Least squares estimators for discretely observed stochastic processes driven by small Levy noises

TL;DR

This paper investigates the properties of least squares estimators for discretely observed stochastic processes influenced by small Lévy noises, establishing their consistency, convergence rates, and asymptotic distribution without moment constraints.

Contribution

It introduces a novel analysis of LSE for processes driven by small Lévy noises, providing new theoretical results under minimal assumptions.

Findings

LSE is consistent as noise diminishes and observations increase.

Convergence rate of the LSE is established.

Asymptotic distribution is a convolution of normal and jump-related distributions.

Abstract

We study the problem of parameter estimation for discretely observed stochastic processes driven by additive small L\'{e}vy noises. We do not impose any moment condition on the driving L\'{e}vy process. Under certain regularity conditions on the drift function, we obtain consistency and rate of convergence of the least squares estimator (LSE) of the drift parameter when a small dispersion coefficient $\varepsilon \to 0$ and $n \to \infty$ simultaneously. The asymptotic distribution of the LSE in our general setting is shown to be the convolution of a normal distribution and a distribution related to the jump part of the L\'evy process.