Title: Estimating occupation time functionals,
Abstract: We study the estimation of integral type functionals $\int_{0}^{t}f(X_{r})dr$
for a function $f$ and a $d$-dimensional càdlàg process $X$ with respect to
discrete observations by a Riemann-sum estimator. Based on novel semimartingale
approximations in the Fourier domain, central limit theorems are proved for
$L^{2}$-Sobolev functions $f$ with fractional smoothness and continuous Itô
semimartingales $X$. General $L^{2}(\mathbb{P})$-upper bounds on the error for
càdlàg processes are given under weak assumptions. These bounds combine and
generalize all previously obtained results in the literature and apply also to
non-Markovian processes. Several detailed examples are discussed. As
application the approximation of local times for fractional Brownian motion is
studied. The optimality of the $L^{2}(\mathbb{P})$-upper bounds is shown by
proving the corresponding lower bounds in case of Brownian motion.