Low-Rank and Sparse Drift Estimation for High-Dimensional L\'evy-Driven Ornstein--Uhlenbeck Processes

Abstract

We study high-dimensional Ornstein--Uhlenbeck processes driven by L\'evy 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 L\'evy process, we derive a non-asymptotic oracle inequality for the Frobenius risk of the estimator. The bound separates a discretization bias term of order $d^2\Delta_n^2$ from a stochastic term of order $\gamma(\Delta_n)T^{-1}(r \log d + s \log d)$, 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 L\'evy regimes.