As a starting point we prove a functional central limit theorem for estimators of the invariant measure of a geometrically ergodic Harris-recurrent Markov chain in a multi-scale space. This allows to construct confidence bands for the invariant density with optimal (up to undersmoothing) L∞-diameter by using wavelet projection estimators. In addition our setting applies to the drift estimation of diffusions observed discretely with fixed observation distance. We prove a functional central limit theorem for estimators of the drift function and finally construct adaptive confidence bands for the drift by using a completely data-driven estimator.@en

Donsker-type functional limit theorems are proved for empirical processes arising from discretely sampled increments of a univariate Lévy process. In the asymptotic regime the sampling frequencies increase to infinity and the limiting object is a Gaussian process that can be obtained from the composition of a Brownian motion with a covariance operator determined by the Lévy measure. The results are applied to derive the asymptotic distribution of natural estimators for the distribution function of the Lévy jump measure. As an application we deduce Kolmogorov–Smirnov type tests and confidence bands.@en

Observing prices of European put and call options, we calibrate exponential Lévy models nonparametrically. We discuss the efficient implementation of the spectral estimation procedures for Lévy models of finite jump activity as well as for self-decomposable Lévy models. Based on finite sample variances, confidence intervals are constructed for the volatility, for the drift and, pointwise, for the jump density. As demonstrated by simulations, these intervals perform well in terms of size and coverage probabilities. We compare the performance of the procedures for finite and infinite jump activity based on options on the German DAX index and find that both methods achieve good calibration results. The stability of the finite activity model is studied when the option prices are observed in a sequence of trading days.@en

We estimate linear functionals in the classical deconvolution problem by kernel estimators. We obtain a uniform central limit theorem with √n-rate on the assumption that the smoothness of the functionals is larger than the ill-posedness of the problem, which is given by the polynomial decay rate of the characteristic function of the error. The limit distribution is a generalized Brownian bridge with a covariance structure that depends on the characteristic function of the error and on the functionals. The proposed estimators are optimal in the sense of semiparametric efficiency. The class of linear functionals is wide enough to incorporate the estimation of distribution functions. The proofs are based on smoothed empirical processes and mapping properties of the deconvolution operator.

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