## Abstract

We derive limit theorems for the empirical distribution function of "devolatilized" increments of an Itô semimartingale observed at high frequencies. These "devolatilized" increments are formed by suitably rescaling and truncating the raw increments to remove the effects of stochastic volatility and "large" jumps. We derive the limit of the empirical c.d.f. of the adjusted increments for any Itô semimartingale whose dominant component at high frequencies has activity index of 1<ß ≤ 2, where ß = 2 corresponds to diffusion. We further derive an associated CLT in the jump-diffusion case. We use the developed limit theory to construct a feasible and pivotal test for the class of Itô semimartingales with nonvanishing diffusion coefficient against Itô semimartingales with no diffusion component.

Original language | English (US) |
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Pages (from-to) | 1850-1888 |

Number of pages | 39 |

Journal | Annals of Applied Probability |

Volume | 24 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2014 |

## Keywords

- High-frequency data
- Itô semimartingale
- Jumps
- Kolmogorov-Smirnov test
- Stable process
- Stochastic volatility

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty