Parametric estimation of L\'evy processes

TL;DR

This chapter explores the theoretical foundations of parametric estimation for Le9vy processes, emphasizing high-frequency data analysis, asymptotic properties, and specific methods like moments for stable processes.

Contribution

It provides a detailed theoretical framework for parametric estimation of infinite activity pure-jump Le9vy processes, including asymptotic results and analysis of estimation methods.

Findings

Asymptotic normality at various convergence rates

Uniform tail-probability estimates for statistical fields

Detailed analysis of method of moments for stable Le9vy processes

Abstract

The main purpose of this chapter is to present some theoretical aspects of parametric estimation of L\'evy processes based on high-frequency sampling, with a focus on infinite activity pure-jump models. Asymptotics for several classes of explicit estimating functions are discussed. In addition to the asymptotic normality at several rates of convergence, a uniform tail-probability estimate for statistical random fields is given. As specific cases, we discuss method of moments for the stable L\'evy processes in much greater detail, with briefly mentioning locally stable L\'evy processes too. Also discussed is, due to its theoretical importance, a brief review of how the classical likelihood approach works or does not, beyond the fact that the likelihood function is not explicit.