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35
The twoparameter PoissonDirichlet distribution derived from a stable subordinator.
, 1995
"... The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov ..."
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Cited by 221 (37 self)
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The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to VershikShmidtIgnatov, are generalized to the twoparameter case. The sizebiased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
A tale of two time scales: determining integrated volatility with noisy highfrequency data
, 2003
"... It is a common financial practice to estimate volatility from the sum of frequentlysampled squared returns. However market microstructure poses challenge to this estimation approach, as evidenced by recent empirical studies in finance. This work attempts to lay out theoretical grounds that reconcil ..."
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Cited by 109 (16 self)
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It is a common financial practice to estimate volatility from the sum of frequentlysampled squared returns. However market microstructure poses challenge to this estimation approach, as evidenced by recent empirical studies in finance. This work attempts to lay out theoretical grounds that reconcile continuoustime modeling and discretetime samples. We propose an estimation approach that takes advantage of the rich sources in tickbytick data while preserving the continuoustime assumption on the underlying returns. Under our framework, it becomes clear why and where the “usual ” volatility estimator fails when the returns are sampled at the highest frequency.
Asymptotic error distributions for the Euler method for stochastic differential equations
 THE ANNALS OF PROBABILITY
, 1998
"... We are interested in the rate of convergence of the Euler scheme approximation of the solution to a stochastic differential equation driven by a general (possibly discontinuous) semimartingale, and by the asymptotic behavior of the associated normalized error. It is well known that for Itô’s equatio ..."
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Cited by 103 (8 self)
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We are interested in the rate of convergence of the Euler scheme approximation of the solution to a stochastic differential equation driven by a general (possibly discontinuous) semimartingale, and by the asymptotic behavior of the associated normalized error. It is well known that for Itô’s equations the rate is 1 / √ n; we provide a necessary and sufficient condition for this rate to be 1 / √ n when the driving semimartingale is a continuous martingale, or a continuous semimartingale under a mild additional assumption; we also prove that in these cases the normalized error processes converge in law. The rate can also differ from 1 / √ n: this is the case for instance if the driving process is deterministic, or if it is a Lévy process without a Brownian component. It is again 1 / √ n when the driving process is Lévy with a nonvanishing Brownian component, but then the normalized error processes converge in law in the finitedimensional sense only, while the discretized normalized error processes converge in law in the Skorohod
Efficient estimation of stochastic volatility using noisy observations: A multiscale approach
, 2004
"... With the availability of high frequency financial data, nonparametric estimation of volatility of an asset return process becomes feasible. A major problem is how to estimate the volatility consistently and efficiently, when the observed asset returns contain error or noise, for example, in the form ..."
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Cited by 85 (10 self)
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With the availability of high frequency financial data, nonparametric estimation of volatility of an asset return process becomes feasible. A major problem is how to estimate the volatility consistently and efficiently, when the observed asset returns contain error or noise, for example, in the form of microstructure noise. The former (consistency) has been addressed heavily in the recent literature, however, the resulting estimator is not quite efficient. In Zhang, Mykland, and AïtSahalia (2003), the best estimator converges to the true volatility only at the rate of n −1/6. In this paper, we propose an efficient estimator which converges to the true at the rate of n −1/4, which is the best attainable. The estimator remains valid when the observation noise is dependent. Some key words and phrases: consistency, dependent noise, discrete observation, efficiency, Ito process, microstructure noise, observation error, rate of convergence, realized volatility
Estimating covariation: Epps effect and microstructure noise
 Journal of Econometrics, forthcoming
, 2009
"... This paper is about how to estimate the integrated covariance 〈X, Y 〉T of two assets over a fixed time horizon [0, T], when the observations of X and Y are “contaminated ” and when such noisy observations are at discrete, but not synchronized, times. We show that the usual previoustick covariance e ..."
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Cited by 26 (3 self)
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This paper is about how to estimate the integrated covariance 〈X, Y 〉T of two assets over a fixed time horizon [0, T], when the observations of X and Y are “contaminated ” and when such noisy observations are at discrete, but not synchronized, times. We show that the usual previoustick covariance estimator is biased, and the size of the bias is more pronounced for less liquid assets. This is an analytic characterization of the Epps effect. We also provide optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms, including nonsynchronous trading, microstructure noise, and time discretization. Finally, a twoscales covariance estimator is provided which simultaneously cancels (to first order) the Epps effect and the effect of microstructure noise. The gain is demonstrated in data.
ANOVA FOR DIFFUSIONS AND ITO PROCESSES
 SUBMITTED TO THE ANNALS OF STATISTICS
"... Ito processes are the most common form of continuous semimartingales, and include diffusion processes. The paper is concerned with the nonparametric regression relationship between two such Ito processes. We are interested in the quadratic variation (integrated volatility) of the residual in this re ..."
