Sparse inference of the drift of a high-dimensional Ornstein-Uhlenbeck process

TL;DR

This paper develops a Lasso-based method for inferring the sparse drift of high-dimensional Ornstein-Uhlenbeck processes, providing theoretical guarantees and demonstrating improved performance over traditional methods.

Contribution

It introduces a novel Lasso-based inference procedure for high-dimensional OU processes with theoretical analysis and practical validation, without requiring restricted eigenvalue conditions.

Findings

The method achieves asymptotic consistency in variable selection.

It provides non-asymptotic and asymptotic error bounds.

Numerical results show improved accuracy over maximum likelihood methods.

Abstract

Given the observation of a high-dimensional Ornstein-Uhlenbeck (OU) process in continuous time, we proceed to the inference of the drift parameter under a row-sparsity assumption. Towards that aim, we consider the negative log-likelihood of the process, penalized by an $\ell^1$-penalization (Lasso and Adaptive Lasso). We provide both non-asymptotic and asymptotic results for this procedure, by means of a sharp oracle inequality, and a limit theorem in the long-time asymptotics, including asymptotic consistency for variable selection. As a by-product, we point out the fact that for the Ornstein-Uhlenbeck process, one does not need an assumption of restricted eigenvalue type in order to derive fast rates for the Lasso, while it is well-known to be mandatory for linear regression for instance. Numerical results illustrate the benefits of this penalized procedure compared to standard maximum likelihood approaches both on simulations and real-world financial data.