  Let $\Sc(\vL)$ be the cyclotomic $q$-Schur algebra associated to the
Ariki-Koike algebra $\He_{n,r}$, introduced by Dipper-James-Mathas. In this
paper, we consider $v$-decomposition numbers of $\Sc(\vL)$, namely
decomposition numbers with respect to the Jantzen filtrations of Weyl modules.
We prove, as a $v$-analogue of the result obtained by Shoji-Wada, a product
formula for $v$-decomposition numbers of $\Sc(\vL)$, which asserts that certain
$v$-decomposition numbers are expressed as a product of $v$-decomposition
numbers for various cyclotomic $q$-Schur algebras associated to Ariki-koike
algebras $\He_{n_i,r_i}$ of smaller rank. Moreover we prove a similar formula
for $v$-decomposition numbers of $\He_{n,r}$ by using a Schur functor.
