decorrelation property of discrete wavelet transform under fixed domain asymptotics theoretical aspects of the decorrelation property of the discrete wavelet transform when applied to stochastic processes have been studied exclusively from the increasing domain perspective in which the distance between neighboring observations stays roughly constant as the number of observations increases to understand the underlying data generating process and to obtain good interpolations fixed domain asymptotics in which the number of observations increases in a fixed region is often more appropriate than increasing domain asymptotics in the fixed domain setting we prove that for a general class of inhomogeneous covariance functions with suitable choice of wavelet filters the wavelet transform of a nonstationary process has mostly asymptotically uncorrelated components

functional data analysis for volatility abstract we introduce a functional volatility process for modeling volatility trajectories for high frequency observations in financial markets and describe functional representations and data based recovery of the process from repeated observations a study of its asymptotic properties as the frequency of observed trades increases is complemented by simulations and an application to the analysis of intra day volatility patterns of the s p 500 index the proposed volatility model is found to be useful to identify recurring patterns of volatility and for successful prediction of future volatility through the application of functional regression and prediction techniques

high frequency returns jumps and the mixture of normals hypothesis abstract previous empirical studies find both evidence of jumps in asset prices and that returns standardized by realized volatility are approximately standard normal these findings appear to be contradictory using a sample of high frequency returns for 20 heavily traded us stocks we show how microstructure noise distorts the standard deviation and kurtosis of returns normalized using realized variance when returns are standardized using a recently developed realized kernel estimator the resulting series is clearly platykurtotic and the standard normal distribution is soundly rejected moreover daily returns standardized using realized bipower variation an estimator for integrated variance that is robust to the presence of jumps are more consistent with the standard normal distribution these results suggest that there is no empirical contradiction jumps should be included in stock price models

the effects of random and discrete sampling when estimating continuous time diffusions high frequency financial data are not only discretely sampled in time but the time separating successive observations is often random we analyze the consequences of this dual feature of the data when estimating a continuous time model in particular we measure the additional effects of the randomness of the sampling intervals over and beyond those due to the discreteness of the data we also examine the effect of simply ignoring the sampling randomness we find that in many situations the randomness of the sampling has a larger impact than the discreteness of the data

the algebra of two scales estimation and the s tsrv high frequency estimation that is robust to sampling times abstract in this paper we derive a new algebraic property of two scales estimation in high frequency data under which the effect of sampling times is canceled to high order this is a particular robustness property of the two scales construction in general irregular asynchronous or endogenous times can cause problems in estimators based on equidistant observation of trade or quote times the new algebraic property can be combined with pre averaging giving rise to the smoothed two scales realized volatility s tsrv we derive a finite sample solution to controlling edge effects and for handling irregular and endogenous observation times and asynchronously observed multivariate data in connection with this development we use the algebraic approach to define a version of the s tsrv which has particularly small edge effect in microstructure noise the main result of the paper is a representation of the statistical error of the estimator in terms of simple components as an application of this representation the paper develops a central limit theory for multivariate volatility estimators the approach can also handle leads and lags in the signal process

bounds for blow up time for the heat equation under nonlinear boundary conditions a differential inequality technique is used to determine a lower bound on the blow up time for solutions to the heat equation subject to a nonlinear boundary condition when blow up of the solution does occur in addition a sufficient condition which implies that blow up does occur is determined

bootstrapping realized volatility we propose bootstrap methods for a general class of nonlinear transformations of realized volatility which includes the raw version of realized volatility and its logarithmic transformation as special cases we consider the independent and identically distributed i i d bootstrap and the wild bootstrap wb and prove their first order asymptotic validity under general assumptions on the log price process that allow for drift and leverage effects we derive edgeworth expansions in a simpler model that rules out these effects the i i d bootstrap provides a second order asymptotic refinement when volatility is constant but not otherwise the wb yields a second order asymptotic refinement under stochastic volatility provided we choose the external random variable used to construct the wb data appropriately none of these methods provides third order asymptotic refinements both methods improve upon the first order asymptotic theory in finite samples

