Optimal convergence rates for the invariant density estimation of jump-diffusion processes

We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for d = 1 and d = 2. We consider a class of fully non-linear jump diffusion processes whose invariant density belongs to some Hölder space. Firstly, in dimension one, we...

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Detalles Bibliográficos
Autores: Amorino, Chiara, Nualart, Eulàlia
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2022
País:España
Institución:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:10230/57624
Acceso en línea:http://hdl.handle.net/10230/57624
http://dx.doi.org/10.1051/ps/2022001
Access Level:acceso abierto
Palabra clave:Minimax risk
convergence rate
non-parametric statistics
ergodic diffusion with jumps
Lévy driven SDE
invariant density estimation
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spelling Optimal convergence rates for the invariant density estimation of jump-diffusion processesAmorino, ChiaraNualart, EulàliaMinimax riskconvergence ratenon-parametric statisticsergodic diffusion with jumpsLévy driven SDEinvariant density estimationWe aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for d = 1 and d = 2. We consider a class of fully non-linear jump diffusion processes whose invariant density belongs to some Hölder space. Firstly, in dimension one, we show that the kernel density estimator achieves the convergence rate 1/T, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in Amorino and Gloter [J. Stat. Plann. Inference 213 (2021) 106–129], which depends on the Blumenthal-Getoor index for d = 1 and is equal to (logT)/T for d = 2. Secondly, when the jump and diffusion coefficients are constant and the jumps are finite, we show that is not possible to find an estimator with faster rates of estimation. Indeed, we get some lower bounds with the same rates {1/T, (logT)/T} in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case for the fully non-linear process.CA gratefully acknowledges financial support of ERC Consolidator Grant 815703 “STAMFORD: Statistical Methods for High Dimensional Diffusions”. EN acknowledges support from the Spanish MINECO grant PGC2018-101643-B-I00 and Ayudas Fundación BBVA a Equipos de Investigación Científica 2017.EDP Sciences202320232022info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/10230/57624http://dx.doi.org/10.1051/ps/2022001reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésESAIM: Probability and Statistics. 2022;26:126-51.info:eu-repo/grantAgreement/EC/H2020/815703info:eu-repo/grantAgreement/ES/2PE/PGC2018-101643-B-I00© The authors. Published by EDP Sciences, SMAI 2022. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.http://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:10230/576242026-05-29T05:05:01Z
dc.title.none.fl_str_mv Optimal convergence rates for the invariant density estimation of jump-diffusion processes
title Optimal convergence rates for the invariant density estimation of jump-diffusion processes
spellingShingle Optimal convergence rates for the invariant density estimation of jump-diffusion processes
Amorino, Chiara
Minimax risk
convergence rate
non-parametric statistics
ergodic diffusion with jumps
Lévy driven SDE
invariant density estimation
title_short Optimal convergence rates for the invariant density estimation of jump-diffusion processes
title_full Optimal convergence rates for the invariant density estimation of jump-diffusion processes
title_fullStr Optimal convergence rates for the invariant density estimation of jump-diffusion processes
title_full_unstemmed Optimal convergence rates for the invariant density estimation of jump-diffusion processes
title_sort Optimal convergence rates for the invariant density estimation of jump-diffusion processes
dc.creator.none.fl_str_mv Amorino, Chiara
Nualart, Eulàlia
author Amorino, Chiara
author_facet Amorino, Chiara
Nualart, Eulàlia
author_role author
author2 Nualart, Eulàlia
author2_role author
dc.subject.none.fl_str_mv Minimax risk
convergence rate
non-parametric statistics
ergodic diffusion with jumps
Lévy driven SDE
invariant density estimation
topic Minimax risk
convergence rate
non-parametric statistics
ergodic diffusion with jumps
Lévy driven SDE
invariant density estimation
description We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for d = 1 and d = 2. We consider a class of fully non-linear jump diffusion processes whose invariant density belongs to some Hölder space. Firstly, in dimension one, we show that the kernel density estimator achieves the convergence rate 1/T, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in Amorino and Gloter [J. Stat. Plann. Inference 213 (2021) 106–129], which depends on the Blumenthal-Getoor index for d = 1 and is equal to (logT)/T for d = 2. Secondly, when the jump and diffusion coefficients are constant and the jumps are finite, we show that is not possible to find an estimator with faster rates of estimation. Indeed, we get some lower bounds with the same rates {1/T, (logT)/T} in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case for the fully non-linear process.
publishDate 2022
dc.date.none.fl_str_mv 2022
2023
2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10230/57624
http://dx.doi.org/10.1051/ps/2022001
url http://hdl.handle.net/10230/57624
http://dx.doi.org/10.1051/ps/2022001
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv ESAIM: Probability and Statistics. 2022;26:126-51.
info:eu-repo/grantAgreement/EC/H2020/815703
info:eu-repo/grantAgreement/ES/2PE/PGC2018-101643-B-I00
dc.rights.none.fl_str_mv http://creativecommons.org/licenses/by/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv EDP Sciences
publisher.none.fl_str_mv EDP Sciences
dc.source.none.fl_str_mv reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
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repository.mail.fl_str_mv
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