Asymptotic properties of realized power variations and related functionals of semimartingales

By Jean Jacod. 2006

295 citations recorded by the source corpus.

This paper is concerned with the asymptotic behavior of sums of the form , where X is a 1-dimensional semimartingale and f a suitable test function, typically f(x)=xr, as [Delta]n-->0. We prove a variety of "laws of large numbers", that is convergence in probability of Un(f)t, sometimes after normalization. We also exhibit in many cases the rate of convergence, as well as associated central limit theorems.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/426bd8887ef5a55a160d51fabefd18c3ab2ee2c9

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Citations in the corpus

Limit Theorems for Functionals of Higher Order Differences of Brownian Semi-Stationary Processes (2009, unknown): “[6], [8], [9], [10], [14], [16], [18], [23] for theory and applications.”
Asymptotic analysis of hedging errors in models with jumps (2009, background): “A related problem arises in the context of asymptotic behavior of realized volatility and power variation [1,2,13].”
Threshold Bipower Variation and the Impact of Jumps on Volatility Forecasting (2010, background): “Jacod (2008); Mancini (2009). A simple special case of TMPV is threshold bipower variation, obtained with γ1 = γ2 = 1 and multiplying by a suitable constant:”
MODELING HIGH-FREQUENCY FINANCIAL DATA BY PURE JUMP PROCESSES (2012, methodology): “Since J is purely discontinuous, we can take the exponent in (64) of Aı̈tSahalia and Jacod (2009) as 1/2, and then by Lemma 1, Pi,2 ≤C∆̟−1/2 n Pti−1(|∆i J | ≥ √ ∆n)≤C∆ n .”
Multipower Variation for Brownian Semistationary Processes (2011, background): “The motivation for the development of the results reported in this paper has been the need to construct tools for studying the probabilistic limit behaviour of (realised) quadratic variation and other multipower variations in relation to the class of Brownian semistationary (BSS) processes.”
Quasi-maximum likelihood estimation of volatility with high frequency data (2010, background): “See Jacod (2008) Hypoth- esis (L-s) for more details.”
Large-dimensional factor modeling based on high-frequency observations (2019, unknown): “(18)A different approach of using perturbed matrices for rank testing has been proposed in Jacod and Podolskij (2013).”
On the jump activity index for semimartingales (2012, background): “…and HKUST6015/08P E-mails: majing@ust.hk, xinbingkong@yahoo.com.cn, liuzhi@ust.hk, per.mykland@gmail.com in semimartingales with discontinuous part (i.e., jumps) in the literature; see AitSahalia (2004), Ait-Sahalia and Jacod (2007), Fan and Wang (2007), Jacod (2008) and references therein.”
Microstructure Noise in the Continuous Case: The Pre-Averaging Approach ∗ (2008, methodology): “Then a standard localization procedure explained in details in [4] for example shows that for proving Theorem 3.”
Designing Realised Kernels to Measure the Ex-Post Variation of Equity Prices in the Presence of Noise (2008, background): “See Jacod (2008) for a general treatment of results of this kind.”
A remark on the rates of convergence for integrated volatility estimation in the presence of jumps (2012, result): “These results are shown in [3], and Mancini in [6] has proved that when the jumps of X are those of a stable process with index β [so (L-r) holds for all r > β , but not for r = β], and when β ≥ 1, the estimator converges exactly at rate n (2−β), in the sense that the sequence n (2−β)(Ĉ(vn)1 − C1) c”
Bipower Variation for Gaussian Processes with Stationary Increments (2009, unknown): “Extensions of the theory to Lévy processes and Itô semimartingales have been obtained, particularly by Jacod [18] (cf. also [5]), and applications to finance of such extensions are discussed in [20] and [28].”