= Solution
If the filtration is stable from degree $\ell$, then $M^*$ is generated over $R^*$ by finite generating sets for $M_0,\ldots,M_\ell$. Conversely, let homogeneous elements of degrees at most $\ell$ generate $M^*$. In every degree $n>\ell$, each expression for an element of $M_n$ uses a positive-degree coefficient from $R^*$, so $M_n=IM_{n-1}$. Hence finite generation is equivalent to stability.
For $M'\subseteq M$, take
$$
M_n=I^nM\cap M'.
$$
Then $M^*$ is a graded submodule of the finite Rees module $\bigoplus I^nM$. Since $R$ is Noetherian and $I$ is finitely generated, $R^*$ is Noetherian; hence $M^*$ is finite and the filtration is stable. Therefore, for some $\ell$ and all $n\geq\ell$,
$$
\boxed{I^nM\cap M'=I^{n-\ell}(I^\ell M\cap M').}
$$
This is the <Artin-Rees lemma>.
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