a higher order weak approximation scheme of multidimensional stochastic differential equations using malliavin weights we show a new higher order weak approximation with malliavin weights for multidimensional stochastic differential equations by extending the method in takahashi and yamada 2016 the estimate of global error of the discretization is based on a sharp small time expansion using a malliavin calculus approach we give explicit malliavin weights for second order discretization as polynomials of brownian motions the effectiveness is illustrated through an example in option pricing

quantifying signals with power law correlations a comparative study of detrended fluctuation analysis and detrended moving average techniques detrended fluctuation analysis dfa and detrended moving average dma are two scaling analysis methods designed to quantify correlations in noisy nonstationary signals we systematically study the performance of different variants of the dma method when applied to artificially generated long range power law correlated signals with an a priori known scaling exponent alpha 0 and compare them with the dfa method we find that the scaling results obtained from different variants of the dma method strongly depend on the type of the moving average filter further we investigate the optimal scaling regime where the dfa and dma methods accurately quantify the scaling exponent alpha 0 and how this regime depends on the correlations in the signal finally we develop a three dimensional representation to determine how the stability of the scaling curves obtained from the dfa and dma methods depends on the scale of analysis the order of detrending and the order of the moving average we use as well as on the type of correlations in the signal

theoretical foundation of detrending methods for fluctuation analysis such as detrended fluctuation analysis and detrending moving average we present a bottom up derivation of fluctuation analysis with detrending for the detection of long range correlations in the presence of additive trends or intrinsic nonstationarities while the well known detrended fluctuation analysis dfa and detrending moving average dma were introduced ad hoc we claim basic principles for such methods where dfa and dma are then shown to be specific realizations the mean squared displacement of the summed time series contains the same information about long range correlations as the autocorrelation function but has much better statistical properties for large time lags however the scaling exponent of its estimator on a single time series is affected not only by trends on the data but also by intrinsic nonstationarities we therefore define the fluctuation function as mean squared displacement with weighting kernel we require that its estimator be unbiased and exhibit the correct scaling behavior for the random component of a signal which is only achieved if the weighting kernel implies detrending we show how dfa and dma satisfy these requirements and we extract their kernel weights

sharp derivative bounds for solutions of degenerate semi linear partial differential equations abstract the paper is a continuation of the kusuoka stroock programme of establishing smoothness properties of solutions of possibly degenerate partial differential equations by using probabilistic methods we analyze here a class of semi linear parabolic partial differential equations for which the linear part is a second order differential operator of the form v 0 i 1 n v i 2 where v 0 v n are first order differential operators that satisfy the so called ufg condition see kusuoka and stroock 1987 16 which is weaker than the hormander one specifically we prove that the bounds of the higher order derivatives of the solution along the vector fields coincide with those obtained in the linear case when the boundary condition is lipschitz continuous but that the asymptotic behavior of the derivatives may change because of the simultaneity of the nonlinearity and of the degeneracy when the boundary condition is of polynomial growth and measurable only

analysis of wiener functionals malliavin calculus and its applications to heat kernels on donne un expose facilement accessible du calcul de malliavin du calcul differentiel de dimension infinie sur l espace de wiener

comparison of detrending methods for fluctuation analysis abstract we examine several recently suggested methods for the detection of long range correlations in data series based on similar ideas as the well established detrended fluctuation analysis dfa in particular we present a detailed comparison between the regular dfa and two recently suggested methods the centered moving average cma method and a modified detrended fluctuation analysis mdfa we find that cma performs the same as dfa in long data with weak trends and is slightly superior to dfa in short data with weak trends when comparing standard dfa to mdfa we observe that dfa performs slightly better in almost all examples we studied we also discuss how several types of trends affect different types of dfa for weak trends in the data the new methods are comparable with dfa in these respects however if the functional form of the trend in data is not a priori known dfa remains the method of choice only a comparison of dfa results using different detrending polynomials yields full recognition of the trends a comparison with independent methods is recommended for proving long range correlations

integrated approach to the assessment of long range correlation in time series data to assess whether a given time series can be modeled by a stochastic process possessing long range correlation one usually applies one of two types of analysis methods the spectral method and the random walk analysis the first objective of this work is to show that each one of these methods used alone can be susceptible to producing false results we thus advocate an integrated approach which requires the use of both methods in a consistent fashion we provide the theoretical foundation of this approach and illustrate the main ideas using examples the second objective relates to the observation of long range anticorrelation hurst exponent hl1 2 in real world time series data the very peculiar nature of such processes is emphasized in light of the stringent condition under which such processes can occur using examples we discuss the possible factors that could contribute to the false claim of long range anticorrelations and demonstrate the particular importance of the integrated approach in this case

