First of all, we send our extremely belated congratulations to Professor Jean Jacod, on such an insightful and well-written article which promoted the prosperity of the field of high-frequency data research over the past two decades. We would also like to congratulate the Journal of Financial Econometrics for recognizing this important article. 1 A General Result of Great Mathematical Beauty The paper “Limit of Random Measures Associated with the Increments of a Brownian Semimartingale” by Jacod [referred to as Jacod (1994) below] focuses on the asymptotic behavior of the following processes: Utn(g)=δn∑i:S(n,i)≤t(g(T(n,i),ξin)−∫g(T(n,i),x)ρT(n,i)(dx)), and Vtn(g)=δn(∑i:S(n,i)≤tg(T(n,i),ξin)−∫0t∫xg(t,x)ρt(dx) dt), where 1. (T(n,i)) is an increasing sequence of stopping times with T(n,0)=0 and limi→∞T(n,i)= ∞. 2. For each i, Δ(n,i)∈(0,∞) is FT(n,i)-measurable such that S(n,i):=T(n,i)+Δ(n,i)≤T(n,i+1). 3. ξin=Δ(n,i)−1/2(XS(n,i)−X(T(n,i)), where (Xt) is either a standard Brownian motion or a Brownian semimartingale of the form dXt=at dWt+bt dt. 4. g(·) is a predictable function satisfying certain regularity conditions (Assumption K in the paper), and ρt(·) stands for the centered Gaussian measure with covariance matrix ct:=atatT. The results are fairly complete in the regular observation time case, namely, when Δ(n,i)≡1/n and T(n,i)=i/n. We start the discussion on this case first, and then move on to the irregular time case which was also covered by Jacod (1994). The paper considers a general multivariate setting. Below, we focus on the univariate case. 1.1 The Case of Regular Observation Times In the regular observation times case, assuming that (at) is a Brownian semimartingale, under rather general conditions, the paper shows that (Theorem 6.5 and Corollary 6.6), with δn=1/n, (Utn(g)) and (Vtn(g)) converge stably in law to a same limiting process, U¯(g) defined by Equation (6.17) in the paper. The results have several applications which have been widely studied in the high-frequency literature: Taking g(x)=x2 yields the following fundamental result in estimating the integrated volatility: n(∑i≤[nt](ΔXi/n)2−∫0tat2 dt) converges stably in law to an F-conditionally Gaussian martingale with quadratic variation 2∫0tas4 ds. More generally, take g(x)=xp for some p∈ℕ, then one has that n(np/2−1∑i≤[nt](ΔXi/n)p−αp∫0tatp dt) converges stably in law, where αp=E(Zp) with Z∼N(0,1). The limiting process is an F-conditionally Gaussian martingale if p is even, and involves a drift term if p is odd. If, on the other hand, one takes g(x)=|x|β for some β>1, then one has the following result about power variation: n(nβ/2−1∑i≤[nt]|ΔXi/n|β−α′β∫0tatβ dt) converges stably in law to an F-conditionally Gaussian martingale with quadratic variation (α′2β−(αβ′)2)∫0tas2β ds, where α′r=E(|Z|r) for any r > 0. The first result has later been established in Barndorff-Nielsen and Shephard (2002) and Bandi and Russell (2008), and the power variation has been extensively studied starting from Barndorff-Nielsen and Shephard (2003). It is worth mentioning that in Jacod (1994) the function g(·) is allowed to depend on time t and even the process X. In Delattre and Jacod (1997), the authors further relaxed the condition by allowing g to depend on n. Such a relaxation enables them to study the pure rounding case with diminishing rounding level. The paper Delattre and Jacod (1997) started the study on rounding effect in high-frequency data, leading to further developments in Rosenbaum (2009) and Li and Mykland (2015), among others. 