nonlinear shrinkage estimation of large integrated covariance matrices integrated covariance matrices arise in intra day models of asset returns which allow volatility to change across the trading day when the number of assets is large the natural estimator of such a matrix suffers from bias contributed from extreme eigenvalues we introduce a novel nonlinear shrinkage estimator for the integrated covariance matrix which shrinks the extreme eigenvalues of a realized covariance matrix back to an acceptable level and enjoys a certain asymptotic efficiency when the number of assets is of the same order as the number of data points novel maximum exposure and actual risk bounds are derived when our estimator is used in constructing the minimum variance portfolio compared to other methods our estimator performs favorably in both simulations and a real data analysis

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

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

econometric analysis of multivariate realised qml estimation of the covariation of equity prices under asynchronous trading estimating the covariance between assets using high frequency data is challenging due to market microstructure effects and asynchronous trading in this paper we develop a multivariate realised quasi likelihood qml approach carrying out inference as if the observations arise from an asynchronously observed vector scaled brownian model observed with error under stochastic volatility the resulting realised qml estimator is positive semi definite uses all available data is consistent and asymptotically mixed normal the quasi likelihood is computed using a kalman filter and optimised using a relatively simple em algorithm which scales well with the number of assets we derive the theoretical properties of the estimator and prove that it achieves the efficient rate of convergence we show how to make itobtain the non parametric efficiency bound for this problem the estimator is also analysed usingmonte carlo methods and applied to equity data with varying levels of liquidity

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

phase diagram for the eigen quasispecies theory with a truncated fitness landscape using methods of statistical physics we present rigorous theoretical calculations of eigen s quasispecies theory with the truncated fitness landscape which dramatically limits the available sequence space of information carriers as the mutation rate is increased from small values to large values one can observe three phases the first i selective also known as ferromagnetic phase the second ii intermediate phase with some residual order and the third iii completely randomized also known as paramagnetic phase we calculate the phase diagram for these phases and the concentration of information carriers in the master sequence also known as peak configuration x 0 and other classes of information carriers as the phase point moves across the boundary between phase i and phase ii x 0 changes continuously as the phase point moves across the boundary between phase ii and phase iii x 0 has a large change our results are applicable for the general case of a fitness landscape

estimating quadratic variation when quoted prices change by a constant increment abstract for financial assets whose best quotes almost always change by jumping by the market s price tick size one cent five cents etc this paper proposes an estimator of quadratic variation which controls for microstructure effects it measures the prevalence of alternations where quotes jump back to their just previous price it defines a simple property called uncorrelated alternation which under conditions implies that the estimator is consistent in an asymptotic limit theory where jumps become very frequent and small feasible limit theory is developed and in simulations works well

positive periodic solutions of periodic neutral lotka volterra system with state dependent delays by using a fixed point theorem of strict set contraction some new criteria are established for the existence of positive periodic solutions of the following periodic neutral lotka volterra system with state dependent delays dxi t dt xi t ri t j 1naij t xj t j 1nbij t xj t ij t x1 t xn t j 1ncij t xj t ij t x1 t xn t i 1 2 n where ri aij bij cij c r r i j 1 2 n are periodic functions and ij ij c rn 1 r i 1 2 n are periodic functions with respect to their first arguments respectively

periodic solution of neutral lotka volterra system with periodic delays abstract a nonautonomous n species lotka volterra system with neutral delays is investigated a set of verifiable sufficient conditions is derived for the existence of at least one strictly positive periodic solution of this lotka volterra system by applying an existence theorem and some analysis techniques where the assumptions of the existence theorem are different from that of gaines and mawhin s continuation theorem r e gaines j l mawhin coincidence degree and nonlinear differential equations springer verlag berlin 1977 and that of abstract continuation theory for k set contraction w petryshyn z yu existence theorem for periodic solutions of higher order nonlinear periodic boundary value problems nonlinear anal 6 1982 943 969 moreover a problem proposed by freedman and wu h i freedman j wu periodic solution of single species models with periodic delay siam j math anal 23 1992 689 701 is answered

evolution equation of phenotype distribution general formulation and application to error catastrophe an equation describing the evolution of phenotypic distribution is derived using methods developed in statistical physics the equation is solved by using the singular perturbation method and assuming that the number of bases in the genetic sequence is large applying the equation to the mutation selection model by eigen provides the critical mutation rate for the error catastrophe phenotypic fluctuation of clones individuals sharing the same gene is introduced into this evolution equation with this formalism it is found that the critical mutation rate is sometimes increased by the phenotypic fluctuations i e noise can enhance robustness of a fitted state to mutation our formalism is systematic and general while approximations to derive more tractable evolution equations are also discussed

a composite coincidence degree with applications to boundary value problems of neutral equations we present a topological degree theory for the nonlinear problem l i b x g x with applications to a class of boundary value problems of neutral equations where l is an unbounded fredholm operator of index zero b is condensing and g is l compact

