Hiding a drift

Abstract

In this article we consider a Brownian motion with drift of the form \[dS_t=μ_t dt+dB_t\qquadfor t\ge0,\] with a specific nontrivial $({\mu }_t{)}_{t\ge 0}$, predictable with respect to ${\mathbb{F}}^B$, the natural filtration of the Brownian motion $B=(B_t{)}_{t\ge 0}$. We construct a process $H=(H_t{)}_{t\ge 0}$, also predictable with respect to ${\mathbb{F}}^B$, such that $((H\cdot S{)}_t{)}_{t\ge 0}$ is a Brownian motion in its own filtration. Furthermore, for any $\delta >0$, we refine this construction such that the drift $({\mu }_t{)}_{t\ge 0}$ only takes values in $]\mu -\delta ,\mu +\delta [$, for fixed $\mu >0$.

Published in at http://dx.doi.org/10.1214/09-AOP469 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)