Nonparametric estimation of the characteristic triplet of a discretely observed L\'evy process

TL;DR

This paper develops nonparametric estimators for the characteristic triplet of a Lévy process from discrete observations, analyzing their asymptotic properties using Fourier inversion and kernel smoothing techniques.

Contribution

It introduces new nonparametric estimators for the Lévy process parameters and provides their asymptotic error bounds, advancing statistical inference methods for Lévy processes.

Findings

Derived upper bounds on mean square errors for b3 and  estimators

Established an upper bound on the mean integrated square error for the jump measure estimator

Analyzed asymptotic behavior of the proposed estimators

Abstract

Given a discrete time sample $X_1,... X_n$ from a L\'evy process $X=(X_t)_{t\geq 0}$ of a finite jump activity, we study the problem of nonparametric estimation of the characteristic triplet $(\gamma,\sigma^2,\rho)$ corresponding to the process $X.$ Based on Fourier inversion and kernel smoothing, we propose estimators of $\gamma,\sigma^2$ and $\rho$ and study their asymptotic behaviour. The obtained results include derivation of upper bounds on the mean square error of the estimators of $\gamma$ and $\sigma^2$ and an upper bound on the mean integrated square error of an estimator of $\rho.$