establishing a direct connection between detrended fluctuation analysis and fourier analysis to understand methodological features of the detrended fluctuation analysis dfa using a higher order polynomial fitting we establish the direct connection between dfa and fourier analysis based on an exact calculation of the single frequency response of the dfa the following facts are shown analytically 1 in the analysis of stochastic processes exhibiting a power law scaling of the power spectral density psd s f f a higher order detrending in the dfa has no adverse effect in the estimation of the dfa scaling exponent which satisfies the scaling relation 1 2 2 the upper limit of the scaling exponents detectable by the dfa depends on the order of polynomial fit used in the dfa and is bounded by m 1 where m is the order of the polynomial fit 3 the relation between the time scale in the dfa and the corresponding frequency in the psd are distorted depending on both the order of the dfa and the frequency dependence of the psd we can improve the scale distortion by introducing the corrected time scale in the dfa corresponding to the inverse of the frequency scale in the psd in addition our analytical approach makes it possible to characterize variants of the dfa using different types of detrending as an application properties of the detrending moving average algorithm are discussed

effect of nonlinear filters on detrended fluctuation analysis when investigating the dynamical properties of complex multiple component physical and physiological systems it is often the case that the measurable system s output does not directly represent the quantity we want to probe in order to understand the underlying mechanisms instead the output signal is often a linear or nonlinear function of the quantity of interest here we investigate how various linear and nonlinear transformations affect the correlation and scaling properties of a signal using the detrended fluctuation analysis dfa which has been shown to accurately quantify power law correlations in nonstationary signals specifically we study the effect of three types of transforms i linear y i a x i b ii nonlinear polynomial y i a x k i and iii nonlinear logarithmic y i log x i delta filters we compare the correlation and scaling properties of signals before and after the transform we find that linear filters do not change the correlation properties while the effect of nonlinear polynomial and logarithmic filters strongly depends on a the strength of correlations in the original signal b the power k of the polynomial filter and c the offset delta in the logarithmic filter we further apply the dfa method to investigate the apparent scaling of three analytic functions i exponential exp x a ii logarithmic log x a and iii power law x a lambda which are often encountered as trends in physical and biological processes while these three functions have different characteristics we find that there is a broad range of values for parameter a common for all three functions where the slope of the dfa curves is identical we further note that the dfa results obtained for a class of other analytic functions can be reduced to these three typical cases we systematically test the performance of the dfa method when estimating long range power law correlations in the output signals for different parameter values in the three types of filters and the three analytic functions we consider

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

modified detrended fluctuation analysis based on empirical mode decomposition for the characterization of anti persistent processes abstract detrended fluctuation analysis dfa is a simple but very efficient method for investigating the power law long term correlations of non stationary time series in which a detrending step is necessary to obtain the local fluctuations at different timescales we propose to determine the local trends through empirical mode decomposition emd and perform the detrending operation by removing the emd based local trends which gives an emd based dfa method similarly we also propose a modified multifractal dfa algorithm called an emd based mfdfa the performance of the emd based dfa and mfdfa methods is assessed with extensive numerical experiments based on fractional brownian motion and multiplicative cascading process we find that the emd based dfa method performs better than the classic dfa method in the determination of the hurst index when the time series is strongly anticorrelated and the emd based mfdfa method outperforms the traditional mfdfa method when the moment order q of the detrended fluctuations is positive we apply the emd based mfdfa to the 1 min data of shanghai stock exchange composite index and the presence of multifractality is confirmed we also analyze the daily austrian electricity prices and confirm its anti persistence

detrending fluctuation analysis based on high pass filtering abstract detrended fluctuation analysis dfa is a scaling method commonly used for detecting long range correlations in non stationary time series the dfa method uses a trend based on polynomial fitting to extract and quantify fluctuations at different time scales basically such procedure acts as a non dynamical high pass filter that removes time series components below a given time scale as an alternative to the polynomial fitting approach this paper proposes a dfa method based on well known high pass filters e g butterworth elliptic etc numerical results show that the proposed dfa approach yields results similar to traditional dfa method maybe the main advantage of the proposed dfa method is that efficient implementations of high pass filters are available commercially

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

detrending moving average algorithm frequency response and scaling performances the detrending moving average dma algorithm has been widely used in its several variants for characterizing long range correlations of random signals and sets one dimensional sequences or high dimensional arrays over either time or space in this paper mainly based on analytical arguments the scaling performances of the centered dma including higher order ones are investigated by means of a continuous time approximation and a frequency response approach our results are also confirmed by numerical tests the study is carried out for higher order dma operating with moving average polynomials of different degree in particular detrending power degree frequency response asymptotic scaling upper limit of the detectable scaling exponent and finite scale range behavior will be discussed

on the structure of simple c algebras tensored with a uhf algebra ii let d be a simple unital c algebra let b be a uhf algebra and put a b d it is proved that if p q a are projections and p q for all quasitraces on a then p q in the sense of murray and von neumann a more general result involving positive operators in a is also proved if d has finitely many extremal quasi traces and the projections in d k separate these then it is proved that a has real rank zero finally it is proved that provided d is stably finite then each positive state on k0 d lifts to a quasi trace

