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 $X=(X_t)_{t\geq 0}$ of a finite jump activity, we study the problem of nonparametric estimation of the characteristic triplet $(γ,σ^2,ρ)$ corresponding to the process $X.$ Based on Fourier inversion and kernel smoothing, we propose estimators of $γ,σ^2$ 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 $σ^2$ and an upper bound on the mean integrated square error of an estimator of $ρ.$