estimating quadratic variation consistently in the presence of endogenous and diurnal measurement error abstract we propose an econometric model that captures the effects of market microstructure on a latent price process in particular we allow for correlation between the measurement error and the return process and we allow the measurement error process to have a diurnal heteroskedasticity we propose a modification of the tsrv estimator of quadratic variation we show that this estimator is consistent with a rate of convergence that depends on the size of the measurement error but is no worse than n 1 6 we investigate in simulation experiments the finite sample performance of various proposed implementations

systems with counterclockwise input output dynamics a new input output property is proposed suitable for the study of positive feedback interconnections both in a linear and nonlinear set up convergence results and small gain theorems are proved as well as criteria for the application of the property to the study of multistability lyapunov like conditions and dissipation inequalities for its detection are also provided

dynamics of cubic siegel polynomials we study the one dimensional parameter space of cubic polynomials in the complex plane which have a fixed siegel disk of rotation number where is a given irrational number of brjuno type the main result of this work is that when is of bounded type the boundary of the siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial we also show that these boundaries vary continuously as one moves in the parameter space this is most nontrivial near the set of cubics with both critical points on the boundary of their siegel disk we prove that this locus is a jordan curve in the parameter space most of the techniques and results can be generalized to polynomials of higher degrees

relaxation in nonconvex optimal control problems described by fractional differential equations abstract we consider the minimization problem of an integral functional with integrand that is not convex in the control on solutions of a control system described by fractional differential equation with mixed nonconvex constraints on the control a relaxation problem is treated along with the original problem it is proved that under general assumptions the relaxation problem has an optimal solution and that for each optimal solution there is a minimizing sequence of the original problem that converges to the optimal solution with respect to the trajectory the control and the functional in appropriate topologies simultaneously

subsampling high frequency data the main contribution of this paper is to propose a novel way of conducting inference for an important general class of estimators that includes many estimators of integrated volatility a subsampling scheme is introduced that consistently estimates the asymptotic variance for an estimator thereby facilitating inference and the construction of valid confidence intervals the new method does not rely on the exact form of the asymptotic variance which is useful when the latter is of complicated form the method is applied to the volatility estimator of ait sahalia et al 2011 in the presence of autocorrelated and heteroscedastic market microstructure noise

every storage function is a state function it is shown that for linear dynamical systems with quadratic supply rates a storage function can always be written as a quadratic function of the state of an associated linear dynamical system this dynamical system is obtained by combining the dynamics of the original system with the dynamics of the supply rate

control of biological models with hysteresis abstract this article presents a control problem in population dynamics when diffusive and hysteretic effects in the evolution of population are taken into account namely we consider a diffusive prey predator model in a bounded domain with a parameter corresponding to the quantity of food for the preys in the model the evolution of the food density is described by a hysteresis operator of the generalized play type whose inputs are the densities of preys predators roughly speaking the rate of change of the food density depends not only on the present state of preys predators but also on their immediately preceding density history our main concern is to extend the previous research of this model by controlling the food supply and to study the well posedness of the corresponding dynamical control problem

a kyp lemma and invariance principle for systems with multiple hysteresis non linearities absolute stability criteria for systems with multiple hysteresis non linearities are given in this paper it is shown that the stability guarantee is achieved with a simple two part test on the linear subsystem if the linear subsystem satis es a particular linear matrix inequality and a simple residue condition then as is proven the non linear system will be asymptotically stable the main stability theorem is developed using a combination of passivity lyapunov and popov stability theories to show that the state describing the linear system dynamics must converge to an equilibrium position of the non linear closed loop system the invariant sets that contain all such possible equilibrium points are described in detail for several common types of hystereses the class of non linearities covered by the analysis is very general and includes multiple slope restricted memoryless non linearities as a special case simple numerical examples are used to demonstrate the e ectiveness of the new analysis in comparison to other recent results and graphically illustrate state asymptotic stability

