Testing the characteristics of a L\'evy process

TL;DR

This paper develops a unified approach using empirical characteristic functions to test hypotheses on the parameters of a Lévy process, addressing both high- and low-frequency data regimes.

Contribution

It introduces a non- or semiparametric testing framework for Lévy process characteristics that works across different observation frequencies, unifying and extending existing theory.

Findings

Asymptotic separation rates are characterized for various regimes.

The empirical characteristic function approach simplifies complex inference problems.

The method applies to both high-frequency and low-frequency data scenarios.

Abstract

For $n$ equidistant observations of a L\'evy process at time distance $\Delta_n$ 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 $n\to\infty$ we allow for both, the high-frequency regime $\Delta_n=\frac1n$ and the low-frequency regime $\Delta_n=1$ as well as intermediate cases. The approach via 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.