  Given a Gaussian random walk (or a Wiener process), possibly with drift,
observed through noise, we consider the problem of estimating its first-passage
time $\tau_\ell$ of a given level $\ell$ with a stopping time $\eta$ defined
over the noisy observation process.
  Main results are upper and lower bounds on the minimum mean absolute
deviation $\inf_\eta \ex|\eta-\tau_\ell|$ which become tight as
$\ell\to\infty$. Interestingly, in this regime the estimation error does not
get smaller if we allow $ \eta$ to be an arbitrary function of the entire
observation process, not necessarily a stopping time.
  In the particular case where there is no drift, we show that it is impossible
to track $\tau_\ell$: $\inf_\eta \ex|\eta-\tau_\ell|^p=\infty$ for any $\ell>0$
and $p\geq1/2$.