bogolyubov type theorem with constraints induced by a control system with hysteresis effect abstract we consider the problem of minimization of an integral functional with nonconvex with respect to the control integrand we minimize our functional over the solution set of a control system described by two ordinary differential equations subject to a control constraint given by a multivalued mapping with closed nonconvex values the coefficients of the equations and the constraint depend on the phase variables one of the equations contains the subdifferential of the indicator function of a closed convex set depending on the unknown phase variable the equation containing the subdifferential describes an input output relation of hysteresis type along with the original problem we also consider the problem of minimizing the integral functional with the convexified with respect to the control integrand over the solution set of the same system with the convexified control constraint under sufficiently general assumptions we prove that this relaxed problem has an optimal solution which is the limit of a minimizing sequence of the original problem the convergence takes place simultaneously with respect to the trajectory the control and the functional and is uniform in appropriate topologies

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

stability of systems with the duhem hysteresis operator in this paper we discuss the dissipativity property of the counterclockwise duhem operator sufficient conditions on the functions which define the duhem operator are given such that the duhem operator has counterclockwise input output dynamics in particular an explicit construction of the storage functions satisfying the counterclockwise dissipation inequality is given the constructed storage function is also related to the underlying anhysteresis function which is commonly used to describe hysteresis in magnetic materials the results can thus facilitate analysis of systems with the counterclockwise duhem operator via the dissipativity approach

a bias corrected estimator of the covariation matrix of multiple security prices when both microstructure effects and sampling durations are persistent and endogenous i propose a bias corrected non parametric estimator of the covariation matrix of log security prices designed as a convex combination of two realized kernels the estimator is simple but possesses desirable statistical properties including consistency asymptotic normality and the parametric rate of convergence in the presence of persistent diurnally heteroskedastic and endogenous microstructure effects it is robust to the asynchronous trading of multiple securities with persistent and endogenous refresh time durations i also prove the consistency of a subsampling based estimator of the asymptotic covariance matrix of the proposed estimator in simulations the non linear functions of the proposed estimator exhibit smaller bias than those based on a realized kernel while only slightly increasing the variance thereby the proposed estimator reduces the mean squared error

all bounded type siegel disks of rational maps are quasi disks we prove that every bounded type siegel disk of a rational map must be a quasi disk with at least one critical point on its boundary this verifies douady sullivan s conjecture in the case of bounded type rotation numbers

renormalization group analysis of quasi periodicity in analytic maps analytic maps of the form f z e 2 pi i omega z mathcal o z 2 display quasiperiodicity when satisfies a diophantine condition quasiperiodic motion is confined to a neighborhood of the origin known as a siegel domain the boundary of this domain obeys universal scaling relations in this paper we investigate these scaling relations through a renormalization group analysis and we discuss singularities and asymptotic form of the scaling function

subsampling realised kernels in a recent paper we have introduced the class of realised kernel estimators of the increments of quadratic variation in the presence of noise we showed that this estimator is consistent and derived its limit distribution under various assumptions on the kernel weights in this paper we extend our analysis looking at the class of subsampled realised kernels and we derive the limit theory for this class of estimators we find that subsampling is highly advantageous for estimators based on discontinuous kernels such as the truncated kernel for kinked kernels such as the bartlett kernel we show that subsampling is impotent in the sense that subsampling has no effect on the asymptotic distribution perhaps surprisingly for the efficientsmooth kernels such as the parzen kernel we show that subsampling is harmful as it increases the asymptotic variance we also study the performance of subsampled realised kernels in simulations and in empirical work

multivariate realised kernels consistent positive semi definite estimators of the covariation of equity prices with noise and non synchronous trading we propose a multivariate realised kernel to estimate the ex post covariation of log prices we show this new consistent estimator is guaranteed to be positive semi definite and is robust to measurement noise of certain types and can also handle non synchronous trading it is the first estimator which has these three properties which are all essential for empirical work in this area we derive the large sample asymptotics of this estimator and assess its accuracy using a monte carlo study we implement the estimator on some us equity data comparing our results to previous work which has used returns measured over 5 or 10 minutes intervals we show the new estimator is substantially more precise

