Hiding a drift

In this article we consider a Brownian motion with drift of the form

dS(t) = mu(t)dt + dB(t) for t >= 0,

with a specific nontrivial (mu(t))(t) >= 0, predictable with respect to F-B, the natural filtration of the Brownian motion B = (B-t)(t >= 0). We construct a process H = (H-t)(t >= 0), also predictable with respect to F-B such that ((H center dot S)(t))(t >= 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 >= 0) only takes values in]mu - delta, mu + delta[, for fixed mu > 0.