Central limit theorems for the nonparametric estimation of time-changed Lévy models

Statistics and Modeling for Complex Data

We consider a time-changed Lévy model of the form X(t)=Z(T(t)), where Z is a Lévy process with Lévy measure F and T(t) is an independent random clock with speed process driven by an ergodic diffusion r(t). By modeling the log return process of a financial asset in terms of X(t), one can incorporate several important stylized features of asset prices, such as leptokurtic return distributions and volatility clustering. In this talk, we propose an estimator for the integral F(g) of a test function g with respect to the Lévy measure F and prove central limit theorems (CLT) for our estimators. The functional parameters F(g) can in turn be used as the building blocks of several nonparametric estimation methods such as sieve-based estimation and kernel estimation. The CLT are valid when both the sampling frequency and the time-horizon of observations get larger. Our results combine the long-run ergodic properties of the diffusion process r(t) with the short-term ergodic properties of the Lévy process Z via central limit theorems for martingale differences.