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

Abstract

Let {Zt}t≥0be a Lévy process with Lévy measure ν, and let τ(t)=∫0tr(u)du, where {r(t)}t≥0is a positive ergodic diffusion independent fromZ. Based upon discrete observations of the time-changed Lévy processXt≔Zτtduring a time interval [0,T], we study the asymptotic properties of certain estimators of the parameters β(φ)≔∫φ(x)ν(dx), 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 ofrand conditions on φ necessary for the standard short-term ergodic property limt→ 0E φ(Zt)/t= β(φ) to hold, consistency and asymptotic normality of the proposed estimators are ensured when the time horizonTincreases in such a way that the sampling frequency is high enough relative toT.

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