testing if the market microstructure noise is fully explained by the informational content of some variables from the limit order book abstract in this paper we build tests for the presence of residual noise in a model where the market microstructure noise is a known parametric function of some variables from the limit order book the tests compare two distinct quasi maximum likelihood estimators of volatility where the related model includes a residual noise in the market microstructure noise or not the limit theory is investigated in a general nonparametric framework in the presence of residual noise we examine the central limit theory of the related quasi maximum likelihood estimation approach

the cauchy problem for heat equations with exponential nonlinearity abstract the cauchy problem for the semilinear heat equations is studied in the orlicz space exp l 2 r n where any power behavior of interaction works as a subcritical nonlinearity we prove the existence of global solutions for the semilinear heat equations with the exponential nonlinearity under the smallness condition on the initial data in exp l 2 r n

pole placement and compensator design of generalized systems abstract the system considered is a generalized system e ax bu y cx with matrix e singular when the impulsive modes of the system are both controllable and observable it is shown in this paper that these modes can be eliminated by almost any constant output feedback for a strongly controllable and observable generalized system after its impulsive modes are eliminated the poles of the system can be assigned arbitrarily by state feedback even if the states are not available the paper adopts an efficient method to design a compensator for the system

blow up phenomena of solutions for a reaction diffusion equation with weighted exponential nonlinearity abstract blow up phenomena for a reaction diffusion equation with weighted exponential reaction term and null dirichlet boundary condition are investigated we establish sufficient conditions to guarantee existence of global solution or blow up solution under appropriate measure sense by virtue of the method of super sub solutions the bernoulli equation and the modified differential inequality techniques moreover upper and lower bounds for the blow up time are found in higher dimensional spaces and some examples for application are presented

asymptotic error distributions for the euler method for stochastic differential equations 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 ito s equations the rate is 1 sqrt n we provide a necessary and sufficient condition for this rate to be 1 sqrt 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 sqrt n this is the case for instance if the driving process is deterministic or if it is a levy process without a brownian component it is again 1 sqrt n when the driving process is levy with a nonvanishing brownian component but then the normalized error processes converge in law in the finite dimensional sense only while the discretized normalized error processes converge in law in the skorohod sense and the limit is given an explicit form

adaptive density estimation from data with small measurement errors in this paper we study the problem of density estimation from data that contains small measurement errors the only assumption on these errors is that the maximal measurement error is bounded by some real number converging to zero for sample size tending to infinity in particular we do not assume that the measurement errors are independent with expectation zero we estimate the density by a standard kernel density estimate applied to data with measurement errors and derive a data driven method to choose its bandwidth we derive an adaptation result for this estimate and analyze the expected l 1 error of our density estimate depending on the smoothness of the density and the size of the maximal measurement error the results are applied in a density estimation problem in a simulation model where we show under suitable assumptions that the l 1 error of our newly proposed estimate converges to zero much faster than the l 1 error of the standard kernel density estimate if both are based on the same number of observations in the simulation model the performance of the method in case of finite sample size is analyzed using simulated data

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

econometrics of co jumps in high frequency data with noise abstract we establish estimation methods to determine co jumps in multivariate high frequency data with non synchronous observations and market microstructure a rate optimal estimator of the entire quadratic covariation of an ito semimartingale is constructed by a locally adaptive spectral approach thresholding allows to disentangle the co jump from the continuous part we derive a feasible limit theorem for a truncated estimator of integrated covolatility which facilitates asymptotically efficient co volatility estimation in the presence of jumps a test for common jumps is presented simulations and an empirical application to intra day tick data from eurex futures demonstrate the practical value of the approach

