Nonparametric volatility estimation in scalar diffusions: Optimality across observation frequencies

The nonparametric volatility estimation problem of a scalar diffusion process observed at equidistant time points is addressed. Using the spectral representation of the volatility in terms of the invariant density and an eigenpair of the infinitesimal generator the first known estimator that attains the minimax optimal convergence rates for both high and low-frequency observations is constructed. The proofs are based on a posteriori error bounds for generalized eigenvalue problems as well as the path properties of scalar diffusions and stochastic analysis. The finite sample performance is illustrated by a numerical example.