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

solvable biological evolution models with general fitness functions and multiple mutations in parallel mutation selection scheme in a recent paper phys rev e 69 046121 2004 we used the suzuki trottere formalism to study a quasispecies biological evolution model in a parallel mutation selection scheme with a single peak fitness function and a point mutation in the present paper we extend such a study to evolution models with more general fitness functions or multiple mutations in the parallel mutation selection scheme we give some analytical equations to define the error thresholds for some general cases of mean field like or symmetric mutation schemes and fitness functions we derive some equations for the dynamics in the case of a point mutation and polynomial fitness functions we derive exact dynamics for two point mutations asymmetric mutations and the four value spin model with a single peak fitness function the same method is applied for the model with a royal road fitness function we derive the steady state distribution for the single peak fitness function

on positive periodic solutions of a nonautonomous neutral delay n species competitive system abstract the purpose of this paper is to investigate a general neutral delay n species competitive nonautonomous system different from gaines and mawhin s continuation theorem r e gaines j l mawhin coincidence degree and nonlinear differential equations springer verlag berlin 1977 and abstract continuation theory for k set contraction w petryshyn z yu existence theorem for periodic solutions of higher order nonlinear periodic boundary value problems nonlinear anal 6 1982 943 969 r e gaines j l mawhin lecture notes in mathematics vol 586 springer verlag berlin 1977 a new existence theorem is introduced with the help of the theorem and some techniques a set of sufficient conditions are derived for the existence of at least one strictly positive componentwise periodic solution as an application we also examine some special cases which have been studied extensively in the literature some known results are improved and generalized

quasispecies theory for multiple peak fitness landscapes we use a path integral representation to solve the eigen and crow kimura molecular evolution models for the case of multiple fitness peaks with arbitrary fitness and degradation functions in the general case we find that the solution to these molecular evolution models can be written as the optimum of a fitness function with constraints enforced by lagrange multipliers and with a term accounting for the entropy of the spreading population in sequence space the results for the eigen model are applied to consider virus or cancer proliferation under the control of drugs or the immune system

solvable biological evolution model with a parallel mutation selection scheme based on the connection between a quantum spin model and an asexual biological evolution model with a single peak fitness function in parallel mutation selection scheme we solve exactly both static and dynamics of the evolution model we find that relaxation in such a parallel scheme is faster than that in a connected scheme of eigen model our method can also be extended to other fitness functions

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

nonparametric pricing of interest rate derivative securities we propose a nonparametric estimation procedure for continuous time stochastic models because prices of derivative securities depend crucially on the form of the instantaneous volatility of the underlying process we leave the volatility function unrestricted and estimate it nonparametrically only discrete data are used but the estimation procedure still does not rely on replacing the continuous time model by some discrete approximation instead the drift and volatility functions are forced to match the densities of the process we estimate the stochastic differential equation followed by the short term interest rate and compute nonparametric prices for bonds and bond options

statistical mechanics of eigen s evolution model the correspondence between eigen s model of macromolecular evolution and the equilibrium statistical mechanics of an inhomogeneous ising system is developed the free energy landscape of random ising systems with the hopfield hamiltonian as a special example is applied to the replication rate coefficient landscape the coupling constants are scaled with 1 l since the maxima of any landscape must not increase with the length of the macromolecules the calculated error threshold relation then agrees with eigen s expression which was derived in a different way it gives an explicit expression for the superiority parameter in terms of the parameters of the landscape the dynamics of selection and evolution is discussed

on the estimation of integrated covariance matrices of high dimensional diffusion processes we consider the estimation of integrated covariance icv matrices of high dimensional diffusion processes based on high frequency observations we start by studying the most commonly used estimator the realized covariance rcv matrix we show that in the high dimensional case when the dimension p and the observation frequency n grow in the same rate the limiting spectral distribution lsd of rcv depends on the covolatility process not only through the targeting icv but also on how the covolatility process varies in time we establish a marcenko pastur type theorem for weighted sample covariance matrices based on which we obtain a marcenko pastur type theorem for rcv for a class mathcal c of diffusion processes the results explicitly demonstrate how the time variability of the covolatility process affects the lsd of rcv we further propose an alternative estimator the time variation adjusted realized covariance tvarcv matrix we show that for processes in class mathcal c the tvarcv possesses the desirable property that its lsd depends solely on that of the targeting icv through the marcenko pastur equation and hence in particular the tvarcv can be used to recover the empirical spectral distribution of the icv by using existing algorithms