assessment of uncertainty in high frequency data the observed asymptotic variance the availability of high frequency financial data has generated a series of estimators based on intra day data improving the quality of large areas of financial econometrics however estimating the standard error of these estimators is often challenging the root of the problem is that traditionally standard errors rely on estimating a theoretically derived asymptotic variance and often this asymptotic variance involves substantially more complex quantities than the original parameter to be estimated standard errors are important they are used to assess the precision of estimators in the form of confidence intervals to create feasible statistics for testing to build forecasting models based on say daily estimates and also to optimize the tuning parameters the contribution of this paper is to provide an alternative and general solution to this problem which we call observed asymptotic variance it is a general nonparametric method for assessing asymptotic variance avar it provides consistent estimators of avar for a broad class of integrated parameters t dt where the spot parameter process can be a general semimartingale with continuous and jump components the observed avar is implemented with the help of a two scales method its construction works well in the presence of microstructure noise and when the observation times are irregular or asynchronous in the multivariate case the methodology is valid for a wide variety of estimators including the standard ones for variance and covariance and also for more complex estimators such as of leverage effects high frequency betas and semivariance

stability of steady states of the cauchy problem for the exponential reaction diffusion equation abstract we consider the cauchy problem u t u e u x r n t 0 t u x 0 u 0 x r n where u 0 c r n and t 0 we first study the radial steady states of the equation and the number of intersections distinguishing four different cases n 1 n 2 3 n 9 and n 10 writing explicitly every steady state for n 1 and n 2 then we study the large time behavior of solutions of the parabolic problem

lower bounds for blow up time in parabolic problems under dirichlet conditions we consider an initial boundary value problem for the semilinear heat equation whose solution may blow up in finite time we use a differential inequality technique to determine a lower bound on blow up time if blow up occurs a second method based on a comparison principle is also presented

on the approximate maximum likelihood estimation for diffusion processes the transition density of a diffusion process does not admit an explicit expression in general which prevents the full maximum likelihood estimation mle based on discretely observed sample paths ait sahalia j finance 54 1999 1361 1395 econometrica 70 2002 223 262 proposed asymptotic expansions to the transition densities of diffusion processes which lead to an approximate maximum likelihood estimation amle for parameters built on ait sahalia s econometrica 70 2002 223 262 ann statist 36 2008 906 937 proposal and analysis on the amle we establish the consistency and convergence rate of the amle which reveal the roles played by the number of terms used in the asymptotic density expansions and the sampling interval between successive observations we find conditions under which the amle has the same asymptotic distribution as that of the full mle a first order approximation to the fisher information matrix is proposed

realised quantile based estimation of the integrated variance in this paper we propose a new jump robust quantile based realised variance measure of ex post return variation that can be computed using potentially noisy data the estimator is consistent for the integrated variance and we present feasible central limit theorems which show that it converges at the best attainable rate and has excellent efficiency asymptotically the quantile based realised variance is immune to finite activity jumps and outliers in the price series while in modified form the estimator is applicable with market microstructure noise and therefore operational on high frequency data simulations show that it has superior robustness properties in finite sample while an empirical application illustrates its use on equity data

efficient estimation of integrated volatility incorporating trading information we consider a setting where market microstructure noise is a parametric function of trading information possibly with a remaining noise component assuming that the remaining noise is op 1 n allowing irregular times and jumps we show that we can estimate the parameters at rate n and propose a volatility estimator which enjoys n convergence rate simulation studies show that our method performs well even with model misspecification and rounding empirical studies demonstrate the practical relevance and advantages of our method furthermore we find that a simple model can account for a high percentage of the total variation in microstructure noise

estimating the integrated volatility with tick observations abstract we develop a volatility estimator that can be directly applied to tick by tick data more specifically we consider a model that allows for i irregular observation times that can be endogenous ii dependent noise that can have diurnal features and be dependent on the latent price process and iii jumps in the latent price process we show that our estimator yields consistent estimates and enjoys the optimal rate of convergence simulation as well as empirical studies demonstrate favorable properties of our proposed estimator

