83 results on '"Semimartingale"'
Search Results
2. Testing the volatility jumps based on the high frequency data.
- Author
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Liu, Guangying, Liu, Meiyao, and Lin, Jinguan
- Subjects
- *
GAUSSIAN distribution , *CENTRAL limit theorem , *NULL hypothesis , *CONTINUOUS processing , *MARKOVIAN jump linear systems - Abstract
This article tests volatility jumps based on the high frequency data. Under the null hypothesis that the volatility process is a continuous semimartingale, our test statistic converges to a normal distribution, and under the alternative hypothesis where the volatility has jumps, the statistic diverges to infinity. Compared to the test statistic of Bibinger et al. (Bibinger et al. (2017). Annals of Statistics 45, 1542–1578), our proposed statistic diverges to infinity at a faster rate, and has a better power. Simulation studies confirm the theoretical results, and an empirical analysis shows that some real financial data possess volatility jumps. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
3. Estimation of Volatility Functionals: The Case of a Window
- Author
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Jacod, Jean, Rosenbaum, Mathieu, Friz, Peter K., editor, Gatheral, Jim, editor, Gulisashvili, Archil, editor, Jacquier, Antoine, editor, and Teichmann, Josef, editor
- Published
- 2015
- Full Text
- View/download PDF
4. Testing the volatility jumps based on the high frequency data
- Author
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Jinguan Lin, Meiyao Liu, and Guangying Liu
- Subjects
Statistics and Probability ,Semimartingale ,Applied Mathematics ,Econometrics ,Frequency data ,Statistics, Probability and Uncertainty ,Volatility (finance) ,Mathematics ,Central limit theorem - Published
- 2021
5. A universal approach to estimate the conditional variance in semimartingale limit theorems
- Author
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Mathias Vetter
- Subjects
Statistics and Probability ,Probability (math.PR) ,05 social sciences ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,01 natural sciences ,Power (physics) ,010104 statistics & probability ,Semimartingale ,0502 economics and business ,Convergence (routing) ,Consistent estimator ,FOS: Mathematics ,Applied mathematics ,Limit (mathematics) ,0101 mathematics ,Heuristics ,Conditional variance ,Mathematics - Probability ,050205 econometrics ,Central limit theorem ,Mathematics - Abstract
The typical central limit theorems in high-frequency asymptotics for semimartingales are results on stable convergence to a mixed normal limit with an unknown conditional variance. Estimating this conditional variance usually is a hard task, in particular when the underlying process contains jumps. For this reason, several authors have recently discussed methods to automatically estimate the conditional variance, i.e. they build a consistent estimator from the original statistics, but computed at various different time scales. Their methods work in several situations, but are essentially restricted to the case of continuous paths always. The aim of this work is to present a new method to consistently estimate the conditional variance which works regardless of whether the underlying process is continuous or has jumps. We will discuss the case of power variations in detail and give insight to the heuristics behind the approach.
- Published
- 2021
6. Small time central limit theorems for semimartingales with applications.
- Author
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Gerhold, Stefan, Kleinert, Max, Porkert, Piet, and Shkolnikov, Mykhaylo
- Subjects
- *
CENTRAL limit theorem , *SEMIMARTINGALES (Mathematics) , *STOCHASTIC convergence , *GAUSSIAN processes , *STOCHASTIC differential equations , *MARKET volatility - Abstract
We give conditions under which the normalized marginal distribution of a semimartingale converges to a Gaussian limit law as time tends to zero. In particular, our result is applicable to solutions of stochastic differential equations with locally bounded and continuous coefficients. The limit theorems are subsequently extended to functional central limit theorems on the process level. We present two applications of the results in the field of mathematical finance: to the pricing of at-the-money digital options with short maturities and short time implied volatility skews. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
7. On the Estimation of Integrated Volatility With Jumps and Microstructure Noise.
- Author
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Jing, Bing-Yi, Liu, Zhi, and Kong, Xin-Bing
- Subjects
MICROSTRUCTURE ,CENTRAL limit theorem ,ASYMPTOTIC efficiencies ,NOISE measurement ,VARIATIONS (Musical composition) - Abstract
In this article, we propose a nonparametric procedure to estimate the integrated volatility of an Itô semimartingale in the presence of jumps and microstructure noise. The estimator is based on a combination of the preaveraging method and threshold technique, which serves to remove microstructure noise and jumps, respectively. The estimator is shown to work for both finite and infinite activity jumps. Furthermore, asymptotic properties of the proposed estimator, such as consistency and a central limit theorem, are established. Simulations results are given to evaluate the performance of the proposed method in comparison with other alternative methods. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
8. Asymptotic properties for multipower variation of semimartingales and Gaussian integral processes with jumps.
- Author
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Liu, Guangying, Wei, Zhengyuan, and Zhang, Xinsheng
- Subjects
- *
SEMIMARTINGALES (Mathematics) , *GAUSSIAN distribution , *GAUSSIAN processes , *DISTRIBUTION (Probability theory) , *CENTRAL limit theorem , *INTEGRALS - Abstract
Abstract: This paper presents limit theorems of realized multipower variation for semimartingales and Gaussian integral processes with jumps observed in high frequency. In particular, we obtain a central limit theorem of realized multipower variation for semimartingale where some of the powers equal one and the others are less one. Convergence in probability and central limit theorems of realized threshold bipower variation for Gaussian integral processes with jumps are also obtained. These results provide new statistical tools to analyze and test the long memory effect in high frequency situation. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
9. Measuring Downside Risk Using High-Frequency Data: Realized Downside Risk Measure.
- Author
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Bi, Tao, Zhang, Bo, and Wu, Huishan
- Subjects
- *
FINANCIAL risk management , *DATA analysis , *CENTRAL limit theorem , *STOCK exchanges , *ECONOMIC models , *MONTE Carlo method , *SEMIMARTINGALES (Mathematics) - Abstract
In this article, we propose a general downside risk measure based on high-frequency downward moves below minimum acceptable target in asset prices. We derive the central limit theorem of this measure, and Monte Carlo simulation experiments support our theoretical results. We also investigate the distributional properties of this measure in China’s stock market. The theoretical and empirical works on realized downside risk measure shed light on the potential of this measure in measuring and modeling financial risk. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
10. Estimation of Correlation for Continuous Semimartingales.
- Author
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VETTER, MATHIAS
- Subjects
- *
PARAMETER estimation , *STATISTICAL correlation , *CONTINUOUS functions , *SEMIMARTINGALES (Mathematics) , *WIENER processes , *ASYMPTOTIC efficiencies , *STOCHASTIC convergence , *CENTRAL limit theorem - Abstract
. In this study we are concerned with inference on the correlation parameter ρ of two Brownian motions, when only high-frequency observations from two one-dimensional continuous Itô semimartingales, driven by these particular Brownian motions, are available. Estimators for ρ are constructed in two situations: either when both components are observed (at the same time), or when only one component is observed and the other one represents its volatility process and thus has to be estimated from the data as well. In the first case it is shown that our estimator has the same asymptotic behaviour as the standard one for i.i.d. normal observations, whereas a feasible estimator can still be defined in the second framework, but with a slower rate of convergence. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
