Hiding a drift

Details Hiding a drift Hiding a drift

2008-02-08

arxiv.org/abs/0802.1152v4

Annals of Probability 2009, Vol. 37, No. 6, 2459-2479

Mathematics

Probability

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

Scientific paper

10.1214/09-AOP469

In this article we consider a Brownian motion with drift of the form \[dS_t=\mu_t dt+dB_t\qquadfor t\ge0,\] with a specific nontrivial $(\mu_t)_{t\geq0}$, predictable with respect to $\mathbb{F}^B$, the natural filtration of the Brownian motion $B=(B_t)_{t\ge0}$. We construct a process $H=(H_t)_{t\ge0}$, 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 $(\mu_t)_{t\ge0}$ only takes values in $]\mu-\delta,\mu+\delta[$, for fixed $\mu>0$.

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