the parameter planes of λz m exp z for m 2 we consider the families of entire transcendental maps given by f m z zm exp z where m 2 all functions f m have a superattracting fixed point at z 0 and a critical point at z m in the parameter planes we focus on the capture zones i e values for which the critical point belongs to the basin of attraction of z 0 denoted by a 0 in particular we study the main capture zone parameter values for which the critical point lies in the immediate basin a 0 and prove that is bounded connected and simply connected all other capture zones are unbounded and simply connected for each parameter in the main capture zone a 0 consists of a single connected component with non locally connected boundary for all remaining values of a 0 is a quasidisk on a different approach we introduce some families of holomorphic maps of mathbb c which serve as a model for f m in the sense that they are related by means of quasiconformal surgery to f m

on the julia set of a typical quadratic polynomial with a siegel disk let 0 1 be an irrational number with continued fraction expan sion a1 a 2 a 3 and consider the quadratic polynomial p z 7 e 2i z z 2 by performing a trans quasiconformal surgery on an associated blaschke product model we prove that if log an o p n a sn 1 then the julia set of p is locally connected and has lebesgue measure zero in particular it follows that for almost every 0 1 the quadratic p has a siegel disk whose boundary is a jordan curve passing through the critical point of p b y standard renormalization theory these results generalize to the quadratics which have siegel disks of higher periods

on siegel disks of a class of entire maps let f c c be an entire map of the form f z p z exp q z where p and q are polynomials of arbitrary degrees we allow the case q 0 building upon a method pioneered by m shishikura we show that if f has a siegel disk of bounded type rotation number centered at the origin then the boundary of this siegel disk is a quasicircle containing at least one critical point of f this unifies and generalizes several previously known results

estimating the quadratic covariation matrix from noisy observations local method of moments and efficiency an efficient estimator is constructed for the quadratic covariation or integrated covolatility matrix of a multivariate continuous martingale based on noisy and non synchronous observations under high frequency asymptotics our approach relies on an asymptotically equivalent continuous time observation model where a local generalised method of moments in the spectral domain turns out to be optimal asymptotic semiparametric efficiency is established in the cramer rao sense main findings are that non synchronicity of observation times has no impact on the asymptotics and that major efficiency gains are possible under correlation simulations illustrate the finite sample behaviour

a phase field model with temperature dependent constraint this paper is concerned with a phase field model with temperature dependent constraint for the order parameter in the kinetic equation of the order parameter we take account of diffusion as well as undercooled superheated state which is modelled by means of a hysteresis input output relation the input is the temperature field and the output is the order parameter in this paper we shall show that there exists a solution globally in time in a physically reasonable sense when the coefficient of interfacial energy is sufficiently small

econometric analysis of realized covariation high frequency based covariance regression and correlation in financial economics this paper analyses multivariate high frequency financial data using realized covariation we provide a new asymptotic distribution theory for standard methods such as regression correlation analysis and covariance it will be based on a fixed interval of time e g a day or week allowing the number of high frequency returns during this period to go to infinity our analysis allows us to study how high frequency correlations regressions and covariances change through time in particular we provide confidence intervals for each of these quantities

empirical bias bandwidths for local polynomial nonparametric regression and density estimation abstract a data based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials the estimator called the empirical bias bandwidth selector ebbs is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function ebbs minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term the variance term used is exact not asymptotic though it involves the conditional variance of the response given the predictors that must be estimated the bias term is estimated empirically not from an asymptotic expression thus ebbs is similar to the double smoothing approach of hardle hall and marron and a local bandwidth selector of schucany but is developed here for a far wider class of estimation problems than what those authors considered ebbs is tested on simulated data and its performance seems quite satisfactory local polynomial smoothing of a histogram is a high

mathematical models for hysteresis the various existing classical models for hysteresis preisach ishlinskii duhem madelung are surveyed as well a more modern treatments by contemporary workers the emphasis is on a clear mathematical description of the formulation and properties of each model in addition the authors try to make the reader aware of the many open questions in the study of hysteresis

parabolic pdes with hysteresis and quasivariational inequalities w0 being prescribed as an initial output of w its input output relation is illustrated in fig 1 1 see 1 part i iii 2 for the precise de nition in the gure fa and fd are continuous and nondecreasing functions on r such that fa fd on r we refer for some results on this sort of model to 1 4 as is well known 1 part i iii 2 1 2 is equivalent to the following variational inequality wt x t iu x t w x t 3 0 x t qt 1 3

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

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

quadratic like dynamics of cubic polynomials a small perturbation of a quadratic polynomial f with a non repelling fixed point gives a polynomial g with an attracting fixed point and a jordan curve julia set on which g acts like angle doubling however there are cubic polynomials with a non repelling fixed point for which no perturbation results into a polynomial with jordan curve julia set motivated by the study of the closure of the cubic principal hyperbolic domain we describe such polynomials in terms of their quadratic like restrictions

bounded type siegel disks of a family of sine families abstract we prove that for any positive integer n there is a polynomial p z with degree n such that the entire function g z p sin z has a bounded type siegel disk bounded by a quasi circle in the plane which passes through at least one critical point of g in addition let 0 1 be an irrational number of bounded type we prove that for any integer k 0 the boundary of the siegel disk of g z 0 sin z e 2 i 1 s 2 k d s centered at the origin is a quasi circle passing through exactly two critical points 2 and 2 with multiplicity k 1 respectively

pre averaging estimators of the ex post covariance matrix in noisy diffusion models with non synchronous data we show how pre averaging can be applied to the problem of measuring the ex post covariance of financial asset returns under microstructure noise and non synchronous trading a pre averaged realised covariance is proposed and we present an asymptotic theory for this new estimator which can be configured to possess an optimal convergence rate or to ensure positive semi definite covariance matrix estimates we also derive a noise robust hayashi yoshida estimator that can be implemented on the original data without prior alignment of prices we uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high frequency equity data

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

knowing factors or factor loadings or neither evaluating estimators of large covariance matrices with noisy and asynchronous data abstract we investigate estimators of factor model based large covariance and precision matrices using high frequency data which are asynchronous and potentially contaminated by the market microstructure noise our estimation strategies rely on the pre averaging method with refresh time to solve the microstructure problems while using three different specifications of factor models with a variety of thresholding methods respectively to battle the curse of dimensionality to estimate a factor model we either adopt the time series regression tsr to recover loadings if factors are known or use the cross sectional regression csr to recover factors from known loadings or use the principal component analysis pca if neither factors nor their loadings are assumed known we compare the convergence rates in these scenarios using the joint in fill and increasing dimensionality asymptotics to evaluate the empirical trade off between robustness to model misspecification and statistical efficiency among all 30 combinations of estimation strategies we run a horse race on the out of sample portfolio allocation with dow jones 30 s p 100 and s p 500 index constituents respectively and find the pre averaging based strategy using tsr or pca with location thresholding dominates especially over the subsampling based alternatives

julia sets in parameter spaces given a complex number of modulus 1 we show that the bifurcation locus of the one parameter family fb z z bz2 z3 b contains quasi conformal copies of the quadratic julia set j z z2 as a corollary we show that when the julia set j z z2 is not locally connected for example when z z z2 has a cremer point at 0 the bifurcation locus is not locally connected to our knowledge this is the first example of complex analytic parameter space of dimension 1 with connected but non locally connected bifurcation locus we also show that the set of complex numbers of modulus 1 for which at least one of the parameter rays has a non trivial accumulation set contains a dense g subset of s1

mating siegel quadratic polynomials 1 1 mating definitions and some history mating quadratic polynomials is a topological construction suggested by douady and hubbard do2 to partially parametrize quadratic rational maps of the riemann sphere by pairs of quadratic polynomials some results on matings of higher degree maps exist but we will not discuss them in this paper while there exist several presumably equivalent ways of describing the construction of mating the following approach is perhaps the most standard consider two monic quadratic polynomials fi and f2 whose filled julia sets k fi are locally connected for each fi let di denote the conformal isomorphism between the basin of infinity c k f and c ed with i 00 00 and v oo 1 these bottcher maps conijugate the polynomials to the squaring map

