Nonparametric estimation of time-changed Lévy models under high-frequency data

Abstract

Let { Z t } t ≥0 be a Lévy process with Lévy measure ν, and let τ(t)=∫ 0 t r(u) d u , where { r(t) } t ≥0 is a positive ergodic diffusion independent from Z . Based upon discrete observations of the time-changed Lévy process X t ≔ Z τ t during a time interval [0, T ], we study the asymptotic properties of certain estimators of the parameters β(φ)≔∫φ( x )ν(d x ), which in turn are well known to be the building blocks of several nonparametric methods such as sieve-based estimation and kernel estimation. Under uniform boundedness of the second moments of r and conditions on φ necessary for the standard short-term ergodic property lim t → 0 E φ(Z t )/ t = β(φ) to hold, consistency and asymptotic normality of the proposed estimators are ensured when the time horizon T increases in such a way that the sampling frequency is high enough relative to T .