the distribution of realized exchange rate volatility using high frequency data on deutschemark and yen returns against the dollar we construct model free estimates of daily exchange rate volatility and correlation that cover an entire decade our estimates termed realized volatilities and correlations are not only model free but also approximately free of measurement error under general conditions which we discuss in detail hence for practical purposes we may treat the exchange rate volatilities and correlations as observed rather than latent we do so and we characterize their joint distribution both unconditionally and conditionally noteworthy results include a simple normality inducing volatility transformation high contemporaneous correlation across volatilities high correlation between correlation and volatilities pronounced and persistent dynamics in volatilities and correlations evidence of long memory dynamics in volatilities and correlations and remarkably precise scaling laws under temporal aggregation

designing realised kernels to measure the ex post variation of equity prices in the presence of noise this paper shows how to use realised kernels to carry out efficient feasible inference on the expost variation of underlying equity prices in the presence of simple models of market frictions the issue is subtle with only estimators which have symmetric weights delivering consistent estimators with mixed gaussian limit theorems the weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which is close to that of the maximum likelihood estimator in the parametric version of this problem realised kernels can also be selected to i be analysed using endogenously spaced data such as that in databases on transactions ii allow for market frictions which are endogenous iii allow for temporally dependent noise the finite sample performance of our estimators is studied using simulation while empirical work illustrates their use in practice

quasi maximum likelihood estimation of volatility with high frequency data this paper investigates the properties of the well known maximum likelihood estimator in the presence of stochastic volatility and market microstructure noise by extending the classic asymptotic results of quasi maximum likelihood estimation when trying to estimate the integrated volatility and the variance of noise this parametric approach remains consistent efficient and robust as a quasi estimator under misspecified assumptions moreover it shares the model free feature with nonparametric alternatives for instance realized kernels while being advantageous over them in terms of finite sample performance in light of quadratic representation this estimator behaves like an iterative exponential realized kernel asymptotically comparisons with a variety of implementations of the tukey hanning2 kernel are provided using monte carlo simulations and an empirical study with the euro us dollar future illustrates its application in practice

estimating the quadratic covariation matrix for asynchronously observed high frequency stock returns corrupted by additive measurement error this paper studies the estimation problem of the covariance matrices of asset returns in the presence of microstructure noise and asynchronicity between the observations across different assets motivated by malliavin and mancino 2002 2009 we propose a new fourier domain based estimator of multivariate ex post volatility which we call the fourier realized kernel frk an advantage of this approach is that no explicit time alignment is required unlike the time domain based methods widely adopted in the existing literature we derive the large sample properties and establish asymptotic normality of our estimator under some general conditions that allow for both temporal and cross sectional correlations in the measurement error process our results can be viewed as frequency domain extension of the asymptotic theories for the multivariate realized kernel estimator of barndorff nielsen et al 2011 we show in extensive simulations that our method outperforms the time domain estimators when two assets with different liquidity are traded asynchronously

predictive inference for integrated volatility numerous volatility based derivative products have been engineered in recent years this has led to interest in constructing conditional predictive densities and confidence intervals for integrated volatility in this article we propose nonparametric estimators of the aforementioned quantities based on model free volatility estimators we establish consistency and asymptotic normality for the feasible estimators and study their finite sample properties through a monte carlo experiment finally using data from the new york stock exchange we provide an empirical application to volatility directional predictability

do high frequency measures of volatility improve forecasts of return distributions many finance questions require the predictive distribution of returns we propose a bivariate model of returns and realized volatility rv and explore which features of that time series model contribute to superior density forecasts over horizons of 1 to 60 days out of sample this term structure of density forecasts is used to investigate the importance of the intraday information embodied in the daily rv estimates the functional form for log rv dynamics the timing of information availability and the assumed distributions of both return and log rv innovations we find that a joint model of returns and volatility that features two components for log rv provides a good fit to s p 500 and ibm data and is a significant improvement over an egarch model estimated from daily returns

statistical properties of microstructure noise we study the estimation of joint moments of microstructure noise based on high frequency data the estimation is conducted under a nonparametric setting which allows the underlying price process to have jumps the observation times to be irregularly spaced and the noise to be dependent on the price process and to have diurnal features estimators of arbitrary orders of joint moments are provided for which we establish consistency as well as central limit theorems in particular we provide estimators of autocovariances and autocorrelations of the noise simulation studies demonstrate excellent performance of our estimators in the presence of jumps irregular observation times and even rounding empirical studies reveal moderate positive autocorrelations of microstructure noise for the stocks tested

