Adaptive Elastic-Net estimation for sparse diffusion processes

TL;DR

This paper introduces an adaptive Elastic-Net estimator for high-dimensional ergodic diffusion processes, improving prediction accuracy and model interpretability while providing theoretical guarantees and demonstrating effectiveness through simulations and real data.

Contribution

It develops a novel adaptive Elastic-Net method for sparse diffusion processes, with finite-sample guarantees and optimal convergence rates in high-frequency sampling regimes.

Findings

Enhanced prediction accuracy in high-dimensional diffusion models

Effective recovery of sparse model structure

Strong empirical performance in correlated variable scenarios

Abstract

Penalized estimation methods for diffusion processes and dependent data have recently gained significant attention due to their effectiveness in handling high-dimensional stochastic systems. In this work, we introduce an adaptive Elastic-Net estimator for ergodic diffusion processes observed under high-frequency sampling schemes. Our method combines the least squares approximation of the quasi-likelihood with adaptive $\ell_1$ and $\ell_2$ regularization. This approach allows to enhance prediction accuracy and interpretability while effectively recovering the sparse underlying structure of the model. In the spirit of analyzing high-dimensional scenarios, we provide finite-sample guarantees for the (block-diagonal) estimator's performance by deriving high-probability non-asymptotic bounds for the $\ell_2$ estimation error. These results complement the established oracle properties in the high-frequency asymptotic regime with mixed convergence rates, ensuring consistent selection of the relevant interactions and achieving optimal rates of convergence. Furthermore, we utilize our results to analyze one-step-ahead predictions, offering non-asymptotic control over the $\ell_1$ prediction error. The performance of our method is evaluated through simulations and real data applications, demonstrating its effectiveness, particularly in scenarios with strongly correlated variables.