1.2 Irregular Observation Time Case In the irregular observation time case, the paper provides some parallel convergence results about the process (Utn(g)), and further points out that the convergence of (Vtn(g)) requires stronger regularity assumptions on the function g and the observation times. Below we provide some comments on the assumptions on time in the paper and some later developments in this regard: The paper assumes that the inter-observation duration Δ(n,i)∈(0,∞) is FT(n,i)-measurable. In the customary setting in high-frequency data where S(n,i)(=T(n,i)+Δ(n,i))=T(n,i+1), this amounts to assume that T(n,i+1) is “strongly predictable” in the sense that T(n,i+1) is FT(n,i)-measurable. Such a setting was later further studied in Hayashi, Jacod, and Yoshida (2011). Observe that while the strong predictability assumption is restrictive, it does cover the cases of deterministic observation times and the independent observation times (Mykland and Zhang, 2006). In particular, assuming the existence of the “quadratic variation of time” process, Mykland and Zhang (2006) establish the convergence of (Vtn(g)) with g(x)=x2. The assumption does rule out the endogenous observation time case considered in Li et al. (2014). A striking feature in this case is that the tricity appears in the CLT for (Vtn(g)) with g(x)=x2. Li et al. (2014) also document that endogeneity is present in financial data. 2 A Significant Pioneer in High-Frequency Data Research Jacod (1994) is definitely a significant pioneer in high-frequency data research. Even as an unpublished manuscript, the paper had been recognized and played a significant role in high-frequency data research. Take, for example, the first batch of papers that propose noise-robust estimators, several had cited Jacod (1994) as an important reference, including Barndorff-Nielsen et al. (2008); Jacod et al. (2009); Xiu (2010); and Aït-Sahalia, Mykland, and Zhang (2011). 3 Beyond Jacod (1994) After Jacod (1994), Professor Jean Jacod continued to make key contributions in the field of high-frequency data. His paper with Protter (Jacod and Protter, 1998) had been fundamental for volatility inference. The pre-averaging method developed in Jacod et al. (2009) was recognized as one of the most effective de-noise methods. Professor Jacod also extensively studied the impact of jumps in high-frequency data (Aït-Sahalia and Jacod, 2009, 2011, 2012; Jacod and Todorov, 2009, 2014, 2016; Aït-Sahalia, Jacod, and Li, 2012; Jacod and Reiss, 2014). More recently, he also made important contributions to the understanding of market microstructure noise (Jacod, Li, and Zheng, 2017). Professor Jacod’s book with Shiryaev (Jacod and Shiryaev, 2003) has been regarded as the “bible” in high-frequency data research. His newer books Jacod and Protter (2012) and Aït-Sahalia and Jacod (2014) provide excellent tool sets and reviews for this field. We feel blessed to have such a great researcher in our field and to have the fortune to collaborate with him. We look forward to his continued achievements in the fields of stochastic processes and high-frequency data. Footnotes * Financial support from the Research Grants Council (RGC) of Hong Kong under Grants [GRF 16502014, GRF 16518716 toY.L.] and [GRF 16305315 to X.Z.] is gratefully acknowledged. References Aït-Sahalia Y. , and Jacod J. . 2009 . Estimating the Degree of Activity of Jumps in High Frequency Data . Annals of Statistics 37 : 2202 – 2244 . Google Scholar Crossref Search ADS Aït-Sahalia Y. , and Jacod J. . 2011 . Testing Whether Jumps Have Finite or Infinite Activity . Annals of Statistics 39 : 1689 – 1719 . Google Scholar Crossref Search ADS Aït-Sahalia Y. , and Jacod J. . 2012 . Identifying the Successive Blumenthal–Getoor Indices of a Discretely Observed Process . Annals of Statistics 40 : 1430 – 1464 . Google Scholar Crossref Search ADS Aït-Sahalia Y. , and Jacod J. . 2014 . High-Frequency Financial Econometrics . Princeton University Press . Aït-Sahalia Y. , Jacod J. , and Li J. . 2012 . Testing for Jumps in Noisy High Frequency Data . Journal of Econometrics 168 : 207 – 222 . Google Scholar Crossref Search ADS Aït-Sahalia Y. , Mykland P. A. , and Zhang L. . 2011 . Ultra High Frequency Volatility Estimation with Dependent Microstructure Noise . Journal of Econometrics 160 : 160 – 175 . Google Scholar Crossref Search ADS Bandi F. M. , and Russell J. R. . 2008 . Microstructure Noise, Realized Variance, and Optimal Sampling . Review of Economic Studies 75 : 339 – 369 . Google Scholar Crossref Search ADS Barndorff-Nielsen O. E. , Hansen P. R. , Lunde A. , and Shephard N. . 