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

Part of: Inference from stochastic processes Markov processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

José E. Figueroa-López
José E. Figueroa-López*
Affiliation:
Purdue University
*
∗ Postal address: Department of Statistics, Purdue University, West Lafayette, IN 47907-2066, USA. Email address: figueroa@stat.purdue.edu
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Abstract

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Let {Zt}t≥0 be a Lévy process with Lévy measure ν, and let τ(t)=∫0tr(u) d u, where {r(t)}t≥0 is a positive ergodic diffusion independent from Z. Based upon discrete observations of the time-changed Lévy process Xt≔Zτt during a time interval [0,T], we study the asymptotic properties of certain estimators of the parameters β(φ)≔∫φ(x)ν(d x), 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 limt→ 0 E φ(Zt)/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.

Keywords

Lévy processnonparametric estimationhigh-frequency-based inferencestochastic volatility

MSC classification

Primary: 60J75: Jump processes 60F05: Central limit and other weak theorems 62M05: Markov processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 4 , December 2009 , pp. 1161 - 1188
Copyright
Copyright © Applied Probability Trust 2009 

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