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

Abstract

Abstract. Let {Zt}t0 be a Lévy process with Lévy measure ν and let be a random clock, where g is a non‐negative function and is an ergodic diffusion independent of Z. Time‐changed Lévy 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 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 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 .

Parametric Estimation of Lévy Processes

“…There, the author suggests using the natural statisticŝ β n (ϕ) := 1 T n n ∑ j=1 ϕ(∆ n j Y ), and provides sets of conditions under which the estimator is asymptotically normal at rate √ T n . To deduce it, as in (4.26) we need information about the rate of convergence of h −1 n E 0 {ϕ(X h n )} to its limit; see [30,Assumption 3]. If in particular Y is a Lévy process with no drift and no Gaussian component and if ϕ is smooth enough with ϕ(0) = ∂ ϕ(0) = 0, then it follows from (4.27) and (4.29) that 1 h n E{ϕ(X h n )} − β (ϕ) h n .…”

“…The paper [30] studied a non-parametric estimation problem of the functional parameter β (ϕ) := ϕ(z)ν(dz) under T n → ∞ for a random time change of a Lévy process Y with Lévy measure ν and for a measurable function ϕ such that the integral β (ϕ) is well-defined. There, the author suggests using the natural statistics βn (ϕ) := 1…”

Statistical estimation of Lévy-type stochastic volatility models

“…The last estimation problem (3) listed at the beginning of this section, namely the non-parametric estimation of the Lévy density s, has been considered in detail in Figueroa [13]- [14]. We point out that this problem cannot be solved in finite-time horizon T < ∞.…”

Asymptotic results for time-changed Lévy processes sampled at hitting times

“…Time-changed Lévy models were introduced into financial literature in [11] and their estimation from high frequency data with deterministic sampling was recently addressed in [16,17].…”

“…In recent years, a large number of papers have been devoted to asymptotic results and statistical procedures for time-changed Lévy processes [16,17,22,41] and more general semimartingales [1,4,3,2,23], under high frequency discrete sampling. The classical high frequency setting consists in observing n values of the process over a fixed time interval [0, T ] at deterministic sampling times 0 = t n 0 < t n 1 < · · · < t n n = T .…”