ultra high frequency volatility estimation with dependent microstructure noise we analyze the impact of time series dependence in market microstructure noise on the properties of estimators of the integrated volatility of an asset price based on data sampled at frequencies high enough for that noise to be a dominant consideration we show that combining two time scales for that purpose will work even when the noise exhibits time series dependence analyze in that context a refinement of this approach is based on multiple time scales and compare empirically our different estimators to the standard realized volatility

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

econometric analysis of realized volatility and its use in estimating stochastic volatility models summary the availability of intraday data on the prices of speculative assets means that we can use quadratic variation like measures of activity in financial markets called realized volatility to study the stochastic properties of returns here under the assumption of a rather general stochastic volatility model we derive the moments and the asymptotic distribution of the realized volatility error the difference between realized volatility and the discretized integrated volatility which we call actual volatility these properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation intensive methods

a quasi maximum likelihood approach for integrated covariance matrix estimation with high frequency data estimating the integrated covariance matrix icm from high frequency financial trading data is crucial to reflect the volatilities and covariations of the underlying trading instruments such an objective is difficult due to contaminated data with microstructure noises asynchronous trading records and increasing data dimensionality in this paper we study a quasi maximum likelihood qml approach for estimating an icm from high frequency financial data we explore a novel multivariate moving average time series device that is convenient for evaluating the estimator both theoretically for its asymptotic properties and numerically for its practical implementations we demonstrate that the qml estimator is consistent to the icm and is asymptotically normally distributed efficiency gain of the qml approach is theoretically quantified and numerically demonstrated via extensive simulation studies an application of the qml approach is illustrated through analyzing a high frequency financial trading data set

estimating covariation epps effect microstructure noise abstract 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 previous tick 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 the 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 two scales 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

jump robust volatility estimation using nearest neighbor truncation abstract we propose two new jump robust estimators of integrated variance that allow for an asymptotic limit theory in the presence of jumps specifically our medrv estimator has better efficiency properties than the tripower variation measure and displays better finite sample robustness to jumps and small zero returns we stress the benefits of local volatility measures using short return blocks as this greatly alleviates the downward biases stemming from rapid fluctuations in volatility including diurnal intraday u shape patterns an empirical investigation of the dow jones 30 stocks and extensive simulations corroborate the robustness and efficiency properties of our nearest neighbor truncation estimators

a tale of two time scales it is a common practice in finance to estimate volatility from the sum of frequently sampled squared returns however market microstructure poses challenges to this estimation approach as evidenced by recent empirical studies in finance the present work attempts to lay out theoretical grounds that reconcile continuous time modeling and discrete time samples we propose an estimation approach that takes advantage of the rich sources in tick by tick data while preserving the continuous time 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 frequencies if the noise is asymptotically small our work provides a way of finding the optimal sampling frequency a better approach the two scales estimator works for any size of the noise

testing for jumps when asset prices are observed with noise a swap variance approach abstract this paper proposes a new test for jumps in asset prices that is motivated by the literature on variance swaps formally the test follows by a direct application of ito s lemma to the semi martingale process of asset prices and derives its power from the impact of jumps on the third and higher order return moments intuitively the test statistic reflects the cumulative gain of a variance swap replication strategy which is known to be minimal in the absence of jumps but substantial in the presence of jumps simulations show that the jump test has nice properties and is generally more powerful than the widely used bi power variation test an important feature of our test is that it can be applied in analytically modified form to noisy high frequency data and still retain power as a by product of our analysis we obtain novel analytical results regarding the impact of noise on bi power variation an empirical illustration using ibm trade data is also included

anova for diffusions and ito processes ito processes are the most common form of continuous semimartingales and include diffusion processes this 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 non parametrically the appropriateness of a one regressor model in general on the other hand it also helps quantify and characterize the trading hedging error in the case of financial applications