Testing the characteristics of a Lévy process

For n equidistant observations of a Lévy process at time distance Δn we consider the problem of testing hypotheses on the volatility, the jump measure and its Blumenthal-Getoor index in a non- or semiparametric manner. Asymptotically as n→∞ we allow for both, the high-frequency regime Δn=1n and the low-frequency regime Δn=1 as well as intermediate cases. The approach via the empirical characteristic function unifies existing theory and sheds new light on diverse results. Particular emphasis is given to asymptotic separation rates which reveal the complexity of these basic, but surprisingly non-standard inference questions.