2008 . Designing Realized Kernels to Measure the Ex Post Variation of Equity Prices in the Presence of Noise . Econometrica 76 : 1481 – 1536 . Google Scholar Crossref Search ADS Barndorff-Nielsen O. E. , and Shephard N. . 2002 . Econometric Analysis of Realized Volatility and Its Use in Estimating Stochastic Volatility Models . Journal of the Royal Statistical Society Series B: Statistical Methodology 64 : 253 – 280 . Google Scholar Crossref Search ADS Barndorff-Nielsen O. E. , and Shephard N. . 2003 . Realized Power Variation and Stochastic Volatility Models . Bernoulli 9 : 243 – 265 . Google Scholar Crossref Search ADS Delattre S. , and Jacod J. . 1997 . A Central Limit Theorem for Normalized Functions of the Increments of a Diffusion Process, in the Presence of Round-Off Errors . Bernoulli 3 : 1 – 28 . Google Scholar Crossref Search ADS Hayashi T. , Jacod J. , and Yoshida N. . 2011 . Irregular Sampling and Central Limit Theorems for Power Variations: The Continuous Case . Annales de l’Institut Henri Poincaré B, Probability and Statistics 47 : 1197 – 1218 . Google Scholar Crossref Search ADS Jacod J. 1994 . Limit of Random Measures Associated with the Increments of a Brownian Semimartingale . Preprint . Jacod J. , Li Y. , Mykland P. A. , Podolskij M. , and Vetter M. . 2009 . Microstructure Noise in the Continuous Case: the Pre-averaging Approach . Stochastic Processes and their Application 119 : 2249 – 2276 . Google Scholar Crossref Search ADS Jacod J. , Li Y. , and Zheng X. . 2017 . Statistical Properties of Microstructure Noise . Econometrica 85 : 1133 – 1174 . Google Scholar Crossref Search ADS Jacod J. , and Protter P. . 1998 . Asymptotic Error Distributions for the Euler Method for Stochastic Differential Equations . Annals of Probability 26 : 267 – 307 . Google Scholar Crossref Search ADS Jacod J. , and Protter P. . 2012 . Discretization of Processes. Vol. 67 of Stochastic Modelling and Applied Probability . Heidelberg : Springer . Jacod J. , and Reiss M. . 2014 . A Remark on the Rates of Convergence for Integrated Volatility Estimation in the Presence of Jumps . Annals of Statistics 42 : 1131 – 1144 . Google Scholar Crossref Search ADS Jacod J. , and Shiryaev A. N. . 2003 . Limit Theorems for Stochastic Processes. Vol. 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] , 2nd edn . Berlin : Springer-Verlag . Jacod J. , and Todorov V. . 2009 . Testing for Common Arrivals of Jumps for Discretely Observed Multidimensional Processes . Annals of Statistics 37 : 1792 – 1838 . Google Scholar Crossref Search ADS Jacod J. , and Todorov V. . 2014 . Efficient Estimation of Integrated Volatility in Presence of Infinite Variation Jumps . Annals of Statistics 42 : 1029 – 1069 . Google Scholar Crossref Search ADS Jacod J. , and Todorov V. . 2016 . “Efficient Estimation of Integrated Volatility in Presence of Infinite Variation Jumps with Multiple Activity Indices.” In The Fascination of Probability, Statistics and Their Applications , pp. 317 – 341 . Cham : Springer . Li Y. , and Mykland P. A. . 2015 . Rounding and Volatility Estimation . Journal of Financial Econometrics 13 : 478 – 504 . Google Scholar Crossref Search ADS Li Y. , Mykland P. A. , Renault E. , Zhang L. , and Zheng X. . 2014 . Realized Volatility When Sampling Times Are Possibly Endogenous . Econometric Theory 30 : 580 – 605 . Google Scholar Crossref Search ADS Mykland P. A. , and Zhang L. . 2006 . ANOVA for Diffusions and Itô Processes . Annals of Statistics 34 : 1931 – 1963 . Google Scholar Crossref Search ADS Rosenbaum M. 2009 . Integrated Volatility and Round-Off Error . Bernoulli 15 : 687 – 720 . Google Scholar Crossref Search ADS Xiu D. 2010 . Quasi-maximum Likelihood Estimation of Volatility with High Frequency Data . Journal of Econometrics 159 : 235 – 250 . Google Scholar Crossref Search ADS © The Author, 2017. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
Journal of Financial Econometrics – Oxford University Press
Published: Sep 1, 2018
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