Parametric Estimation of Lévy Processes

Abstract

The main purpose of this chapter is to present some theoretical aspects of parametric estimation of Lévy 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 Lévy processes in much greater detail, with briefly mentioning locally stable Lévy 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.

Efficient estimation of stable Lévy process with symmetric jumps

“…Theorem 3.1 remedies the incorrect uses of the delta method for the asymptotic normality results given in Sections 3.3 and 3.4 of [18], together with the relevant references cited therein.…”

“…where ∆ (1) X ≤ · · · ≤ ∆ (n) X denote the order statistics of (∆ j X) n j=1 . From theory of order statistics or that of the least absolute deviation estimation, we know thatμ 0 n is rate-efficient ( [16], [18]):…”

“…(recall that we are fixing a true value θ = (β, σ, µ)); this form is enough to cover the example g q in Section 3.4, where we will demonstrate that the restriction (26) does not force us to narrow the parameter space of β beforehand. Under the conditions (24) and (25) we can apply the median-adjusted central limit theorem [18,Theorem 3.7] for √ n(G n (β) − G 0 (β, σ)), hence in particular G n (β) p − → G 0 (β, σ). Assume that the random equation…”

Statistical inference for misspecified ergodic Lévy driven stochastic differential equation models

“…X is a Lévy process with drift), it may be interest in its own right and enormous papers have addressed this problem so far. We refer to [31] for comprehensive accounts under Z being assumed to have a certain parametric structure. As for the situation where just a few information on Z is available, one of plausible attempts is the method of moments proposed in [14], [15], and [37], for example.…”

Nonparametric inference on Lévy measures of compound Poisson-driven Ornstein-Uhlenbeck processes under macroscopic discrete observations

“…We refer to recent contributions by Woerner (2001), Masuda (2011, 2013), and Brouste and Masuda (2018) on parametric inference on Lévy processes. We also find an overview of recent developments on the parametric inference on Lévy processes in Masuda (2015). Some authors have studied statistical inference on Lévy process under macroscopic observations.…”