Nonsynchronous covariation process and limit theorems

Takaki Hayashi, Nakahiro Yoshida

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

An asymptotic distribution theory of the nonsynchronous covariation process for continuous semimartingales is presented. Two continuous semimartingales are sampled at stopping times in a nonsynchronous manner. Those sampling times possibly depend on the history of the stochastic processes and themselves. The nonsynchronous covariation process converges to the usual quadratic covariation of the semimartingales as the maximum size of the sampling intervals tends to zero. We deal with the case where the limiting variation process of the normalized approximation error is random and prove the convergence to mixed normality, or convergence to a conditional Gaussian martingale. A class of consistent estimators for the asymptotic variation process based on kernels is proposed, which will be useful for statistical applications to high-frequency data analysis in finance. As an illustrative example, a Poisson sampling scheme with random change point is discussed.

Original languageEnglish
Pages (from-to)2416-2454
Number of pages39
JournalStochastic Processes and their Applications
Volume121
Issue number10
DOIs
Publication statusPublished - 2011 Oct

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Semimartingale
Limit Theorems
Process Variation
Sampling
High-frequency Data
Distribution Theory
Random errors
Stopping Time
Consistent Estimator
Change Point
Asymptotic Theory
Approximation Error
Finance
Random processes
Martingale
Normality
Asymptotic distribution
Stochastic Processes
Data analysis
Siméon Denis Poisson

Keywords

  • Discrete sampling
  • High-frequency data
  • Martingale central limit theorem
  • Nonsynchronicity
  • Quadratic variation
  • Realized volatility
  • Semimartingale
  • Stable convergence

ASJC Scopus subject areas

  • Modelling and Simulation
  • Statistics and Probability
  • Applied Mathematics

Cite this

Nonsynchronous covariation process and limit theorems. / Hayashi, Takaki; Yoshida, Nakahiro.

In: Stochastic Processes and their Applications, Vol. 121, No. 10, 10.2011, p. 2416-2454.

Research output: Contribution to journalArticle

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