In this article we consider a Brownian motion with drift of the form
dSt=μt dt+dBt for t≥0,
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with a specific nontrivial (μt)t≥0, predictable with respect to $\mathbb{F}^{B}$ , the natural filtration of the Brownian motion B=(Bt)t≥0. We construct a process H=(Ht)t≥0, also predictable with respect to $\mathbb{F}^{B}$ , such that ((H⋅S)t)t≥0 is a Brownian motion in its own filtration. Furthermore, for any δ>0, we refine this construction such that the drift (μt)t≥0 only takes values in ]μ−δ, μ+δ[, for fixed μ>0.