11. Estimation of the instantaneous volatility.
- Author
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Alvarez, Alexander, Panloup, Fabien, Pontier, Monique, and Savy, Nicolas
- Abstract
This paper is concerned with the estimation of the volatility process in a stochastic volatility model of the following form: dX = a dt + σ dW, where X denotes the log-price and σ is a càdlàg semi-martingale. In the spirit of a series of recent works on the estimation of the cumulated volatility, we here focus on the instantaneous volatility for which we study estimators built as finite differences of the power variations of the log-price. We provide central limit theorems with an optimal rate depending on the local behavior of σ. In particular, these theorems yield some confidence intervals for σ. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
12. Understanding limit theorems for semimartingales: a short survey.
- Author
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Podolskij, Mark and Vetter, Mathias
- Subjects
- *
LIMIT theorems , *FUNCTIONALS , *SEMIMARTINGALES (Mathematics) , *STOCHASTIC processes , *SURVEYS - Abstract
This paper presents a short survey on limit theorems for certain functionals of semimartingales that are observed at high frequency. Our aim is to explain the main ideas of the theory to a broader audience. We introduce the concept of stable convergence, which is crucial for our purpose. We show some laws of large numbers (for the continuous and the discontinuous case) that are the most interesting from a practical point of view, and demonstrate the associated stable central limit theorems. Moreover, we state a simple sketch of the proofs and give some examples. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
13. Limit theorems for bipower variation of semimartingales
- Author
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Vetter, Mathias
- Subjects
- *
LIMIT theorems , *SEMIMARTINGALES (Mathematics) , *MATHEMATICAL variables , *STOCHASTIC convergence , *CENTRAL limit theorem , *FUNCTIONAL analysis - Abstract
Abstract: This paper presents limit theorems for certain functionals of semimartingales observed at high frequency. In particular, we extend results from Jacod (2008) to the case of bipower variation, showing under standard assumptions that one obtains a limiting variable, which is in general different from the case of a continuous semimartingale. In a second step a truncated version of bipower variation is constructed, which has a similar asymptotic behaviour as standard bipower variation for a continuous semimartingale and thus provides a feasible central limit theorem for the estimation of the integrated volatility even when the semimartingale exhibits jumps. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
14. Asymptotic properties of realized power variations and related functionals of semimartingales
- Author
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Jacod, Jean
- Subjects
- *
STOCHASTIC analysis , *ASYMPTOTIC distribution , *PROBABILITY theory , *MATHEMATICAL analysis - Abstract
Abstract: This paper is concerned with the asymptotic behavior of sums of the form , where is a 1-dimensional semimartingale and a suitable test function, typically , as . We prove a variety of “laws of large numbers”, that is convergence in probability of , sometimes after normalization. We also exhibit in many cases the rate of convergence, as well as associated central limit theorems. [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
15. Efficient estimation of integrated volatility functionals via multiscale jackknife
- Author
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Jia Li, Dacheng Xiu, and Yunxiao Liu
- Subjects
Statistics and Probability ,Statistics::Theory ,Stochastic volatility ,High-frequency data ,spot volatility ,Nonparametric statistics ,Estimator ,semimartingale ,01 natural sciences ,jackknife ,010104 statistics & probability ,Delta method ,Semimartingale ,60F05 ,Econometrics ,Applied mathematics ,Statistics::Methodology ,60G44 ,0101 mathematics ,Statistics, Probability and Uncertainty ,Volatility (finance) ,62F12 ,Jackknife resampling ,Mathematics ,Central limit theorem - Abstract
We propose semiparametrically efficient estimators for general integrated volatility functionals of multivariate semimartingale processes. A plug-in method that uses nonparametric estimates of spot volatilities is known to induce high-order biases that need to be corrected to obey a central limit theorem. Such bias terms arise from boundary effects, the diffusive and jump movements of stochastic volatility and the sampling error from the nonparametric spot volatility estimation. We propose a novel jackknife method for bias correction. The jackknife estimator is simply formed as a linear combination of a few uncorrected estimators associated with different local window sizes used in the estimation of spot volatility. We show theoretically that our estimator is asymptotically mixed Gaussian, semiparametrically efficient, and more robust to the choice of local windows. To facilitate the practical use, we introduce a simulation-based estimator of the asymptotic variance, so that our inference is derivative-free, and hence is convenient to implement.
- Published
- 2019
16. Mixed-normal limit theorems for multiple Skorohod integrals in high-dimensions, with application to realized covariance
- Author
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Yuta Koike
- Subjects
Statistics and Probability ,Chernozhukov-Chetverikov-Kato theory ,Malliavin calculus ,central limit theorem ,Polytope ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,01 natural sciences ,010104 statistics & probability ,60H07 ,Dimension (vector space) ,60F05 ,0502 economics and business ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,050205 econometrics ,Central limit theorem ,Mathematics ,Covariance matrix ,multiple testing ,05 social sciences ,Probability (math.PR) ,Regular polygon ,Covariance ,Bootstrap ,high-frequency data ,Semimartingale ,62H15 ,60F05, 60H07, 62H15 ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
This paper develops mixed-normal approximations for probabilities that vectors of multiple Skorohod integrals belong to random convex polytopes when the dimensions of the vectors possibly diverge to infinity. We apply the developed theory to establish the asymptotic mixed normality of the realized covariance matrix of a high-dimensional continuous semimartingale observed at a high-frequency, where the dimension can be much larger than the sample size. We also present an application of this result to testing the residual sparsity of a high-dimensional continuous-time factor model., 67 pages, 1 figure. Simulation results have been changed slightly due to a mistake in the original program. To appear in the Electronic Journal of Statistics
- Published
- 2019
17. Functional stable limit theorems for quasi-efficient spectral covolatility estimators
- Author
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Randolf Altmeyer and Markus Bibinger
- Subjects
Statistics and Probability ,Multivariate statistics ,Mathematical optimization ,Applied Mathematics ,Estimator ,Spectral density estimation ,Mathematics - Statistics Theory ,62G05, 62G20, 62M10 ,Statistics Theory (math.ST) ,Bivariate analysis ,Semimartingale ,Modeling and Simulation ,FOS: Mathematics ,Applied mathematics ,Volatility (finance) ,Smoothing ,Mathematics ,Central limit theorem - Abstract
We consider noisy non-synchronous discrete observations of a continuous semimartingale with random volatility. Functional stable central limit theorems are established under high-frequency asymptotics in three setups: one-dimensional for the spectral estimator of integrated volatility, from two-dimensional asynchronous observations for a bivariate spectral covolatility estimator and multivariate for a local method of moments. The results demonstrate that local adaptivity and smoothing noise dilution in the Fourier domain facilitate substantial efficiency gains compared to previous approaches. In particular, the derived asymptotic variances coincide with the benchmarks of semiparametric Cram\'er-Rao lower bounds and the considered estimators are thus asymptotically efficient in idealized sub-experiments. Feasible central limit theorems allowing for confidence are provided., Comment: to appear, Stochastic Processes and their Applications, 2015
- Published
- 2015
18. Limit theorems for integrated local empirical characteristic exponents from noisy high-frequency data with application to volatility and jump activity estimation
- Author
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Viktor Todorov and Jean Jacod
- Subjects
Itô semimartingale ,Statistics and Probability ,Characteristic function (probability theory) ,central limit theorem ,01 natural sciences ,Quadratic variation ,stable process ,integrated volatility ,Stable process ,010104 statistics & probability ,Blumenthal–Getoor index ,jump activity ,60F05 ,60G07 ,0502 economics and business ,empirical characteristic function ,Statistical physics ,0101 mathematics ,050205 econometrics ,Central limit theorem ,Mathematics ,jumps ,05 social sciences ,irregular sampling ,quadratic variation ,Semimartingale ,microstructure noise ,60F17 ,Jump ,Statistics, Probability and Uncertainty ,Volatility (finance) ,60G51 ,Characteristic exponent - Abstract
We derive limit theorems for functionals of local empirical characteristic functions constructed from high-frequency observations of Itô semimartingales contaminated with noise. In a first step, we average locally the data to mitigate the effect of the noise, and then in a second step, we form local empirical characteristic functions from the pre-averaged data. The final statistics are formed by summing the local empirical characteristic exponents over the observation interval. The limit behavior of the statistics is governed by the observation noise, the diffusion coefficient of the Itô semimartingale and the behavior of its jump compensator around zero. Different choices for the block sizes for pre-averaging and formation of the local empirical characteristic function as well as for the argument of the characteristic function make the asymptotic role of the diffusion, the jumps and the noise differ. The derived limit results can be used in a wide range of applications and in particular for doing the following in a noisy setting: (1) efficient estimation of the time-integrated diffusion coefficient in presence of jumps of arbitrary activity, and (2) efficient estimation of the jump activity (Blumenthal–Getoor) index.
