Extensions of Valuations to the Henselization and Completion

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