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

摘要

Let {Z_t}_(t≥0) be a Levy process with Levy measure v and let τ(t):=∫_0~t g(r(u)) du be a random clock, where g is a non-negative function and {r(t)}_(t≥)0 is an ergodic diffusion independent of Z. Time-changed Levy models of the form Xt := Zτ_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)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 r 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 Cox-Ingersoll-Ross process r.