The Annals of Statistics

Jump filtering and efficient drift estimation for Lévy-driven SDEs

Arnaud Gloter, Dasha Loukianova, and Hilmar Mai

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Abstract

The problem of drift estimation for the solution $X$ of a stochastic differential equation with Lévy-type jumps is considered under discrete high-frequency observations with a growing observation window. An efficient and asymptotically normal estimator for the drift parameter is constructed under minimal conditions on the jump behavior and the sampling scheme. In the case of a bounded jump measure density, these conditions reduce to $n\Delta_{n}^{3-\varepsilon}\rightarrow 0$, where $n$ is the number of observations and $\Delta_{n}$ is the maximal sampling step. This result relaxes the condition $n\Delta_{n}^{2}\rightarrow 0$ usually required for joint estimation of drift and diffusion coefficient for SDEs with jumps. The main challenge in this estimation problem stems from the appearance of the unobserved continuous part $X^{c}$ in the likelihood function. In order to construct the drift estimator, we recover this continuous part from discrete observations. More precisely, we estimate, in a nonparametric way, stochastic integrals with respect to $X^{c}$. Convergence results of independent interest are proved for these nonparametric estimators.

Article information

Source
Ann. Statist., Volume 46, Number 4 (2018), 1445-1480.

Dates
Received: March 2016
Revised: May 2017
First available in Project Euclid: 27 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1530086422

Digital Object Identifier
doi:10.1214/17-AOS1591

Mathematical Reviews number (MathSciNet)
MR3819106

Zentralblatt MATH identifier
06936467

Subjects
Primary: 60J75: Jump processes 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation

Keywords
Lévy-driven SDE efficient drift estimation maximum likelihood estimation high frequency data ergodic properties

Citation

Gloter, Arnaud; Loukianova, Dasha; Mai, Hilmar. Jump filtering and efficient drift estimation for Lévy-driven SDEs. Ann. Statist. 46 (2018), no. 4, 1445--1480. doi:10.1214/17-AOS1591. https://projecteuclid.org/euclid.aos/1530086422


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Supplemental materials

  • Jump filtering and efficient drift estimation for Lévy-driven SDEs. The supplement contains the two additional Sections 7–8. In Section 7, we investigate the numerical performance of the estimator for the finite sample. We consider the case of Ornstein–Uhlbenbeck and “Hyperbolic” diffusion models, with finite or infinite activity jump measure. We compare the results for different choices of the threshold constants $a_{i}^{n}$ [recall (3.3)]. In Section 8, we give a proof the Lemma 2.1 about the ergodicity of the process, and of the LAN property (Theorem 5.3). Next, we gather the proofs of the technical Lemmas 6.1, 6.3 and 6.4, and show that the identifiability Assumption 6 is equivalent to (2.1).