Central Limit Theorems for the Non-Parametric Estimation of Time-Changed Levy Models

Let {Zt}(t >= 0) be a Levy process with Levy measure v and let be a random clock, where g is a non-negative function and is an ergodic diffusion independent of Z. Time-changed Levy models of the form are known to incorporate several important stylized features of asset prices, such as leptokurtic distributions and volatility clustering. In this article, we prove central limit theorems for a type of estimators of the integral parameter beta(phi):=integral phi(x)v(dx), valid when both the sampling frequency and the observation time-horizon of the process get larger. Our results combine the long-run ergodic properties of the diffusion process with the short-term ergodic properties of the Levy 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 CoxIngersollRoss process .