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

Let {Zt}t≥0 be a Lévy process with Lévy measure ν and let τ(t) = ∫ t 0 r(u)du where {r(t)}t≥0 is a positive ergodic diffusion independent from Z. Based upon discrete observations of the time-changed Lévy process Xt := Zτt during 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 uniformly boundedness of the second moments of r and conditions on φ necessary for the standard short-term ergodic property limt→0 Eφ(Zt)/t = β(φ) to hold, consistency and asymptotic normality of the proposed estimators are ensured when the time horizon T increases in such as way that the sampling frequency is high enough relative to T . AMS 2000 subject classifications: 60J75; 60F05; 62M05.