error threshold in simple landscapes we consider the quasispecies description of a population evolving in both the master sequence landscape where a single sequence is evolutionarily preferred over all others and the rem landscape where the fitness of different sequences is an independent identically distributed random variable we show that in both cases the error threshold is analogous to a first order thermodynamical transition where the overlap between the average genotype and the optimal one drops discontinuously to zero

on the data driven cos method in this paper we present the data driven cos method ddcos it is a fourier based financial option valuation method which assumes the availability of asset data samples a characteristic function of the underlying asset probability density function is not required as such the presented technique represents a generalization of the well known cos method 1 the convergence of the proposed method is o 1 n in line with monte carlo methods for pricing financial derivatives the ddcos method is then particularly interesting for density recovery and also for the efficient computation of the options sensitivities delta and gamma these are often used in risk management and can be obtained at a higher accuracy with ddcos than with plain monte carlo methods

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

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

spin coherent state representation of the crow kimura and eigen models of quasispecies theory we present a spin coherent state representation of the crow kimura and eigen models of biological evolution we deal with quasispecies models where the fitness is a function of hamming distances from one or more reference sequences in the limit of large sequence length n we find exact expressions for the mean fitness and magnetization of the asymptotic quasispecies distribution in symmetric fitness landscapes the results are obtained by constructing a path integral for the propagator on the coset su 2 u 1 and taking the classical limit the classical limit gives a hamiltonian function on a circle for one reference sequence and on the product of 2m 1 circles for m reference sequences we apply our representation to study the schuster swetina phenomena where a wide lower peak is selected over a narrow higher peak the quadratic landscape with two reference sequences is also analyzed specifically and we present the phase diagram on the mutation fitness parameter phase space furthermore we use our method to investigate more biologically relevant system a model of escape from adaptive conflict through gene duplication and find three different phases for the asymptotic population distribution

efficient asymptotic variance reduction when estimating volatility in high frequency data abstract this paper shows how to carry out efficient asymptotic variance reduction when estimating volatility in the presence of stochastic volatility and microstructure noise with the realized kernels rk from barndorff nielsen et al 2008 and the quasi maximum likelihood estimator qmle studied in xiu 2010 to obtain such a reduction we chop the data into b blocks compute the rk or qmle on each block and aggregate the block estimates the ratio of asymptotic variance over the bound of asymptotic efficiency converges as b increases to the ratio in the parametric version of the problem i e 1 0025 in the case of the fastest rk tukey hanning 16 and 1 for the qmle the impact of stochastic sampling times and jump in the price process is examined carefully the finite sample performance of both estimators is investigated in simulations while empirical work illustrates the gain in practice

the realized laplace transform of volatility we introduce and derive the asymptotic behavior of a new measure constructed from high frequency data which we call the realized laplace transform of volatility the statistic provides a nonparametric estimate for the empirical laplace transform function of the latent stochastic volatility process over a given interval of time and is robust to the presence of jumps in the price process with a long span of data that is under joint long span and infill asymptotics the statistic can be used to construct a nonparametric estimate of the volatility laplace transform as well as of the integrated joint laplace transform of volatility over different points of time we derive feasible functional limit theorems for our statistic both under fixed span and infill asymptotics as well as under joint long span and infill asymptotics which allow us to quantify the precision in estimation under both sampling schemes

existence and global attractivity of positive periodic solutions for delay lotka volterra competition patch systems with stocking abstract by means of mawhin s continuation theorem of coincidence degree and lyapunov functional a set of easily verifiable criteria are established for the existence and global attractivity of positive periodic solutions for delay lotka volterra competition patch system with stocking x 1 t h 1 t x 1 t a 1 t b 1 t x 1 t c 1 t y t d 1 t x 2 t 1 x 1 t s 1 t x 2 t h 2 t x 2 t a 2 t b 2 t x 2 t d 2 t x 1 t 2 x 2 t s 2 t y t h 3 t y t a 3 t b 3 t y t t 0 k s y t s ds c 3 t x 1 t s 3 t some known results are generalized and improved

periodicity on a class of neutral impulsive delay system we present some sufficient conditions for the existence of positive periodic solutions of a class of neutral impulsive delay differential system when our results applied to the differential system without impulse a known result in the literature is corrected moreover one example is given to illustrate the results obtained

fully nonparametric estimation of scalar diffusion models we propose a functional estimation procedure for homogeneous stochastic differential equations based on a discrete sample of observations and with minimal requirements on the data generating process we show how to identify the drift and diffusion function in situations where one or the other function is considered a nuisance parameter the asymptotic behavior of the estimators is examined as the observation frequency increases and as the time span lengthens we prove almost sure consistency and weak convergence to mixtures of normal laws where the mixing variates depend on the chronological local time of the underlying diffusion process that is the random time spent by the process in the vicinity of a generic spatial point the estimation method and asymptotic results apply to both stationary and nonstationary recurrent processes

