Low-rank diffusion matrix estimation for high-dimensional time-changed L\'evy processes

TL;DR

This paper develops a spectral, nuclear-norm-penalized estimator for the low-rank diffusion matrix of high-dimensional, time-changed Lévy processes, with proven optimal convergence rates and demonstrated via simulations.

Contribution

It introduces a novel spectral estimation method with nuclear norm penalization for low-rank diffusion matrices in high-dimensional Lévy processes.

Findings

Convergence rates depend on the rank of the diffusion matrix.

The estimator achieves minimax optimality in fixed dimensions.

Theoretical results are supported by simulation studies.

Abstract

The estimation of the diffusion matrix $\Sigma$ of a high-dimensional, possibly time-changed L\'evy process is studied, based on discrete observations of the process with a fixed distance. A low-rank condition is imposed on $\Sigma$. Applying a spectral approach, we construct a weighted least-squares estimator with nuclear-norm-penalisation. We prove oracle inequalities and derive convergence rates for the diffusion matrix estimator. The convergence rates show a surprising dependency on the rank of $\Sigma$ and are optimal in the minimax sense for fixed dimensions. Theoretical results are illustrated by a simulation study.