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