- Published
- 2018
19. The null hypothesis of common jumps in case of irregular and asynchronous observations
- Author
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Ole Martin and Mathias Vetter
- Subjects
Statistics and Probability ,05 social sciences ,Univariate ,Sample (statistics) ,Mathematics - Statistics Theory ,Bivariate analysis ,Statistics Theory (math.ST) ,01 natural sciences ,Power (physics) ,010104 statistics & probability ,Semimartingale ,0502 economics and business ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Null hypothesis ,Statistic ,050205 econometrics ,Mathematics ,Central limit theorem - Abstract
This paper proposes novel tests for the absence of jumps in a univariate semimartingale and for the absence of common jumps in a bivariate semimartingale. Our methods rely on ratio statistics of power variations based on irregular observations, sampled at different frequencies. We develop central limit theorems for the statistics under the respective null hypotheses and apply bootstrap procedures to assess the limiting distributions. Further we define corrected statistics to improve the finite sample performance. Simulations show that the test based on our corrected statistic yields good results and even outperforms existing tests in the case of regular observations.
- Published
- 2017
20. Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian–fractional Brownian model
- Author
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Tommi Sottinen, Lauri Viitasaari, and Ehsan Azmoodeh
- Subjects
Statistics and Probability ,Central limit theorem ,Malliavin calculus ,fractional Brownian motion ,multiple Wiener integrals ,Asymptotic distribution ,Quadratic variation ,randomized periodogram ,Normal distribution ,FOS: Mathematics ,Applied mathematics ,Mathematics ,Hurst exponent ,lcsh:T57-57.97 ,lcsh:Mathematics ,Probability (math.PR) ,Estimator ,lcsh:QA1-939 ,Fractional part ,quadratic variation ,Semimartingale ,Modeling and Simulation ,lcsh:Applied mathematics. Quantitative methods ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian--fractional Brownian model. In the semimartingale case, that is, where the Hurst parameter $H$ of the fractional part satisfies $H\in(3/4,1)$, the central limit theorem holds. In the nonsemimartingale case, that is, where $H\in(1/2,3/4]$, the convergence toward the normal distribution with a nonzero mean still holds if $H=3/4$, whereas for the other values, that is, $H\in(1/2,3/4)$, the central convergence does not take place. We also provide Berry--Esseen estimates for the estimator., Comment: Published at http://dx.doi.org/10.15559/15-VMSTA24 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)
- Published
- 2015
21. On Integrated Volatility of Itô Semimartingales when Sampling Times are Endogenous
- Author
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Jin-Yuan Chen, Bing-Yi Jing, Zhi Liu, and Cui-Xia Li
- Subjects
Statistics and Probability ,Semimartingale ,Finite variation ,Econometrics ,Estimator ,Endogeneity ,Volatility (finance) ,Mathematics ,Central limit theorem - Abstract
In this paper, we estimate the integrated volatility of Ito semimartingale when sampling times are endogenous. The estimator is proved to be consistent, and is robust to jumps, regardless of whether they are finite and infinite activity jumps. We also establish a central limit theorem for the estimator in a general endogenous time setting when the jumps have finite variation. Simulation is also included to illustrate the performance of the proposed procedure.
- Published
- 2014
22. On U- and V-statistics for discontinuous Itô semimartingales
- Author
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Christian Schmidt, Mark Podolskij, and Mathias Vetter
- Subjects
Statistics and Probability ,Asymptotic analysis ,Semimartingales ,Gaussian ,Structure (category theory) ,01 natural sciences ,Limit theorems ,010104 statistics & probability ,symbols.namesake ,Mathematics::Probability ,High frequency data ,Law of large numbers ,60H05 ,60F05 ,60G48 ,FOS: Mathematics ,Limit (mathematics) ,Stable convergence ,0101 mathematics ,Mathematics ,Central limit theorem ,010102 general mathematics ,Mathematical analysis ,Probability (math.PR) ,Semimartingale ,Jump ,symbols ,U-statistics ,Statistics, Probability and Uncertainty ,62F12 ,Mathematics - Probability - Abstract
In this paper we examine the asymptotic theory for U-statistics and V-statistics of discontinuous Ito semimartingales that are observed at high frequency. For different types of kernel functions we show laws of large numbers and associated stable central limit theorems. In most of the cases the limiting process will be conditionally centered Gaussian. The structure of the kernel function determines whether the jump and/or the continuous part of the semimartingale contribute to the limit.
- Published
- 2017
23. A central limit theorem for the realised covariation of a bivariate Brownian semistationary process
- Author
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Andrea Granelli, Almut E. D. Veraart, and Commission of the European Communities
- Subjects
multivariate setting ,Statistics and Probability ,Asymptotic analysis ,FUNCTIONALS ,Statistics & Probability ,GAUSSIAN-PROCESSES ,central limit theorem ,Mathematics - Statistics Theory ,moving average process ,Statistics Theory (math.ST) ,Bivariate analysis ,BIPOWER VARIATION ,Malliavin calculus ,01 natural sciences ,POWER VARIATION ,010104 statistics & probability ,symbols.namesake ,Mathematics::Probability ,Law of large numbers ,1403 Econometrics ,FOS: Mathematics ,MULTIPOWER VARIATION ,Applied mathematics ,fourth moment theorem ,0101 mathematics ,Gaussian process ,Brownian motion ,Central limit theorem ,Mathematics ,Science & Technology ,0104 Statistics ,Probability (math.PR) ,010102 general mathematics ,16. Peace & justice ,60F05, 60F15, 60G15 ,bivariate Brownian semistationary process ,Semimartingale ,Physical Sciences ,symbols ,VOLATILITY ,stable convergence ,Mathematics - Probability ,high frequency data - Abstract
This article presents a weak law of large numbers and a central limit theorem for the scaled realised covariation of a bivariate Brownian semistationary process. The novelty of our results lies in the fact that we derive the suitable asymptotic theory both in a multivariate setting and outside the classical semimartingale framework. The proofs rely heavily on recent developments in Malliavin calculus.
