### 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 language | English |
---|---|

Pages (from-to) | 2416-2454 |

Number of pages | 39 |

Journal | Stochastic Processes and their Applications |

Volume | 121 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2011 Oct |

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### 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

*Stochastic Processes and their Applications*,

*121*(10), 2416-2454. https://doi.org/10.1016/j.spa.2010.12.005

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

Research output: Contribution to journal › Article

*Stochastic Processes and their Applications*, vol. 121, no. 10, pp. 2416-2454. https://doi.org/10.1016/j.spa.2010.12.005

}

TY - JOUR

T1 - Nonsynchronous covariation process and limit theorems

AU - Hayashi, Takaki

AU - Yoshida, Nakahiro

PY - 2011/10

Y1 - 2011/10

N2 - 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.

AB - 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.

KW - Discrete sampling

KW - High-frequency data

KW - Martingale central limit theorem

KW - Nonsynchronicity

KW - Quadratic variation

KW - Realized volatility

KW - Semimartingale

KW - Stable convergence

UR - http://www.scopus.com/inward/record.url?scp=79961007731&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79961007731&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2010.12.005

DO - 10.1016/j.spa.2010.12.005

M3 - Article

AN - SCOPUS:79961007731

VL - 121

SP - 2416

EP - 2454

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 10

ER -