Central limit theorems for discretized occupation time functionals

The approximation of integral type functionals is studied for discrete observations a continuous Itô semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved L 2 -Sobolev functions with fractional smoothness. An explicit ( P ) -lower bound shows that already lower order quadrature rules, such as trapezoidal rule and classical Riemann estimator, rate optimal, but only efficient, achieving minimal asymptotic variance.