- Published
- 2017
24. Testing for time-varying jump activity for pure jump semimartingales
- Author
-
Viktor Todorov
- Subjects
Itô semimartingale ,Statistics and Probability ,Central limit theorem ,Asymptotic distribution ,jump activity index ,01 natural sciences ,Stable process ,010104 statistics & probability ,62M05 ,nonparametric test ,0502 economics and business ,Statistics ,Test statistic ,stochastic volatility ,0101 mathematics ,050205 econometrics ,Mathematics ,jumps ,05 social sciences ,Mathematical analysis ,Estimator ,high-frequency data ,Semimartingale ,power variation ,Jump ,60H10 ,60J75 ,Statistics, Probability and Uncertainty ,62F12 ,Constant (mathematics) - Abstract
In this paper, we propose a test for deciding whether the jump activity index of a discretely observed Itô semimartingale of pure-jump type (i.e., one without a diffusion) varies over a fixed interval of time. The asymptotic setting is based on observations within a fixed time interval with mesh of the observation grid shrinking to zero. The test is derived for semimartingales whose “spot” jump compensator around zero is like that of a stable process, but importantly the stability index can vary over the time interval. The test is based on forming a sequence of local estimators of the jump activity over blocks of shrinking time span and contrasting their variability around a global activity estimator based on the whole data set. The local and global jump activity estimates are constructed from the real part of the empirical characteristic function of the increments of the process scaled by local power variations. We derive the asymptotic distribution of the test statistic under the null hypothesis of constant jump activity and show that the test has asymptotic power of one against fixed alternatives of processes with time-varying jump activity.
- Published
- 2017
25. A limit theorem for moments in space of the increments of Brownian local time
- Author
-
Simon Campese
- Subjects
Statistics and Probability ,Pure mathematics ,central limit theorem ,Space (mathematics) ,01 natural sciences ,010104 statistics & probability ,60H05 ,60F05 ,FOS: Mathematics ,60G44 ,Limit (mathematics) ,0101 mathematics ,Brownian motion ,Brownian local time ,Variable (mathematics) ,Mathematics ,Central limit theorem ,Conjecture ,Probability (math.PR) ,asymptotic Ray–Knight theorem ,010102 general mathematics ,Semimartingale ,60F05, 60G44, 60H05 ,Local time ,Statistics, Probability and Uncertainty ,Kailath–Segall identity ,Mathematics - Probability - Abstract
We proof a limit theorem for moments in space of the increments of Brownian local time. As special cases for the second and third moments, previous results by Chen et al. (Ann. Prob. 38, 2010, no. 1) and Rosen (Stoch. Dyn. 11, 2011, no. 1), which were later reproven by Hu and Nualart (Electron. Commun. Probab. 14, 2009; Electron. Commun. Probab. 15, 2010) and Rosen (S\'eminaire de Probabilit\'es XLIII, Springer, 2011) are included. Furthermore, a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a fundamentally different approach by exclusively working in the space variable of the Brownian local time, which allows to give a unified argument for arbitrary orders. The main ingredients are Perkins' semimartingale decomposition, the Kailath-Segall identity and an asymptotic Ray-Knight Theorem by Pitman and Yor., Comment: 28 pages; to appear in Annals of Probability
- Published
- 2017
26. A note on central limit theorems for quadratic variation in case of endogenous observation times
- Author
-
Tobias Zwingmann and Mathias Vetter
- Subjects
Statistics and Probability ,realized variance ,Realized variance ,High-frequency observations ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,01 natural sciences ,62M09 ,Quadratic variation ,Regular grid ,010104 statistics & probability ,60F05 ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Special case ,Mathematics ,Central limit theorem ,irregular data ,60F05, 60G51, 62M09 ,010102 general mathematics ,quadratic variation ,Semimartingale ,Statistics, Probability and Uncertainty ,stable convergence ,60G51 - Abstract
This paper is concerned with a central limit theorem for quadratic variation when observations come as exit times from a regular grid. We discuss the special case of a semimartingale with deterministic characteristics and finite activity jumps in detail and illustrate technical issues in more general situations., Comment: 16 pages, 1 figure
- Published
- 2017
27. Efficient Estimation of Integrated Volatility Functionals via Multiscale Jacknife
- Author
-
Dacheng Xiu, Jia Li, and Yunxiao Liu
- Subjects
Delta method ,Semimartingale ,Stochastic volatility ,Nonparametric statistics ,Estimator ,Applied mathematics ,Volatility (finance) ,Jackknife resampling ,Central limit theorem ,Mathematics - Abstract
We propose semi-parametrically efficient estimators for general integrated volatility functionals of multivariate semimartingale processes. It is known that a plug-in method that uses nonparametric estimates of spot volatilities induces high-order biases which need to be corrected to obey a central limit theorem. Such bias terms arise from boundary effects, the diffusive and jump movements of stochastic volatility, and the sampling error from the nonparametric spot volatility estimation. We propose a novel jackknife method for bias-correction. The jackknife estimator is simply formed as a linear combination of a few uncorrected estimators associated with different local window sizes used in the estimation of spot volatility. We show theoretically that our estimator is asymptotically mixed Gaussian, semi-parametrically efficient, and more robust to the choice of local windows. To facilitate the practical use, we introduce a simulation-based estimator of the asymptotic variance, so that our inference is derivative-free and, hence, is very convenient to implement.
- Published
- 2017
28. On the Estimation of Integrated Volatility With Jumps and Microstructure Noise
- Author
-
Xin-Bing Kong, Bing-Yi Jing, and Zhi Liu
- Subjects
Statistics and Probability ,Alternative methods ,Economics and Econometrics ,Mathematical optimization ,Nonparametric statistics ,Estimator ,Microstructure ,Quadratic variation ,Semimartingale ,Applied mathematics ,Statistics, Probability and Uncertainty ,Volatility (finance) ,Social Sciences (miscellaneous) ,Central limit theorem ,Mathematics - Abstract
In this article, we propose a nonparametric procedure to estimate the integrated volatility of an Ito semimartingale in the presence of jumps and microstructure noise. The estimator is based on a combination of the preaveraging method and threshold technique, which serves to remove microstructure noise and jumps, respectively. The estimator is shown to work for both finite and infinite activity jumps. Furthermore, asymptotic properties of the proposed estimator, such as consistency and a central limit theorem, are established. Simulations results are given to evaluate the performance of the proposed method in comparison with other alternative methods.
