Nonparametric inference for the spectral measure of a bivariate pure-jump semimartingale

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2 Scopus citations


We develop a nonparametric estimator for the spectral density of a bivariate pure-jump Itô semimartingale from high-frequency observations of the process on a fixed time interval with asymptotically shrinking mesh of the observation grid. The process of interest is locally stable, i.e., its Lévy measure around zero is like that of a time-changed stable process. The spectral density function captures the dependence between the small jumps of the process and is time invariant. The estimation is based on the fact that the characteristic exponent of the high-frequency increments, up to a time-varying scale, is approximately a convolution of the spectral density and a known function depending on the jump activity. We solve the deconvolution problem in Fourier transform using the empirical characteristic function of locally studentized high-frequency increments and a jump activity estimator.

Original languageEnglish (US)
Pages (from-to)419-451
Number of pages33
JournalStochastic Processes and their Applications
Issue number2
StatePublished - Feb 2019


  • Deconvolution
  • Fourier transform
  • High-frequency data
  • Itô semimartingale
  • Nonparametric inference
  • Spectral density

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


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