A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes

We present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Lévy process onto an Ornstein-Uhlenbeck process. The construction resembles the procedure to build an AR(p) from an AR(1). We show that this family is in fact a subfamily...

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Autores: Arratia Quesada, Argimiro Alejandro|||0000-0003-1551-420X, Cabaña, Ana Alejandra, Cabaña Perez, Enrique
Formato: artículo
Fecha de publicación:2016
País:España
Recursos:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/102108
Acesso em linha:https://hdl.handle.net/2117/102108
https://dx.doi.org/10.2436/20.8080.02.44
Access Level:acceso abierto
Palavra-chave:Lévy processes
Ornstein-Uhlenbeck process
Stochastic processes
Lévy process
Continuous ARMA
Stationary process
Lévy, Processos de
Processos estocàstics
Àrees temàtiques de la UPC::Matemàtiques i estadística
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oai_identifier_str oai:upcommons.upc.edu:2117/102108
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repository_id_str
spelling A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processesArratia Quesada, Argimiro Alejandro|||0000-0003-1551-420XCabaña, Ana AlejandraCabaña Perez, EnriqueLévy processesOrnstein-Uhlenbeck processStochastic processesOrnstein-Uhlenbeck processLévy processContinuous ARMAStationary processLévy, Processos deProcessos estocàsticsÀrees temàtiques de la UPC::Matemàtiques i estadísticaWe present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Lévy process onto an Ornstein-Uhlenbeck process. The construction resembles the procedure to build an AR(p) from an AR(1). We show that this family is in fact a subfamily of the well-known CARMA(p,q) processes, with several interesting advantages, including a smaller number of parameters. The resulting processes are linear combinations of Ornstein-Uhlenbeck processes all driven by the same L´evy process. This provides a straightforward computation of covariances, a state-space model representation and methods for estimating parameters. Furthermore, the discrete and equally spaced sampling of the process turns to be an ARMA(p, p-1) process. We propose methods for estimating the parameters of the iterated Ornstein-Uhlenbeck process when the noise is either driven by a Wiener or a more general Lévy process, and show simulations and applications to real data.Peer ReviewedInstitut d'Estadística de Catalunya20162016-12-0120172017-03-08journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/102108https://dx.doi.org/10.2436/20.8080.02.44reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2http://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/1021082026-05-27T15:37:01Z
dc.title.none.fl_str_mv A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes
title A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes
spellingShingle A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes
Arratia Quesada, Argimiro Alejandro|||0000-0003-1551-420X
Lévy processes
Ornstein-Uhlenbeck process
Stochastic processes
Ornstein-Uhlenbeck process
Lévy process
Continuous ARMA
Stationary process
Lévy, Processos de
Processos estocàstics
Àrees temàtiques de la UPC::Matemàtiques i estadística
title_short A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes
title_full A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes
title_fullStr A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes
title_full_unstemmed A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes
title_sort A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes
dc.creator.none.fl_str_mv Arratia Quesada, Argimiro Alejandro|||0000-0003-1551-420X
Cabaña, Ana Alejandra
Cabaña Perez, Enrique
author Arratia Quesada, Argimiro Alejandro|||0000-0003-1551-420X
author_facet Arratia Quesada, Argimiro Alejandro|||0000-0003-1551-420X
Cabaña, Ana Alejandra
Cabaña Perez, Enrique
author_role author
author2 Cabaña, Ana Alejandra
Cabaña Perez, Enrique
author2_role author
author
dc.subject.none.fl_str_mv Lévy processes
Ornstein-Uhlenbeck process
Stochastic processes
Ornstein-Uhlenbeck process
Lévy process
Continuous ARMA
Stationary process
Lévy, Processos de
Processos estocàstics
Àrees temàtiques de la UPC::Matemàtiques i estadística
topic Lévy processes
Ornstein-Uhlenbeck process
Stochastic processes
Ornstein-Uhlenbeck process
Lévy process
Continuous ARMA
Stationary process
Lévy, Processos de
Processos estocàstics
Àrees temàtiques de la UPC::Matemàtiques i estadística
description We present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Lévy process onto an Ornstein-Uhlenbeck process. The construction resembles the procedure to build an AR(p) from an AR(1). We show that this family is in fact a subfamily of the well-known CARMA(p,q) processes, with several interesting advantages, including a smaller number of parameters. The resulting processes are linear combinations of Ornstein-Uhlenbeck processes all driven by the same L´evy process. This provides a straightforward computation of covariances, a state-space model representation and methods for estimating parameters. Furthermore, the discrete and equally spaced sampling of the process turns to be an ARMA(p, p-1) process. We propose methods for estimating the parameters of the iterated Ornstein-Uhlenbeck process when the noise is either driven by a Wiener or a more general Lévy process, and show simulations and applications to real data.
publishDate 2016
dc.date.none.fl_str_mv 2016
2016-12-01
2017
2017-03-08
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/102108
https://dx.doi.org/10.2436/20.8080.02.44
url https://hdl.handle.net/2117/102108
https://dx.doi.org/10.2436/20.8080.02.44
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2

http://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2

http://creativecommons.org/licenses/by-nc-nd/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Institut d'Estadística de Catalunya
publisher.none.fl_str_mv Institut d'Estadística de Catalunya
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869418102824894464
score 15,301603

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