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Sparse Estimation for High-Dimensional Lévy-driven Ornstein--Uhlenbeck Processes from Discrete Observations

Niklas Dexheimer, Natalia Jeszka

TL;DR

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, and demonstrate that Lasso and Slope remain competitive for jump-driven systems.

Abstract

We study high-dimensional drift estimation for Lévy-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 Lévy 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 Lévy processes are a natural modeling choice.

Sparse Estimation for High-Dimensional Lévy-driven Ornstein--Uhlenbeck Processes from Discrete Observations

TL;DR

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, and demonstrate that Lasso and Slope remain competitive for jump-driven systems.

Abstract

We study high-dimensional drift estimation for Lévy-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 Lévy 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 Lévy processes are a natural modeling choice.
Paper Structure (18 sections, 17 theorems, 168 equations, 7 figures, 1 table)

This paper contains 18 sections, 17 theorems, 168 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Related Work
  4. Structure
  5. Preliminaries and Notation
  6. Mathematical Model
  7. The Lasso and Slope Estimators
  8. Main Results
  9. Proof of the Main Result
  10. Discretization Error
  11. Concentration of the Empirical Covariance Matrix
  12. Stochastic Error
  13. Truncation Bias
  14. Proof of Theorem \ref{['thm: main']}
  15. Simulation Study
  16. ...and 3 more sections

Key Result

Theorem 3.1

Assume ass: ergodicity. Let $\varepsilon_0\in(0,1)$ be given, assume $b\geq1,\eta\geq 2b(\exp(\Delta_n\Vert\boldsymbol{A}_0\Vert)-1)$ and define where $\boldsymbol{\nu}_2$ is defined in eq: def nu mat. Then, there exists a universal constant $c_{\star}>0$ such that for any $T\geq T_\star(\varepsilon_0,\eta)$ the following statements hold true.

Figures (7)

  • Figure 5.1: Comparison of the true parameter $A_0$ with Lasso and Slope estimators, as well as true MLE and truncated MLE
  • Figure 5.2: Support recovery of the investigated estimators. Green entries correspond to correctly classified coefficients of $\boldsymbol{A}_0$, meaning that the true and estimated values are either both zero or both non-zero. Red indicates entries where the estimator sets a coefficient to zero although the true value is non-zero. Orange indicates entries where a non-zero value is estimated even though the true coefficient is zero.
  • Figure 5.3: $L_1$ (top) and $L_2$ (bottom) errors of true and truncated MLE, Lasso and Slope $\pm$ one standard deviation.
  • Figure 5.4: $L_1$ (top) and $L_2$ (bottom) errors of true and truncated MLE, Lasso and Slope $\pm$ one standard deviation. Simulations are conducted in a low-frequency setting.
  • Figure 5.5: $L_2$ errors (top) and proportion of used observations (bottom) for truncated MLE and Lasso ($\pm$ one standard deviation) for different values of $b$. The $L_2$ error of the true MLE is added as a reference in the plot on the top.
  • ...and 2 more figures

Theorems & Definitions (36)

  • Theorem 3.1
  • Remark 3.2
  • Corollary 3.3
  • Remark 3.4
  • Lemma 4.1
  • proof : Proof of Lemma \ref{['lemma: L2 frobenius disc']}
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 26 more