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

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

Abstract

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.

Suggested Citation

  • Marko Voutilainen & Lauri Viitasaari & Pauliina Ilmonen & Soledad Torres & Ciprian Tudor, 2022. "Vector‐valued generalized Ornstein–Uhlenbeck processes: Properties and parameter estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 992-1022, September.
    • Handle: RePEc:bla:scjsta:v:49:y:2022:i:3:p:992-1022
    DOI: 10.1111/sjos.12552
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    References listed on IDEAS

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    7. Ehsan Azmoodeh & Lauri Viitasaari, 2015. "Parameter estimation based on discrete observations of fractional Ornstein–Uhlenbeck process of the second kind," Statistical Inference for Stochastic Processes, Springer, vol. 18(3), pages 205-227, October.
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    9. Viitasaari, Lauri, 2016. "Representation of stationary and stationary increment processes via Langevin equation and self-similar processes," Statistics & Probability Letters, Elsevier, vol. 115(C), pages 45-53.
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