- Published
- 2014
29. Asymptotic properties for multipower variation of semimartingales and Gaussian integral processes with jumps
- Author
-
Xinsheng Zhang, Zhengyuan Wei, and Guangying Liu
- Subjects
Statistics and Probability ,Applied Mathematics ,Mathematical analysis ,Variation (game tree) ,symbols.namesake ,Semimartingale ,Convergence of random variables ,Long memory ,Gaussian integral ,symbols ,Limit (mathematics) ,Statistics, Probability and Uncertainty ,Central limit theorem ,Mathematics - Abstract
This paper presents limit theorems of realized multipower variation for semimartingales and Gaussian integral processes with jumps observed in high frequency. In particular, we obtain a central limit theorem of realized multipower variation for semimartingale where some of the powers equal one and the others are less one. Convergence in probability and central limit theorems of realized threshold bipower variation for Gaussian integral processes with jumps are also obtained. These results provide new statistical tools to analyze and test the long memory effect in high frequency situation.
- Published
- 2013
30. Measuring the relevance of the microstructure noise in financial data
- Author
-
Cecilia Mancini
- Subjects
Statistics and Probability ,Finance ,Semimartingales with jumps, integrated variance, threshold estimation, test to select optimal sampling frequency ,Mean squared error ,Realized variance ,business.industry ,Applied Mathematics ,Variance (accounting) ,Measure (mathematics) ,Noise ,Semimartingale ,Modeling and Simulation ,Path (graph theory) ,business ,Mathematics ,Central limit theorem - Abstract
We show that the Truncated Realized Variance (TRV) of a SemiMartingale (SM) converges to zero when observations are contaminated by noise. Under the additive i.i.d. noise assumption, a central limit theorem is also proved. In consequence it is possible to construct a feasible test allowing us to measure, for a given path of a given data generating process at a given observation frequency, the relevance of the noise in the data when we want to estimate the efficient process integrated variance I V . We thus can optimally select the observation frequency at which we can “safely” use TRV. The performance of our test is verified on simulated data. We are especially interested in the application of the test to financial data, and a comparison conducted with Bandi and Russel (2008) and Ait-Sahalia, Mykland and Zhang (2005) mean square error criteria shows that, in order to estimate IV, in many cases we can rely on TRV for lower observation frequencies than previously indicated when using Realized Variance (RV). The advantages of our method are at least two: on the one hand the underlying model for the efficient data generating process is less restrictive in that jumps are allowed (in the form of an Ito SM). On the other hand our criterion is pathwise, rather than based on an average estimation error, allowing for a more precise estimation of IV because the choice of the optimal frequency is based on the observed path. Further analysis on both simulated and empirical financial data is conducted in Lorenzini (2012) [15] and is also still in progress.
- Published
- 2013
31. Power variation from second order differences for pure jump semimartingales
- Author
-
Viktor Todorov
- Subjects
Statistics and Probability ,Applied Mathematics ,Lévy process ,Semimartingale ,Control theory ,Modeling and Simulation ,Range (statistics) ,Jump ,Statistical physics ,Limit (mathematics) ,Jump process ,Statistic ,Central limit theorem ,Mathematics - Abstract
We introduce power variation constructed from powers of the second-order differences of a discretely observed pure-jump semimartingale processes. We derive the asymptotic behavior of the statistic in the setting of high-frequency observations of the underlying process with a fixed time span. Unlike the standard power variation (formed from the first-order differences of the process), the limit of our proposed statistic is determined solely by the jump component of the process regardless of the activity of the latter. We further show that an associated Central Limit Theorem holds for a wider range of activity of the jump process than for the standard power variation. We apply these results for estimation of the jump activity as well as the integrated stochastic scale.
- Published
- 2013
32. Central Limit Theorems for approximate quadratic variations of pure jump Itô semimartingales
- Author
-
Viktor Todorov, Jean Jacod, and Assane Diop
- Subjects
Statistics and Probability ,Applied Mathematics ,Mathematical analysis ,Limiting ,Quadratic variation ,Semimartingale ,Quadratic equation ,Mathematics::Probability ,Modelling and Simulation ,Modeling and Simulation ,Jump ,Martingale (probability theory) ,Central limit theorem ,Mathematics - Abstract
We derive Central Limit Theorems for the convergence of approximate quadratic variations, computed on the basis of regularly spaced observation times of the underlying process, toward the true quadratic variation. This problem was solved in the case of an Ito semimartingale having a non-vanishing continuous martingale part. Here we focus on the case where the continuous martingale part vanishes and find faster rates of convergence, as well as very different limiting processes.
- Published
- 2013
33. A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales
- Author
-
Neil Shephard, Svend Erik Graversen, Ole E. Barndorff–Nielsen, Mark Podolskij, Jean Jacod, Dept. of Mathematical Science, Aarhus University [Aarhus], Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Dept. of Probability and Statistics, Ruhr-Universität Bochum [Bochum], Nuffield College, University of Oxford, Benassù, Serena, University of Oxford [Oxford], Kabanov, Yuri, Liptser, Robert, and Stoyanov, Jordan
- Subjects
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,05 social sciences ,central limit theorem ,Poisson random measure ,01 natural sciences ,Quadratic variation ,Combinatorics ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,Semimartingale ,quadratic variation ,Bounded function ,0502 economics and business ,0101 mathematics ,Martingale (probability theory) ,Predictable process ,60F17 60G44 ,bipower variation ,Brownian motion ,050205 econometrics ,Mathematics ,Central limit theorem - Abstract
Consider a semimartingale of the form $Y_t=Y_0+\int_0^ta_sds+\int_0^t\si_{s-}~dW_s$, where $a$ is a locally bounded predictable process and $\si$ (the ``volatility'') is an adapted right--continuous process with left limits and $W$ is a Brownian motion. We define the realised bipower variation process $V(Y;r,s)^n_t=n^{{r+s\over2}-1}\sum_{i=1}^{[nt]} |Y_{i\over n}-Y_{i-1\over n}|^r|Y_{i+1\over n}-Y_{i\over n}|^s$, where $r$ and $s$ are nonnegative reals with $r+s>0$. We prove that $V(Y;r,s)^n_t$ converges locally uniformly in time, in probability, to a limiting process $V(Y;r,s)_t$ (the ''bipower variation process''). If further $\si$ is a possibly discontinuous semimartingale driven by a Brownian motion which may be correlated with $W$ and by a Poisson random measure, we prove a central limit theorem, in the sense that $\rn~(V(Y;r,s)^n-V(Y;r,s))$ converges in law to a process which is the stochastic integral with respect to some other Brownian motion $W'$, which is independent of the driving terms of $Y$ and $\si$. We also provide a multivariate version of these results.
