Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Low-rank diffusion matrix estimation for high-dimensional time-changed Lévy processes

Denis Belomestny and Mathias Trabs

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Abstract

The estimation of the diffusion matrix $\Sigma$ of a high-dimensional, possibly time-changed Lévy 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.

Résumé

Nous étudions le problème de l’estimation de la matrice de diffusion $\Sigma$ d’un processus de Lévy en grande dimension, qui peut être changé de temps, en se basant sur des observations discrètes du processus à une distance fixée. Nous imposons une condition de faible rang sur $\Sigma$. À l’aide d’une méthode spectrale, nous construisons un estimateur pondéré des moindres carrés avec une pénalisation par une norme nucléaire. Nous prouvons des inégalités oracle et obtenons des vitesses de convergence pour l’estimateur de la matrice de diffusion. Nous constatons que ces vitesses dépendent du rang de $\Sigma$ d’une façon surprenante, et qu’elles sont optimales au sens minimax pour une dimension fixée. Ces résultats théoriques sont illustrés par une étude de simulations.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1583-1621.

Dates
Received: 15 June 2016
Revised: 3 April 2017
Accepted: 13 June 2017
First available in Project Euclid: 11 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1531296030

Digital Object Identifier
doi:10.1214/17-AIHP849

Mathematical Reviews number (MathSciNet)
MR3825892

Zentralblatt MATH identifier
06976086

Subjects
Primary: 62M05: Markov processes: estimation
Secondary: 60G51: Processes with independent increments; Lévy processes 62G05: Estimation 62M15: Spectral analysis

Keywords
Volatility estimation Lasso-type estimator Minimax convergence rates Nonlinear inverse problem Oracle inequalities Time-changed Lévy process

Citation

Belomestny, Denis; Trabs, Mathias. Low-rank diffusion matrix estimation for high-dimensional time-changed Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1583--1621. doi:10.1214/17-AIHP849. https://projecteuclid.org/euclid.aihp/1531296030


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