  We consider a model $Y\_t=\sigma\_t\eta\_t$ in which $(\sigma\_t)$ is not
independent of the noise process $(\eta\_t)$, but $\sigma\_t$ is independent of
$\eta\_t$ for each $t$. We assume that $(\sigma\_t)$ is stationary and we
propose an adaptive estimator of the density of $\ln(\sigma^2\_t)$ based on the
observations $Y\_t$. Under various dependence structures, the rates of this
nonparametric estimator coincide with the minimax rates obtained in the i.i.d.
case when $(\sigma\_t)$ and $(\eta\_t)$ are independent, in all cases where
these minimax rates are known. The results apply to various linear and non
linear ARCH processes.
