In this article we consider a Brownian motion with drift of the formdS_t=μ_t dtdB_tfor t≤0,with a specific nontrivial (μ_t)t≥0, predictable with respect toF~B,the nat-ural filtration of the Brownian motion B = (B_t)t>0.We construct a processH = (H_t)t>0,also predictable with respect toF~B,such that((H · S)_0isa Brownian motion in its own filtration. Furthermore, for any δ > 0, we re-fine this construction such that the drift (μ_t)t>0 only takes values in ]μ— δ, δ[,for fixed μ >0.
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