We consider a model Y_t=σ_tη_t in which (σ_t) is not
independent of the noise process (η_t), but σ_t is independent of
η_t for each t. We assume that (σ_t) is stationary and we
propose an adaptive estimator of the density of ln(σ2_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 (σ_t) and (η_t) are independent, in all cases where
these minimax rates are known. The results apply to various linear and non
linear ARCH processes