asymptotic equivalence for inference on the volatility from noisy observations we consider discrete time observations of a continuous martingale under measurement error this serves as a fundamental model for high frequency data in finance where an efficient price process is observed under microstructure noise it is shown that this nonparametric model is in le cam s sense asymptotically equivalent to a gaussian shift experiment in terms of the square root of the volatility function and a nonstandard noise level as an application new rate optimal estimators of the volatility function and simple efficient estimators of the integrated volatility are constructed

on the dynamics of a family of generated renormalization transformations abstract we study the family of renormalization transformations of the generalized d dimensional diamond hierarchical potts model in statistical mechanic and prove that their julia sets and non escaping loci are always connected where d 2 in particular we prove that their julia sets can never be a sierpinski carpet if the parameter is real we show that the julia set is a quasicircle if and only if the parameter lies in the unbounded capture domain of these models moreover the asymptotic formula of the hausdorff dimension of the julia set is calculated as the parameter tends to infinity

on the characterization of the duhem hysteresis operator with clockwise input output dynamics abstract in this paper we investigate the dissipativity property of a certain class of duhem hysteresis operator which has clockwise cw input output i o behavior in particular we provide sufficient conditions on the duhem operator such that it is cw and propose an explicit construction of the corresponding function satisfying dissipation inequality of cw systems the result is used to analyze the stability of a second order system with hysteretic friction which is described by a dahl model

high frequency covariance estimates with noisy and asynchronous financial data this article proposes a consistent and efficient estimator of the high frequency covariance quadratic covariation of two arbitrary assets observed asynchronously with market microstructure noise this estimator is built on the marriage of the quasi maximum likelihood estimator of the quadratic variation and the proposed generalized synchronization scheme and thus is not influenced by the epps effect moreover the estimation procedure is free of tuning parameters or bandwidths and is readily implementable monte carlo simulations show the advantage of this estimator by comparing it with a variety of estimators with specific synchronization methods the empirical studies of six foreign exchange future contracts illustrate the time varying correlations of the currencies during the 2008 global financial crisis demonstrating the similarities and differences in their roles as key currencies in the global market

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

semilinear duhem model for rate independent and rate dependent hysteresis the classical duhem model provides a finite dimensional differential model of hysteresis in this paper we consider rate independent and rate dependent semilinear duhem models with provable properties the vector field is given by the product of a function of the input rate and linear dynamics if the input rate function is positively homogeneous then the resulting input output map of the model is rate independent yielding persistent nontrivial input output closed curve that is hysteresis at arbitrarily low input frequency if the input rate function is not positively homogeneous the input output map is rate dependent and can be approximated by a rate independent model for low frequency inputs sufficient conditions for convergence to a limiting input output map are developed for rate independent and rate dependent models finally the reversal behavior and orientation of the rate independent model are discussed

necessary and sufficient conditions for l 1 strong weak lower semicontinuity of integral functionals soit t 7b f un espace fini de mesure nonatomique x ii ii un espace de banach non separable et v i i un espace de banach reflexif separable dont v est le dual soit f txxxv une fonction 7b f 7b b x v mesurable on etudie l integrale fonctionnelle associee i f l x 1 l v 1 definie par if x v f r x t v t dt

relaxation and optimisation of a phase field control system with hysteresis abstractthe present paper is concerned with a nonlinear partial differential control system arising in phase transition modelling the control constraint is given by a state dependent nonconvex valued multivalued mapping we study the problem of minimisation of a given integral cost functional over solutions to the above system the cost integrand is nonconvex in the control variable relaxation results of bogolyubov type are obtained for this problem

extending hysteresis operators to spaces of piecewise continuous functions we consider continuous time hysteresis operators defined to be causal and rate independent operators mapping input signals u r r to output signals v r r we show how a hysteresis operator defined on the set of continuous piecewise monotone functions can be naturally extended to piecewise continuous piecewise monotone functions we prove that the extension is also a hysteresis operator and that a number of important properties of the original operator are inherited by the extension moreover we define the concept of a discrete time hysteresis operator and we show that discretizing continuous time hysteresis operators using standard sampling and hold operations leads to discrete time hysteresis operators we apply the concepts and results described above in the context of sampled data feedback control of linear systems with input hysteresis 2003 elsevier science usa all rights reserved