  We show that if $R$ is a local domain which is dominated by a valuation
$\nu$, then there does not always exist a regular local ring $R'$ which
birationally dominates R and is dominated by $\nu$ and an extension of $\nu$ to
the Henselization $(R')^h$ of $R'$ such that the associated graded rings of
$R'$ and $(R')^h$ along the valuations are equal. We also show that there does
not always exist $R'$, a prime ideal $p$ of the completion $\widehat{R'}$ such
that $p\cap R'=(0)$ and an extension of $\nu$ to $\widehat{R'}$ such that the
associated graded rings of $R'$ and $\widehat{R'}/p$ along the valuation are
equal.
