Nonparametric estimation of time-changed Lévy models under high-frequency data

Abstract

Let {Z t } t≥0 be a Lévy process with Lévy measure ν, and let τ (t) = t 0 r(u) du, where {r(t)} t≥0 is a positive ergodic diffusion independent from Z. Based upon discrete observations of the time-changed Lévy process X t := Z τ t during a time interval [0, T ], we study the asymptotic properties of certain estimators of the parameters β(ϕ) := ϕ(x)ν(dx), which in turn are well known to be the building blocks of several nonparametric methods such as sieve-based estimation and kernel estimation. Under uniform boundedness of the second moments of r and conditions on ϕ necessary for the standard short-term ergodic property lim t→0 E ϕ(Z t )/t = β(ϕ) to hold, consistency and asymptotic normality of the proposed estimators are ensured when the time horizon T increases in such a way that the sampling frequency is high enough relative to T .

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 < ∞.…”

“…However, for any finite time horizon [0, T ], there will be finitely-many of such jumps and consistency won't be achievable no matter how frequently the process is sampled. In [13], we assume that the rate process r is ergodic, and combine its longrun ergodic properties with the short-term ergodic properties of Z to attain consistent estimation of the integral parameters β(ϕ) := ϕ(x)s(x)dx. These integral parameters can subsequently be combined with a numerical approximation method for the function s to yield a non-parametric estimator for s. The estimators for β(ϕ) are given by the realized ϕ-variations of X per unit time:…”

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

“…Hence, the rate process r controls the volatility of the price process S . For further motivation and justification of the model (1–2) we refer to the introductory section in our former paper Figueroa‐López (2009b), the original paper of Carr et al. (2003), or the last chapter of the monograph of Cont & Tankov (2004).…”

“…Following Figueroa‐López (2009b), we are assuming that the Lévy measure of Z admits a density s and we are interested in estimating it, in a non‐parametric fashion, based on discrete observations of X . As a key intermediate step we consider estimators for the integral parameter where ϕ is a given ‘test function’.…”

“…In this article, following our former paper Figueroa‐López (2009b), the main building blocks for our estimation method are the statistics for a given deterministic test function ϕ . We can think of (4) as the realized ϕ ‐variation of the process X per unit time, based on the sampling observations .…”