Least squares estimators based on the Adams method for stochastic differential equations with small L\'evy noise

TL;DR

This paper introduces a novel least squares estimation method for stochastic differential equations driven by small Lévy noise, utilizing the Adams numerical method for improved accuracy over traditional Euler-based approaches.

Contribution

It proposes a new estimation technique based on the Adams method for SDEs with Lévy noise, demonstrating its consistency and superior finite-sample performance.

Findings

Estimates are consistent and asymptotically normal.

The Adams-based estimators outperform Euler-based estimators in finite samples.

The method effectively handles small Lévy noise in SDEs.

Abstract

We consider stochastic differential equations (SDEs) driven by small L\'evy noise with some unknown parameters, and propose a new type of least squares estimators based on discrete samples from the SDEs. To approximate the increments of a process from the SDEs, we shall use not the usual Euler method, but the Adams method, that is, a well-known numerical approximation of the solution to the ordinary differential equation appearing in the limit of the SDE. We show the consistency of the proposed estimators as well as the asymptotic distribution in a suitable observation scheme. We also show that our estimators can be better than the usual LSE based on the Euler method in the finite sample performance.