Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Estimating functions for SDE driven by stable Lévy processes

Emmanuelle Clément and Arnaud Gloter

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Abstract

This paper is concerned with parametric inference for a stochastic differential equation driven by a pure-jump Lévy process, based on high frequency observations on a fixed time period. Assuming that the Lévy measure of the driving process behaves like that of an $\alpha$-stable process around zero, we propose an estimating functions based method which leads to asymptotically efficient estimators for any value of $\alpha\in(0,2)$ and does not require any integrability assumptions on the process. The main limit theorems are derived thanks to a control in total variation distance between the law of the normalized process, in small time, and the $\alpha$-stable distribution. This method is an alternative to the non Gaussian quasi-likelihood estimation method proposed by Masuda (Stochastic Process. Appl. (2018) To appear) where the Blumenthal–Getoor index $\alpha$ is restricted to belong to the interval $[1,2)$.

Résumé

Dans cet article, nous étudions l’estimation des paramètres d’une équation différentielle stochastique dirigée par un processus de saut pur, à partir d’observations hautes fréquences du processus sur un intervalle de temps fixe. En supposant que la mesure de Lévy du processus de saut qui dirige l’équation se comporte autour de zéro comme la mesure de Lévy d’un processus $\alpha$-stable, nous proposons une méthode d’estimation basée sur des fonctions estimantes qui conduit à des estimateurs asymptotiquement efficaces des paramètres de tendance et d’échelle, pour toute valeur de $\alpha\in(0,2)$, et qui ne nécessite pas de conditions d’intégrabilité du processus. Les principaux résultats asymptotiques sont obtenus grâce à un contrôle en variation totale entre la loi du processus renormalisé, en temps petit, et la loi $\alpha$-stable. Cette méthode est une alternative à la méthode de quasi-vraisemblance non gaussienne proposée par Masuda (Stochastic Process. Appl. (2018) To appear), où l’indice de Blumenthal–Getoor $\alpha$ appartient à l’interval $[1,2)$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1316-1348.

Dates
Received: 16 November 2017
Revised: 13 June 2018
Accepted: 11 July 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1569398871

Digital Object Identifier
doi:10.1214/18-AIHP920

Mathematical Reviews number (MathSciNet)
MR4010937

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60G52: Stable processes 60J75: Jump processes 62F12: Asymptotic properties of estimators
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60F05: Central limit and other weak theorems

Keywords
Lévy process Stable process Stochastic differential equation Parametric inference Estimating functions Malliavin calculus

Citation

Clément, Emmanuelle; Gloter, Arnaud. Estimating functions for SDE driven by stable Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1316--1348. doi:10.1214/18-AIHP920. https://projecteuclid.org/euclid.aihp/1569398871


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