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Low-Rank and Sparse Drift Estimation for High-Dimensional Lévy-Driven Ornstein--Uhlenbeck Processes

Marina Palaisti

Abstract

We study high-dimensional Ornstein--Uhlenbeck processes driven by Lévy noise and consider drift matrices that decompose into a low-rank plus sparse component, capturing a few latent factors together with a sparse network of direct interactions. For discrete-time observations under the localized, truncated contrast of Dexheimer and Jeszka, we analyze a convex estimator that minimizes this contrast with a combined nuclear-norm and $\ell_1$-penalty on the low-rank and sparse parts, respectively. Under a restricted strong convexity condition, a rank--sparsity incoherence assumption, and regime-specific choices of truncation level, horizon, and sampling mesh for the background driving Lévy process, we derive a non-asymptotic oracle inequality for the Frobenius risk of the estimator. The bound separates a discretization bias term of order $d^2Δ_n^2$ from a stochastic term of order $γ(Δ_n)T^{-1}(r \log d + s \log d)$, thereby showing that the low-rank-plus-sparse structure improves the dependence on the ambient dimension relative to purely sparse estimators while retaining the same discretization and truncation behavior across the four Lévy regimes.

Low-Rank and Sparse Drift Estimation for High-Dimensional Lévy-Driven Ornstein--Uhlenbeck Processes

Abstract

We study high-dimensional Ornstein--Uhlenbeck processes driven by Lévy noise and consider drift matrices that decompose into a low-rank plus sparse component, capturing a few latent factors together with a sparse network of direct interactions. For discrete-time observations under the localized, truncated contrast of Dexheimer and Jeszka, we analyze a convex estimator that minimizes this contrast with a combined nuclear-norm and -penalty on the low-rank and sparse parts, respectively. Under a restricted strong convexity condition, a rank--sparsity incoherence assumption, and regime-specific choices of truncation level, horizon, and sampling mesh for the background driving Lévy process, we derive a non-asymptotic oracle inequality for the Frobenius risk of the estimator. The bound separates a discretization bias term of order from a stochastic term of order , thereby showing that the low-rank-plus-sparse structure improves the dependence on the ambient dimension relative to purely sparse estimators while retaining the same discretization and truncation behavior across the four Lévy regimes.
Paper Structure (7 sections, 9 theorems, 56 equations)

This paper contains 7 sections, 9 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Model and Estimator
  3. Abstract Decomposable-Penalty Framework
  4. Verification of Assumptions B1--B3 for the OU/Lévy Contrast
  5. Main Result: Oracle Inequality
  6. Oracle inequality for low-rank-plus-sparse drift
  7. Corollaries for Different Lévy Regimes

Key Result

Theorem 3.2

Suppose: Let $(\hat{L},\hat{S})$ be any solution of eq:abstract_estimator and $\hat{A} = \hat{L} + \hat{S}$. Then there exist universal constants $C_1,C_2>0$ such that

Theorems & Definitions (19)

  • Definition 2.1: Low-rank-plus-sparse drift class
  • Remark 2.2: Relation to standard incoherence conditions
  • Definition 3.1: Low-rank-plus-sparse error cone
  • Theorem 3.2: Abstract low-rank-plus-sparse oracle inequality
  • proof
  • Lemma 4.1: Second-order lower bound for $\ell_n$
  • proof
  • Lemma 4.2: Dual norm bounds for the gradient
  • proof
  • Lemma 4.3: Restricted strong convexity on the low-rank-plus-sparse cone
  • ...and 9 more