Sparse Estimation for High-Dimensional L\'evy-driven Ornstein--Uhlenbeck Processes from Discrete Observations

TL;DR

This paper develops high-dimensional estimation methods for Lévydriven Ornstein-Uhlenbeck processes from discrete data, providing sharp theoretical guarantees and practical insights for jump process models.

Contribution

It introduces Lasso and Slope estimators with nonasymptotic bounds for high-dimensional jump processes, extending statistical theory to broader noise types.

Findings

Achieves minimax optimal convergence rates under high-frequency sampling.

Quantifies sample complexity depending on Lévydriven noise characteristics.

Demonstrates practical competitiveness of Lasso and Slope in jump systems.

Abstract

We study high-dimensional drift estimation for L\'evy-driven Ornstein--Uhlenbeck processes based on discrete observations. Assuming sparsity of the drift matrix, we analyze Lasso and Slope estimators constructed from approximate likelihoods and derive sharp nonasymptotic oracle inequalities. Our bounds disentangle the contributions of discretization error and stochastic fluctuations, and establish minimax optimal convergence rates under suitable choices of tuning parameters in a high-frequency regime. We further quantify the sample complexity required to attain these rates depending on the L\'evy noise. The results extend the theory of high-dimensional statistics for stochastic processes to a substantially broader class of noise mechanisms, in particular pure jump processes. They also demonstrate that Lasso and Slope remain competitive for jump-driven systems, providing practical guidance for inference in applications where L\'evy processes are a natural modeling choice.