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^tr(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_τt 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 $\varphi$ 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.