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

Let {Zt}t≥0 be a Lévy process with Lévy measure ν and let $\tau \left(t\right)≔{\int }_0^{t}g\left(\tilde{r}\left(u\right)\right)\text{d}u$ be a random clock, where g is a non-negative function and $\{\tilde{r}\left(t\right){\}}_{t⩾0}$ is an ergodic diffusion independent of Z. Time-changed Lévy models of the form $X_t≔Z_{{\tau }_t}$ 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 β(φ)≔∫φ(x)ν(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 &rtilde; 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 Cox–Ingersoll–Ross process &rtilde;.