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

Argimiro Arratia, Alejandra Cabaña, Enrique M. Cabaña

Research output: Contribution to journalArticleResearchpeer-review

6 Citations (Scopus)


© 2016, Institut d'Estadistica de Catalunya. All rights reserved. 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 Lévy 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.
Original languageEnglish
Pages (from-to)267-302
Issue number2
Publication statusPublished - 1 Jul 2016


  • Continuous ARMA
  • Lévy process
  • Ornstein-Uhlenbeck process
  • Stationary process


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