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Cited by 25 (11 self)
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Ito processes are the most common form of continuous semimartingales, and include diffusion processes. The paper is concerned with the nonparametric regression relationship between two such Ito processes. We are interested in the quadratic variation (integrated volatility) of the residual in this regression, over a unit of time (such as a day). A main conceptual finding is that this quadratic variation can be estimated almost as if the residual process were observed, the difference being that there is also a bias which is of the same asymptotic order as the mixed normal error term. The proposed methodology, “ANOVA for diffusions and Ito processes”, can be used to measure the statistical quality of a parametric model, and, nonparametrically, the appropriateness of a oneregressor model in general. On the other hand, it also helps quantify and characterize the trading (hedging) error in the case of financial applications.
Inference for Continuous Semimartingales Observed at High Frequency: A General Approach
, 2008
"... The econometric literature of high frequency data often relies on moment estimators which are derived from assuming local constancy of volatility and related quantities. We here study this localconstancy approximation as a general approach to estimation in such data. We show that the technique yiel ..."
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Cited by 22 (8 self)
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The econometric literature of high frequency data often relies on moment estimators which are derived from assuming local constancy of volatility and related quantities. We here study this localconstancy approximation as a general approach to estimation in such data. We show that the technique yields asymptotic properties (consistency, normality) that are correct subject to an ex post adjustment involving asymptotic likelihood ratios. These adjustments are given. Several examples of estimation are provided: powers of volatility, leverage effect, integrated betas, bipower, and covariance under asynchronous observation. The first order approximations in this study can be over the period of one observation, or over blocks of successive observations. The advantage of blocking is a gain in transparency in defining and analyzing estimators. The theory relies heavily on the interplay between stable convergence and measure change, and on asymptotic expansions for martingales.
Limit theorems for weighted samples with applications to Sequential Monte Carlo Methods. eprint arXiv:math.ST/0507042
, 2005
"... In the last decade, sequential MonteCarlo methods (SMC) emerged as a key tool in computational statistics (see for instance Doucet et al. (2001), Liu (2001), Künsch (2001)). These algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighte ..."
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Cited by 16 (7 self)
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In the last decade, sequential MonteCarlo methods (SMC) emerged as a key tool in computational statistics (see for instance Doucet et al. (2001), Liu (2001), Künsch (2001)). These algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles. These particles and weights are generated recursively according to elementary transformations: mutation and selection. Examples of applications include the sequential MonteCarlo techniques to solve optimal nonlinear filtering problems in statespace models, molecular simulation, genetic optimization, etc. Despite many theoretical advances (see for instance Gilks and Berzuini (2001), Künsch (2003), Del Moral (2004), Chopin (2004)), the asymptotic property of these approximations remains of course a question of central interest. In this paper, we analyze sequential Monte Carlo methods from an asymptotic perspective, that is, we establish law of large numbers and invariance principle as the number of particles gets large. We introduce the concepts of weighted sample consistency and asymptotic normality, and derive conditions under which the mutation and the selection procedure used in the sequential MonteCarlo buildup preserve these properties. To illustrate our findings, we analyze SMC algorithms to approximate the filtering distribution in statespace models. We show how our techniques allow to relax restrictive technical conditions used in previously reported works and provide grounds to analyze more sophisticated sequential sampling strategies. Short title: Limit theorems for SMC. 1
A Gaussian calculus for inference from high frequency data
, 2006
"... In the econometric literature of high frequency data, it is often assumed that one can carry out inference conditionally on the underlying volatility processes. In other words, conditionally Gaussian systems are considered. This is often referred to as the assumption of “no leverage effect”. This is ..."
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Cited by 12 (3 self)
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In the econometric literature of high frequency data, it is often assumed that one can carry out inference conditionally on the underlying volatility processes. In other words, conditionally Gaussian systems are considered. This is often referred to as the assumption of “no leverage effect”. This is often a reasonable thing to do, as general estimators and results can often be conjectured from considering the conditionally Gaussian case. The purpose of this paper is to try to give some more structure to the things one can do with the Gaussian assumption. We shall argue in the following that there is a whole treasure chest of tools that can be brought to bear on high frequency data problems in this case. We shall in particular consider approximations involving locally constant volatility processes, and develop a general theory for this approximation. As applications of the theory, we propose an improved estimator of quarticity, an ANOVA for processes with multiple regressors, and an estimator for error bars on the HayashiYoshida estimator of quadratic covariation Some key words and phrases: consistency, cumulants, contiguity, continuity, discrete observation, efficiency, Itô process, likelihood inference, realized volatility, stable convergence
A note on the central limit theorem for bipower variation of general functions
 Stochastic Processes and Their Applications 118
"... In this paper we present the central limit theorem for general functions of the increments of Brownian semimartingales. This provides a natural extension of the results derived in BarndorffNielsen, Graversen, Jacod, Podolskij & Shephard (2006), who showed the central limit theorem for even function ..."
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Cited by 8 (5 self)
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In this paper we present the central limit theorem for general functions of the increments of Brownian semimartingales. This provides a natural extension of the results derived in BarndorffNielsen, Graversen, Jacod, Podolskij & Shephard (2006), who showed the central limit theorem 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.