  We study the non-parametric estimation of an unknown stationary density fV of
an unobserved strictly stationary volatility process $(\bm V_t)_{t\geq 0}$ on
$\IRp^2 := (0,\infty)^2$ based on discrete-time observations in a stochastic
volatility model. We identify the underlying multiplicative measurement error
model and build an estimator based on the estimation of the Mellin transform of
the scaled, integrated volatility process and a spectral cut-off regularisation
of the inverse of the Mellin transform. We prove that the proposed estimator
leads to a consistent estimation strategy. A fully data-driven choice of $\bm k
\in \IRp^2$ is proposed and upper bounds for the mean integrated squared risk
are provided. Throughout our study, regularity properties of the volatility
process are necessary for the analsysis of the estimator. These assumptions are
fulfilled by several examples of volatility processes which are listed and used
in a simulation study to illustrate a reasonable behaviour of the proposed
estimator.
