Nonparametric inference for Lévy-driven Ornstein - Uhlenbeck processes

G. Jongbloed*, F. H. Van Der Meulen, A. W. Van Der Vaart

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

65 Citations (Scopus)


We consider nonparametric estimation of the Lévy measure of a hidden Lévy process driving a stationary Omstein-Uhlenbeck process which is observed at discrete time points. This Lévy measure can be expressed in terms of the canonical function of the stationary distribution of the Omstein-Uhlenbeck process, which is known to be self-decomposable. We propose an estimator for this canonical function based on a preliminary estimator of the characteristic function of the stationary distribution. We provide a suppport-reduction algorithm for the numerical computation of the estimator, and show that the estimator is asymptotically consistent under various sampling schemes. We also define a simple consistent estimator of the intensity parameter of the process. Along the way, a nonparametric procedure for estimating a self-decomposable density function is constructed, and it is shown that the Oenstein-Uhlenbeck process is β-mixing. Some general results on uniform convergence of random characteristic functions are included.

Original languageEnglish
Pages (from-to)759-791
Number of pages33
Issue number5
Publication statusPublished - 1 Oct 2005
Externally publishedYes


  • Lévy process
  • Self-decomposability
  • Support-reduction algorithm
  • Uniform convergence of characteristic functions


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