Vector-valued generalized Ornstein–Uhlenbeck processes: Properties and parameter estimation

Marko Voutilainen, Lauri Viitasaari, Pauliina Ilmonen, Soledad Torres, Ciprian Tudor

Research output: Contribution to journalArticleScientificpeer-review

4 Citations (Scopus)
25 Downloads (Pure)


Generalizations of the Ornstein-Uhlenbeck process defined through Langevin equations, such as fractional Ornstein-Uhlenbeck processes, have recently received a lot of attention. However, most of the literature focuses on the one-dimensional case with Gaussian noise. In particular, estimation of the unknown parameter is widely studied under Gaussian stationary increment noise. In this article, we consider estimation of the unknown model parameter in the multidimensional version of the Langevin equation, where the parameter is a matrix and the noise is a general, not necessarily Gaussian, vector-valued process with stationary increments. Based on algebraic Riccati equations, we construct an estimator for the parameter matrix. Moreover, we prove the consistency of the estimator and derive its limiting distribution under natural assumptions. In addition, to motivate our work, we prove that the Langevin equation characterizes essentially all multidimensional stationary processes.

Original languageEnglish
Pages (from-to)992-1022
Number of pages31
JournalScandinavian Journal of Statistics
Issue number3
Early online date8 Aug 2021
Publication statusPublished - Sept 2022
MoE publication typeA1 Journal article-refereed


  • algebraic Riccati equations
  • consistency
  • Langevin equation
  • multivariate Ornstein–Uhlenbeck process
  • nonparametric estimation
  • stationary processes


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