Nonparametric estimation of the characteristic triplet of a discretely observed Lévy process

Abstract

Given a discrete time sample X ₁, … X _n from a Lévy process X=(X _t ) _t≥0 of a finite jump activity, we study the problem of nonparametric estimation of the characteristic triplet (γ, σ², ρ) corresponding to the process X. Based on Fourier inversion and kernel smoothing, we propose estimators of γ, σ² and ρ and study their asymptotic behaviour. The obtained results include derivation of upper bounds on the mean square error of the estimators of γ and σ² and an upper bound on the mean integrated square error of an estimator of ρ.

Keywords:

• characteristic triplet
• Fourier inversion
• kernel smoothing
• Lévy density
• Lévy process
• mean integrated square error
• mean square error

AMS Subject Classification :

• 62G07
• 62G20

Acknowledgements

The author would like to thank Bert van Es and Peter Spreij for discussions on various parts of the draft version of the paper. Part of the research was done while the author was at Korteweg-de Vries Institute for Mathematics in Amsterdam. The research at Korteweg-de Vries Institute for Mathematics was financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).