  For $i : Z \to X$ a closed immersion of smooth varieties, we study how the
$V$-filtration along $Z$ and the Hodge filtration on a mixed Hodge module $M$
on $X$ interact with each other. We also give a formula for the functors $i^*$,
$i^!$ in terms of this $V$-filtration. As applications, we obtain results on
the Hodge filtration of monodromic mixed Hodge modules and we give a Hodge
theoretic proof of Skoda's theorem on multiplier ideals. Finally, we use the
results to study the Fourier-Laplace transform of a monodromic mixed Hodge
module.
