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

Let {Z_t}_t≥0 be a Lévy process with Lévy measure ν, and let τ(t) = ∫^t₀ 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 X_t := Z_τt during a time interval [0, T ], we study the asymptotic properties of certain estimators of the parameters β(ψ) := ∫ ψ(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 of r and conditions on ψ necessary for the standard short-term ergodic property lim_t→0 E ψ(Z_t )/t = β(ψ) 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 .