- Published
- 2016
34. Testing for simultaneous jumps in case of asynchronous observations
- Author
-
Ole Martin and Mathias Vetter
- Subjects
Statistics and Probability ,Itô semimartingale ,Statistics::Theory ,Correlation coefficient ,Asymptotic distribution ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Bivariate analysis ,01 natural sciences ,Synchronization ,010104 statistics & probability ,62G10, 62M05 (primary), 60J60, 60J75 (secondary) ,0502 economics and business ,FOS: Mathematics ,common jumps ,Applied mathematics ,0101 mathematics ,050205 econometrics ,Central limit theorem ,Mathematics ,05 social sciences ,Power (physics) ,asynchronous observations ,Semimartingale ,Asynchronous communication ,high-frequency statistics ,stable convergence - Abstract
This paper proposes a novel test for simultaneous jumps in a bivariate It\^o semimartingale when observation times are asynchronous and irregular. Inference is built on a realized correlation coefficient for the jumps of the two processes which is estimated using bivariate power variations of Hayashi-Yoshida type without an additional synchronization step. An associated central limit theorem is shown whose asymptotic distribution is assessed using a bootstrap procedure. Simulations show that the test works remarkably well in comparison with the much simpler case of regular observations., Comment: 35 pages, 4 figures, 1 table
- Published
- 2016
35. Estimation of Correlation for Continuous Semimartingales
- Author
-
Mathias Vetter
- Subjects
Statistics and Probability ,Semimartingale ,Rate of convergence ,Stochastic volatility ,Calculus ,Estimator ,Inference ,Applied mathematics ,Statistics, Probability and Uncertainty ,Volatility (finance) ,Brownian motion ,Central limit theorem ,Mathematics - Abstract
In this paper we are concerned with inference on the correlation parameter of two Brownian motions, when only high-frequency observations from two one-dimensional continuous It^ o semimartingales, driven by these particular Brownian motions, are available. Estimators for are constructed in two situations: Either when both components are observed (at the same time), or when only one component is observed and the other one represents its volatility process and thus has to be estimated from the data as well. In the rst case it is shown that our estimator has the same asymptotic behaviour as the standard one for i.i.d. observations, whereas a feasible estimator can still be dened in the second framework, but with a slower rate of convergence.
- Published
- 2012
36. IDENTIFYING THE BROWNIAN COVARIATION FROM THE CO-JUMPS GIVEN DISCRETE OBSERVATIONS
- Author
-
Fabio Gobbi and Cecilia Mancini
- Subjects
Economics and Econometrics ,Integrated covariation, finite activity jumps, threshold method ,Stochastic process ,Monte Carlo method ,co-jumps, integrated covariation, integrated variance, finite activity jumps, infinite activity jumps, threshold estimator ,Estimator ,threshold method ,Covariance ,Delta method ,Integrated covariation ,Semimartingale ,Econometrics ,finite activity jumps ,Statistical physics ,Social Sciences (miscellaneous) ,Brownian motion ,Central limit theorem ,Mathematics - Abstract
When the covariance between the risk factors of asset prices is due to both Brownian and jump components, the realized covariation (RC) approaches the sum of the integrated covariation (IC) with the sum of the co-jumps, as the observation frequency increases to infinity, in a finite and fixed time horizon. In this paper the two components are consistently separately estimated within a semimartingale framework with possibly infinite activity jumps. The threshold (or truncated) estimator $I\hat C_n $ is used, which substantially excludes from RC all terms containing jumps. Unlike in Jacod (2007, Universite de Paris-6) and Jacod (2008, Stochastic Processes and Their Applications 118, 517–559), no assumptions on the volatilities’ dynamics are required. In the presence of only finite activity jumps: 1) central limit theorems (CLTs) for $I\hat C_n $ and for further measures of dependence between the two Brownian parts are obtained; the estimation error asymptotic variance is shown to be smaller than for the alternative estimators of IC in the literature; 2) by also selecting the observations as in Hayashi and Yoshida (2005, Bernoulli 11, 359–379), robustness to nonsynchronous data is obtained. The proposed estimators are shown to have good finite sample performances in Monte Carlo simulations even with an observation frequency low enough to make microstructure noises’ impact on data negligible.
- Published
- 2012
37. Understanding limit theorems for semimartingales: a short survey
- Author
-
Mathias Vetter and Mark Podolskij
- Subjects
Statistics and Probability ,Semimartingale ,Law of large numbers ,Simple (abstract algebra) ,Convergence (routing) ,Calculus ,Limit (mathematics) ,Statistics, Probability and Uncertainty ,Mathematical proof ,Sketch ,Mathematics ,Central limit theorem - Abstract
This paper presents a short survey on limit theorems for certain functionals of semimartingales, which are observed at high frequency. Our aim is to explain the main ideas of the theory to a broader audience. We introduce the concept of stable convergence, which is crucial for our purpose. We show some laws of large numbers (for the continuous and the discontinuous case) that are the most interesting from a practical point of view, and demonstrate the associated stable central limit theorems. Moreover, we state a simple sketch of the proofs and give some examples.
- Published
- 2010
38. Limit theorems for bipower variation of semimartingales
- Author
-
Mathias Vetter
- Subjects
Statistics and Probability ,Stochastic process ,Applied Mathematics ,Central limit theorem ,High-frequency observations ,Mathematical analysis ,Bipower variation ,Semimartingale ,Variation (linguistics) ,Mathematics::Probability ,Modelling and Simulation ,Modeling and Simulation ,Convergence (routing) ,Stable convergence ,Limit (mathematics) ,Volatility (finance) ,Mathematics ,Variable (mathematics) - Abstract
This paper presents limit theorems for certain functionals of semimartingales observed at high frequency. In particular, we extend results from Jacod (2008) [5] to the case of bipower variation, showing under standard assumptions that one obtains a limiting variable, which is in general different from the case of a continuous semimartingale. In a second step a truncated version of bipower variation is constructed, which has a similar asymptotic behaviour as standard bipower variation for a continuous semimartingale and thus provides a feasible central limit theorem for the estimation of the integrated volatility even when the semimartingale exhibits jumps.
- Published
- 2010
39. New tests for jumps in semimartingale models
- Author
-
Mark Podolskij and Daniel Ziggel
- Subjects
Statistics and Probability ,Statistics::Theory ,Mathematical optimization ,media_common.quotation_subject ,Infinity ,Power (physics) ,Normal distribution ,Noise ,Semimartingale ,Mathematics::Probability ,Applied mathematics ,Statistical hypothesis testing ,Mathematics ,Central limit theorem ,media_common - Abstract
In this paper we propose a test to determine whether jumps are present in a discretely sampled process or not. We use the concept of truncated power variation to construct our test statistics for (i) semimartingale models and (ii) semimartingale models with noise. The test statistics diverge to infinity if jumps are present and have a normal distribution otherwise. Our method is valid (under very weak assumptions) for all semimartingales with absolute continuous characteristics and rather general model for the noise process. We finally implement the test and present the simulation results. Our simulations suggest that for semimartingale models the new test is much more powerful than tests proposed by Barndorff-Nielsen and Shephard (J Fin Econ 4:1–30, 2006) and Ait-Sahalia and Jacod (Ann Stat 371:184–222, 2009).
