu>0$.