00 8 Hiding the Drift

In this article we consider a Brownian motion with drift, denoted by S = (St)t≥0, of the form dSt = μtdt+ dBt for t ≥ 0, with a specific non-trivial drift predictable with respect to F , the natural filtration of the Brownian motion B = (Bt)t≥0. We construct a process H = (Ht)t≥0, also predictable with respect to F , 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. AMS Mathematics Subject Classification [2000]:60H05 and 60G44 (primary); 60G05 and 60H10 (secondary)