- Published
- 2009
40. A note on the central limit theorem for bipower variation of general functions
- Author
-
Mark Podolskij and Silja Kinnebrock
- Subjects
Statistics and Probability ,Statistics::Theory ,Picard–Lindelöf theorem ,High-frequency data ,Stochastic process ,Bipower Variation ,Central Limit Theorem ,Diffusion Models ,High-Frequency Data ,Semimartingale Theory ,Applied Mathematics ,Mathematical finance ,Central limit theorem ,Mathematical analysis ,Semimartingale theory ,Bipower variation ,Arzelà–Ascoli theorem ,Semimartingale ,Mathematics::Probability ,Modelling and Simulation ,Modeling and Simulation ,Even and odd functions ,Applied mathematics ,Diffusion models ,Empirical process ,Mathematics - Abstract
In this paper we present a central limit theorem for general functions of the increments of Brownian semimartingales. This provides a natural extension of the results derived in [O.E. Barndorff-Nielsen, S.E. Graversen, J. Jacod, M. Podolskij, N. Shephard, A central limit theorem for realised power and bipower variations of continuous semimartingales, in: From Stochastic Analysis to Mathematical Finance, Festschrift for Albert Shiryaev, Springer, 2006], where the central limit theorem was shown for even functions. We prove an infeasible central limit theorem for general functions and state some assumptions under which a feasible version of our results can be obtained. Finally, we present some examples from the literature to which our theory can be applied.
- Published
- 2008
41. Long time asymptotics for constrained diffusions in polyhedral domains
- Author
-
Chihoon Lee and Amarjit Budhiraja
- Subjects
Statistics and Probability ,Stochastic process ,Applied Mathematics ,Mathematical analysis ,Ergodicity ,Dynamical system ,Orthant ,Semimartingale ,Modelling and Simulation ,Modeling and Simulation ,Ergodic theory ,Applied mathematics ,Invariant measure ,Mathematics ,Central limit theorem - Abstract
We study long time asymptotic properties of constrained diffusions that arise in the heavy traffic analysis of multiclass queueing networks. We first consider the classical diffusion model with constant coefficients, namely a semimartingale reflecting Brownian motion (SRBM) in a d -dimensional positive orthant. Under a natural stability condition on a related deterministic dynamical system [P. Dupuis, R.J. Williams, Lyapunov functions for semimartingale reflecting brownian motions, Annals of Probability 22 (2) (1994) 680–702] showed that an SRBM is ergodic. We strengthen this result by establishing geometric ergodicity for the process. As consequences of geometric ergodicity we obtain finiteness of the moment generating function of the invariant measure in a neighborhood of zero, uniform time estimates for polynomial moments of all orders, and functional central limit results. Similar long time properties are obtained for a broad family of constrained diffusion models with state dependent coefficients under a natural condition on the drift vector field. Such models arise from heavy traffic analysis of queueing networks with state dependent arrival and service rates.
- Published
- 2007
42. Jump activity estimation for pure-jump semimartingales via self-normalized statistics
- Author
-
Viktor Todorov
- Subjects
Itô semimartingale ,Statistics and Probability ,Characteristic function (probability theory) ,Central limit theorem ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,jump activity index ,01 natural sciences ,010104 statistics & probability ,62M05 ,0502 economics and business ,FOS: Mathematics ,Applied mathematics ,Limit (mathematics) ,stochastic volatility ,0101 mathematics ,050205 econometrics ,Mathematics ,Stochastic volatility ,jumps ,05 social sciences ,Estimator ,high-frequency data ,Delta method ,Semimartingale ,power variation ,Jump ,60H10 ,60J75 ,Statistics, Probability and Uncertainty ,62F12 - Abstract
We derive a nonparametric estimator of the jump-activity index $\beta$ of a "locally-stable" pure-jump It\^{o} semimartingale from discrete observations of the process on a fixed time interval with mesh of the observation grid shrinking to zero. The estimator is based on the empirical characteristic function of the increments of the process scaled by local power variations formed from blocks of increments spanning shrinking time intervals preceding the increments to be scaled. The scaling serves two purposes: (1) it controls for the time variation in the jump compensator around zero, and (2) it ensures self-normalization, that is, that the limit of the characteristic function-based estimator converges to a nondegenerate limit which depends only on $\beta$. The proposed estimator leads to nontrivial efficiency gains over existing estimators based on power variations. In the L\'{e}vy case, the asymptotic variance decreases multiple times for higher values of $\beta$. The limiting asymptotic variance of the proposed estimator, unlike that of the existing power variation based estimators, is constant. This leads to further efficiency gains in the case when the characteristics of the semimartingale are stochastic. Finally, in the limiting case of $\beta=2$, which corresponds to jump-diffusion, our estimator of $\beta$ can achieve a faster rate than existing estimators., Comment: Published at http://dx.doi.org/10.1214/15-AOS1327 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
- Published
- 2015
43. Estimation of Volatility Functionals: The Case of a $$\sqrt{n}$$ Window
- Author
-
Jean Jacod and Mathieu Rosenbaum
- Subjects
Combinatorics ,Delta method ,Semimartingale ,Statistics ,Estimator ,Integral element ,Volatility (finance) ,Mathematics ,Central limit theorem - Abstract
We consider a multidimensional Ito semimartingale regularly sampled on [0, t] at high frequency \(1/\Delta _n\), with \(\Delta _n\) going to zero. The goal of this paper is to provide an estimator for the integral over [0, t] of a given function of the volatility matrix, with the optimal rate \(1/\sqrt{\Delta _n}\) and minimal asymptotic variance. To achieve this, we use spot volatility estimators based on observations within time intervals of length \(k_n\Delta _n\). In [5], this was done with \(k_n\rightarrow \infty \) and \(k_n \sqrt{\Delta _n}\rightarrow 0\), and a central limit theorem was given after suitable de-biasing. Here we do the same with the choice \(k_n\asymp 1/\sqrt{\Delta _n}\). This results in a smaller bias, although more difficult to eliminate.
- Published
- 2015
44. High-frequency asymptotics for path-dependent functionals of Ito semimartingales
- Author
-
Mark Podolskij and Moritz Duembgen
- Subjects
Statistics and Probability ,Asymptotic analysis ,Class (set theory) ,Applied Mathematics ,Mathematical analysis ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,math.ST ,Range (mathematics) ,Semimartingale ,Mathematics::Probability ,Law of large numbers ,Modeling and Simulation ,FOS: Mathematics ,Applied mathematics ,stat.TH ,Limit (mathematics) ,Central limit theorem ,Path dependent ,Mathematics - Abstract
The estimation of local characteristics of Ito semimartingales has received a great deal of attention in both academia and industry over the past decades. In various papers limit theorems were derived for functionals of increments and ranges in the infill asymptotics setting. In this paper we establish the asymptotic theory for a wide class of statistics that are built from the incremental process of an Ito semimartingale. More specifically, we will show the law of large numbers and the associated stable central limit theorem for the path dependent functionals in the continuous setting, and discuss the asymptotic theory for range-based statistics in the discontinuous framework. Some examples from economics and physics demonstrate the potential applicability of our theoretical results in practice.
