On penalized estimation for dynamical systems with small noise

TL;DR

This paper investigates Lasso-type penalized estimators for small noise dynamical systems, focusing on their consistency, asymptotic behavior, and oracle properties, with a special emphasis on the $l^1$ penalty and adaptive methods.

Contribution

It introduces a Lasso-based estimation framework for dynamical systems with small noise, analyzing its theoretical properties and extending to adaptive Lasso with oracle properties.

Findings

Proves consistency of Lasso estimators in this setting.

Derives asymptotic distributions for different $p$ values.

Shows adaptive Lasso achieves oracle properties for $p=1$.

Abstract

We consider a dynamical system with small noise for which the drift is parametrized by a finite dimensional parameter. For this model we consider minimum distance estimation from continuous time observations under $l^p$-penalty imposed on the parameters in the spirit of the Lasso approach with the aim of simultaneous estimation and model selection. We study the consistency and the asymptotic distribution of these Lasso-type estimators for different values of $p$. For $p=1$ we also consider the adaptive version of the Lasso estimator and establish its oracle properties.