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

dynamics of the eigen and the crow kimura models for molecular evolution we introduce an alternative way to study molecular evolution within well established hamilton jacobi formalism showing that for a broad class of fitness landscapes it is possible to derive dynamics analytically within the 1n accuracy where n is the genome length for a smooth and monotonic fitness function this approach gives two dynamical phases smooth dynamics and discontinuous dynamics the latter phase arises naturally with no explicite singular fitness function counterintuitively the hamilton jacobi method yields straightforward analytical results for the models that utilize fitness as a function of hamming distance from a reference genome sequence we also show the way in which this method gives dynamical phase structure for multipeak fitness

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

eigen model as a quantum spin chain exact dynamics we map the eigen model of biological evolution naturwissenschaften 58 465 1971 into a quantum spin model with non hermitian hamiltonian based on such a connection we derive exact relaxation periods for the eigen model to approach static energy landscape from various initial conditions we also study a simple case of dynamic fitness function

diploid biological evolution models with general smooth fitness landscapes and recombination using a hamilton jacobi equation approach we obtain analytic equations for steady state population distributions and mean fitness functions for crow kimura and eigen type diploid biological evolution models with general smooth hypergeometric fitness landscapes our numerical solutions of diploid biological evolution models confirm the analytic equations obtained we also study the parallel diploid model for the simple case of recombination and calculate the variance of distribution which is consistent with numerical results

nonparametric estimation of jump diffusion models abstract this paper develops the asymptotics for nonparametric kernel estimators of local time drift and volatilities and levy measure in jump diffusion models our asymptotics are developed in a very general set up allowing the sample span to increase as the sampling interval decreases and without assuming stationarity for drift and volatilities we analyze both local constant and local linear estimators we consider not only estimators for instantaneous conditional second moment but also threshold estimators to disentangle diffusive and jump volatilities the optimal bandwidths are provided for all these estimators

biological evolution in a multidimensional fitness landscape we considered a multiblock molecular model of biological evolution in which fitness is a function of the mean types of alleles located at different parts blocks of the genome we formulated an infinite population model with selection and mutation and calculated the mean fitness for the case of recombination we formulated a model with a multidimensional fitness landscape the dimension of the space is equal to the number of blocks and derived a theorem about the dynamics of initially narrow distribution we also considered the case of lethal mutations we also formulated the finite population version of the model in the case of lethal mutations our models derived for the virus evolution are interesting also for the statistical mechanics and the hamilton jacobi equation as well

continuous record asymptotics for rolling sample variance estimators it is widely known that conditional covariances of asset returns change over time researchers doing empirical work have adopted many strategies for accommodating conditional heteroskedasticity among the popular strategies are a chopping the available data into short blocks of time and assuming homoskedasticity within the blocks b performing one sided rolling regressions in which only data from say the preceding five year period is used to estimate the conditional covariance of returns at a given date and c performing two sided rolling regressions in which covariances are estimated for each date using say five years of lags and five years of leads another model garch amounts to a one sided weighted rolling regression we develop continuous record asymptotic approximations for the measurement error in conditional variances and covariances when using these methods we derive asymptotically optimal window lengths for standard rolling regressions and optimal weights for weighted rolling regressions as an empirical example we estimate volatility on the s p 500 stock index using daily data from 1928 to 1990

a new method for the solution of models of biological evolution derivation of exact steady state distributions we investigate well known models of biological evolution and address the open problem of how construct a correct continuous analog of mutations in discrete sequence space we deal with models where the fitness is a function of a hamming distance from the reference sequence the mutation selection master equation in the discrete sequence space is replaced by a hamilton jacobi equation for the logarithm of relative frequencies of different sequences the steady state distribution mean fitness and the variance of fitness are derived all our results are asymptotic in the large genome limit a variety of important biological and biochemical models can be solved by this new approach

a fourier transform method for nonparametric estimation of multivariate volatility we provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semi martingales which is based on fourier analysis the co volatility is reconstructed as a stochastic function of time by establishing a connection between the fourier transform of the prices process and the fourier transform of the co volatility process a non parametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes the asymptotic properties of the random estimator are studied namely consistency in probability uniformly in time and convergence in law to a mixture of gaussian distributions