Nonparametric estimation for irregularly sampled L\'evy processes

TL;DR

This paper develops nonparametric methods for estimating jump dynamics and distributional density of irregularly sampled Le9vy processes at low frequency, providing theoretical guarantees and practical illustrations.

Contribution

It introduces new nonparametric estimators for Le9vy processes under irregular sampling, with proven minimax optimality and explicit risk bounds.

Findings

Derived non-asymptotic risk bounds for estimators.

Established minimax optimality of jump measure estimator.

Provided numerical examples demonstrating practical performance.

Abstract

We consider nonparametric statistical inference for L\'evy processes sampled irregularly, at low frequency. The estimation of the jump dynamics as well as the estimation of the distributional density are investigated. Non-asymptotic risk bounds are derived and the corresponding rates of convergence are discussed under global as well as local regularity assumptions. Moreover, minimax optimality is proved for the estimator of the jump measure. Some numerical examples are given to illustrate the practical performance of the estimation procedure.