- Published
- 2015
45. The Heckman–Opdam Markov processes
- Author
-
Bruno Schapira
- Subjects
Statistics and Probability ,Pure mathematics ,Mathematical finance ,Mathematical analysis ,Markov process ,symbols.namesake ,Semimartingale ,Probability theory ,Law of large numbers ,Symmetric space ,symbols ,Decomposition method (constraint satisfaction) ,Statistics, Probability and Uncertainty ,Analysis ,Central limit theorem ,Mathematics - Abstract
We introduce and study the natural counterpart of the Dunkl Markov processes in a negatively curved setting. We give a semimartingale decomposition of the radial part, and some properties of the jumps. We prove also a law of large numbers, a central limit theorem, and the convergence of the normalized process to the Dunkl process. Eventually we describe the asymptotic behavior of the infinite loop as it was done by Anker, Bougerol and Jeulin in the symmetric spaces setting in [1].
- Published
- 2006
46. Limit theorems for multipower variation in the presence of jumps
- Author
-
Matthias Winkel, Ole E. Barndorff-Nielsen, and Neil Shephard
- Subjects
Statistics and Probability ,Multipower variation ,Semimartingales ,Economics ,Quadratic variation ,Bipower variation ,Mathematics::Probability ,semimartingales ,Modelling and Simulation ,Stochastic volatility ,Limit (mathematics) ,stochastic volatility ,Infinite activity ,Brownian motion ,Central limit theorem ,Mathematics ,Power variation ,Stochastic process ,Applied Mathematics ,Mathematical analysis ,multipower variation ,quadratic variation ,Semimartingale ,power variation ,Modeling and Simulation ,Bipower variation, Infinite activity, Multipower variation, Power variation, Quadratic variation, Semimartingales, Stochastic volatility ,Jump ,infinite activity ,bipower variation ,Jump process - Abstract
In this paper we provide a systematic study of the robustness of probability limits and central limit theory for realised multipower variation when we add finite activity and infinite activity jump processes to an underlying Brownian semimartingale. X-Classification-JEL
- Published
- 2006
- Full Text
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47. Statistique sur la variance et régularisation des trajectoires de diffusions
- Author
-
Gonzalo Perera and Mario Wschebor
- Subjects
Statistics and Probability ,Stochastic differential equation ,Semimartingale ,Basis (linear algebra) ,Differential equation ,Mathematical analysis ,Applied mathematics ,Inference ,Statistics, Probability and Uncertainty ,Smoothing ,Statistical hypothesis testing ,Central limit theorem ,Mathematics - Abstract
We give a hypothesis testing method to fit the diffusion coefficient σ of a d-dimensional stochastic differential equation on the basis of the observation of certain functionals of regularizations of the solution.
- Published
- 2002
48. Moderate deviations for martingale differences and applications to φ -mixing sequences
- Author
-
Hacène Djellout
- Subjects
Combinatorics ,Semimartingale ,Convergence of random variables ,Mathematical analysis ,Large deviations theory ,Martingale difference sequence ,Stationary sequence ,Martingale (probability theory) ,Rate function ,Central limit theorem ,Mathematics - Abstract
For a R d -valued sequence of martingale differences { m k } k S 1 , we obtain a moderate deviation principle for the sequence of partial sums { Z n ( t ) 1 ~ k =1 [ nt ] m k / b n , t ] [0,1]}, in the space of cadlag functions equipped with the Skorohod topology, under the following conditions: a Chen-Ledoux type condition, an exponential convergence in probability of the associated quadratic variation process of the martingale, and a condition of "Lindeberg" type. For the small jumps of Z n (·), we apply the general result of Puhalskii [Puhalskii, A. (1994). "Large deviations of semimartingales via convergence of the predictable characteristics". Stoch. Stoch. Rep. , 49 , pp. 27-85]. Following the method of Ledoux [Ledoux, M. (1992). "Sur les deviations moderees des sommes de variables aleatoires vectorielles independantes de meme loi". Ann. Inst. H. Poincare , 28 , pp. 267-280] and Arcones [Arcones, A. (1999). "The large deviation principle for stochastic processes", Submitted for publication], we prov...
- Published
- 2002
49. Efficient estimation of integrated volatility in presence of infinite variation jumps
- Author
-
Jean Jacod and Viktor Todorov
- Subjects
Statistics and Probability ,Itô semimartingale ,Quadratic variation ,central limit theorem ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,01 natural sciences ,Lévy process ,Stable process ,integrated volatility ,010104 statistics & probability ,60F05 ,60G07 ,0502 economics and business ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,050205 econometrics ,Central limit theorem ,Mathematics ,05 social sciences ,Nonparametric statistics ,Estimator ,Semimartingale ,60F17 ,Statistics, Probability and Uncertainty ,Volatility (finance) ,60G51 - Abstract
We propose new nonparametric estimators of the integrated volatility of an It\^{o} semimartingale observed at discrete times on a fixed time interval with mesh of the observation grid shrinking to zero. The proposed estimators achieve the optimal rate and variance of estimating integrated volatility even in the presence of infinite variation jumps when the latter are stochastic integrals with respect to locally "stable" L\'{e}vy processes, that is, processes whose L\'{e}vy measure around zero behaves like that of a stable process. On a first step, we estimate locally volatility from the empirical characteristic function of the increments of the process over blocks of shrinking length and then we sum these estimates to form initial estimators of the integrated volatility. The estimators contain bias when jumps of infinite variation are present, and on a second step we estimate and remove this bias by using integrated volatility estimators formed from the empirical characteristic function of the high-frequency increments for different values of its argument. The second step debiased estimators achieve efficiency and we derive a feasible central limit theorem for them., Comment: Published in at http://dx.doi.org/10.1214/14-AOS1213 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
- Published
- 2014
50. A test for the rank of the volatility process: The random perturbation approach
- Author
-
Mark Podolskij, Jean Jacod, and 社会科学の高度統計・実証分析拠点構築 = Research Unit for Statistical and Empirical Analysis in Social Sciences
- Subjects
Statistics and Probability ,Alternative hypothesis ,Central limit theorem ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Homoscedasticity testing ,62M07, 60F05, 62E20, 60F17 ,Limit theory ,62M07 ,High frequency data ,Homoscedasticity ,60F05 ,Rank estimation ,FOS: Mathematics ,Test statistic ,Applied mathematics ,homoscedasticity testing ,Stable convergence ,rank estimation ,central limit theorem, high frequency data, homoscedasticity testing, Ito semimartingales, rank estimation, stable convergence ,Mathematics ,62E20 ,Random perturbation ,Itô semimartingales ,Semimartingale ,60F17 ,Statistics, Probability and Uncertainty ,Volatility (finance) ,stable convergence ,high frequency data - Abstract
December 2012, In this paper we present a test for the maximal rank of the matrix-valued volatility process in the continuous Ito semimartingale framework. Our idea is based upon a random perturbation of the original high frequency observations of an Ito semimartingale, which opens the way for rank testing. We develop the complete limit theory for the test statistic and apply it to various null and alternative hypotheses. Finally, we demonstrate a homoscedasticity test for the rank process., グローバルCOEプログラム = Global COE Program
- Published
- 2013
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