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Sparse estimation for the drift of high-dimensional Ornstein--Uhlenbeck processes with i.i.d. paths

Shogo Nakakita

TL;DR

This work addresses the problem of estimating the drift matrix $oldsymbol{A}$ of a high-dimensional Ornstein–Uhlenbeck process from $N$ i.i.d. paths on a finite horizon, without relying on ergodicity. It analyzes sparsity-regularized maximum-likelihood estimators, specifically Lasso ($ orm{oldsymbol{A}}_1$) and Slope ($ orm{oldsymbol{A}}_*$), and proves that they achieve minimax-optimal convergence rates under mild assumptions, with the effective sample size playing the role of the long-time horizon in ergodic settings. The paper derives non-asymptotic high-probability error bounds for both estimators, compares their statistical and computational trade-offs, and establishes a matching minimax lower bound, complemented by a novel concentration bound for non-centered sample covariances. Numerical experiments on OU systems with dimensions up to $d=25$ confirm that sparse estimators outperform the maximum likelihood estimator and effectively recover zero drift entries, supporting the theoretical findings and providing practical guidance for high-dimensional drift estimation in non-ergodic regimes.

Abstract

We study sparsity-regularized maximum likelihood estimation for the drift parameter of high-dimensional non-stationary Ornstein--Uhlenbeck processes given repeated measurements of i.i.d. paths. In particular, we show that Lasso and Slope estimators can achieve the minimax optimal rate of convergence. We exhibit numerical experiments for sparse estimation methods and show their performance.

Sparse estimation for the drift of high-dimensional Ornstein--Uhlenbeck processes with i.i.d. paths

TL;DR

This work addresses the problem of estimating the drift matrix of a high-dimensional Ornstein–Uhlenbeck process from i.i.d. paths on a finite horizon, without relying on ergodicity. It analyzes sparsity-regularized maximum-likelihood estimators, specifically Lasso () and Slope (), and proves that they achieve minimax-optimal convergence rates under mild assumptions, with the effective sample size playing the role of the long-time horizon in ergodic settings. The paper derives non-asymptotic high-probability error bounds for both estimators, compares their statistical and computational trade-offs, and establishes a matching minimax lower bound, complemented by a novel concentration bound for non-centered sample covariances. Numerical experiments on OU systems with dimensions up to confirm that sparse estimators outperform the maximum likelihood estimator and effectively recover zero drift entries, supporting the theoretical findings and providing practical guidance for high-dimensional drift estimation in non-ergodic regimes.

Abstract

We study sparsity-regularized maximum likelihood estimation for the drift parameter of high-dimensional non-stationary Ornstein--Uhlenbeck processes given repeated measurements of i.i.d. paths. In particular, we show that Lasso and Slope estimators can achieve the minimax optimal rate of convergence. We exhibit numerical experiments for sparse estimation methods and show their performance.
Paper Structure (21 sections, 12 theorems, 95 equations, 1 figure)

This paper contains 21 sections, 12 theorems, 95 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Literature reviews
  3. Notation
  4. Paper organization
  5. Problem setup
  6. Theoretical results
  7. Assumptions
  8. Upper bounds for errors
  9. Differences between Lasso and Slope
  10. Lower bounds for errors
  11. Numerical experiments
  12. Setup
  13. Results
  14. Concluding remarks
  15. Proofs of preliminary results
  16. ...and 6 more sections

Key Result

Proposition 3.1

Suppose that Assumption assm:eigen holds, and let $s=\|\mathbf{A}_{0}\|_{0}$. There exists a universal constant $c_{\textnormal{L}}>0$ such that if $\lambda_{\textnormal{L}}$ satisfies then, for some $c\ge 1$ dependent only on $T$, $\mathfrak{a}_{0}$, $\mathfrak{p}_{0}$, $K$ and $\|\mathbf{\Sigma}\|_{\textnormal{op}}$, for any $\epsilon_{0}\in(0,1)$, $N\in\mathbb{N}$, and $\mathbf{A}\in\mathbb{R}

Figures (1)

  • Figure 1: The comparison of $d^{-1}$-scaled squared $\ell^{2}$ distances and (top) and $d^{-1}$-scaled $\ell^{1}$ (bottom) distances between the true value and the MLE (left), Lasso (middle), and Slope (right). The coloured areas represent (mean) $\pm$ (standard deviation).

Theorems & Definitions (24)

  • Proposition 3.1
  • Corollary 3.2
  • Remark 1
  • Proposition 3.3
  • Corollary 3.4
  • Proposition 3.5
  • Proposition A.1: pointwise concentration
  • proof
  • Lemma A.2: uniform concentration
  • proof
  • ...and 14 more