Central limit theorems for the non-parametric estimation of time-changed Le19 evy models

Let {Zt}t≥0 be a Lévy process with Lévy measure ν and let τ(t) := ∫ t 0 g(r̃(u))du be a random clock, where g is a non-negative function and {r̃(t)}t≥0 is an ergodic diffusion independent of Z. Time-changed Lévy models of the form Xt := Zτt are known to be good models to capture several stylized features of asset prices such as leptokurtic distributions and volatility clustering. In our previous work [19], we proposed consistent estimators for the integral parameter β(φ) := ∫ φ(x)ν(dx) based on highfrequency discrete observations of the process X. In this paper, we prove central limit theorems for our estimators, valid when both the sampling frequency and time-horizon get larger. Our results combine the long-run ergodic properties of the diffusion r̃ with the short-term ergodic properties of the Lévy process Z via central limit theorems for martingale differences. The performance of the estimators are illustrated numerically for Normal Inverse Gaussian process Z and a CIR process r̃. We found that the bias of the estimators is affected mainly by the sampling frequency, while the standard error is affected mainly by the time-horizon T . AMS 2000 subject classifications: 60J75; 60F05; 62M05.