Hiding a drift

Abstract

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

dS_t=μ_t dt+dB_t for t≥0,

with a specific nontrivial (μ_t)_t≥0, predictable with respect to $\mathbb{F}^{B}$, 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 $\mathbb{F}^{B}$, 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 (μ_t)_t≥0 only takes values in ]μ−δ, μ+δ[, for fixed μ>0.