fast algorithm for scaling analysis with higher order detrending moving average method among scaling analysis methods based on the root mean square deviation from the estimated trend it has been demonstrated that centered detrending moving average dma analysis with a simple moving average has good performance when characterizing long range correlation or fractal scaling behavior furthermore higher order dma has also been proposed it is shown to have better detrending capabilities removing higher order polynomial trends than original dma however a straightforward implementation of higher order dma requires a very high computational cost which would prevent practical use of this method to solve this issue in this study we introduce a fast algorithm for higher order dma which consists of two techniques 1 parallel translation of moving averaging windows by a fixed interval 2 recurrence formulas for the calculation of summations our algorithm can significantly reduce computational cost monte carlo experiments show that the computational time of our algorithm is approximately proportional to the data length although that of the conventional algorithm is proportional to the square of the data length the efficiency of our algorithm is also shown by a systematic study of the performance of higher order dma such as the range of detectable scaling exponents and detrending capability for removing polynomial trends in addition through the analysis of heart rate variability time series we discuss possible applications of higher order dma

probabilistic properties of detrended fluctuation analysis for gaussian processes detrended fluctuation analysis dfa is one of the most widely used tools for the detection of long range dependence in time series although dfa has found many interesting applications and has been shown to be one of the best performing detrending methods its probabilistic foundations are still unclear in this paper we study probabilistic properties of dfa for gaussian processes our main attention is paid to the distribution of the squared error sum of the detrended process we use a probabilistic approach to derive general formulas for the expected value and the variance of the squared fluctuation function of dfa for gaussian processes we also get analytical results for the expected value of the squared fluctuation function for particular examples of gaussian processes such as gaussian white noise fractional gaussian noise ordinary brownian motion and fractional brownian motion our analytical formulas are supported by numerical simulations the results obtained can serve as a starting point for analyzing the statistical properties of dfa based estimators for the fluctuation function and long memory parameter

effect of trends on detrended fluctuation analysis scaling behavior we find that crossovers result from the competition between the scaling of the noise and the apparent scaling of the trend we study how the characteristics of these crossovers depend on i the slope of the linear trend ii the amplitude and period of the periodic trend iii the amplitude and power of the power law trend and iv the length as well as the correlation properties of the noise surprisingly we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws i e long range power law dependence of the position of the crossover on the parameters of the trends we show that the dfa result of noise with a trend can be exactly determined by the superposition of the separate results of the dfa on the noise and on the trend assuming that the noise and the trend are not correlated if this superposition rule is not followed this is an indication that the noise and the superposed trend are not independent so that removing the trend could lead to changes in the correlation properties of the noise in addition we show how to use dfa appropriately to minimize the effects of trends how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation or if the crossover is due to a trend without any transition in the dynamical properties of the noise

the law of the euler scheme for stochastic differential equations i convergence rate of the distribution function we study the approximation problem ofef x t byef x t n where x t is the solution of a stochastic differential equation x t n is defined by the euler discretization scheme with stept n andf is a given function for smoothf s talay and tubaro have shown that the erroref x t f x t n can be expanded in powers of 1 n which permits to construct romberg extrapolation precedures to accelerate the convergence rate here we prove that the expansion exists also whenf is only supposed measurable and bounded under an additional nondegeneracy condition of hormander type for the infinitesimal generator of x t to obtain this result we use the stochastic variations calculus in the second part of this work we will consider the density of the law ofx t n and compare it to the density of the law ofx t

tensor products of c algebras with the ideal property abstract combining a construction of dadarlat of a unital simple non exact c algebra c of real rank zero and stable rank one which is shape equivalent to a uhf algebra with results of kirchberg and a result obtained by dadarlat and the first named author we show that b h c contains an ideal that is not generated by its projections we also find a unital separable sub c algebra a of b h such that a is of real rank zero and a c has an ideal that is not generated by its projections

detecting long range correlations with detrended fluctuation analysis abstract we examine the detrended fluctuation analysis dfa which is a well established method for the detection of long range correlations in time series we show that deviations from scaling which appear at small time scales become stronger in higher orders of dfa and suggest a modified dfa method to remove them the improvement is necessary especially for short records that are affected by non stationarities furthermore we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of dfa

log amplitude statistics of intermittent and non gaussian time series in studies of hydrodynamic turbulence and nonequilibrium systems it has been demonstrated that the observed non gaussian probability density functions are often described effectively by a superposition of gaussian distributions with fluctuating variances based on this framework we propose a general method to characterize intermittent and non gaussian time series in our approach an observed time series is assumed to be described by the multiplication of gaussian and amplitude random variables where the amplitude variable describes the variance fluctuation it is shown analytically that statistical properties of the log amplitude fluctuations can be estimated using the logarithmic absolute moments of the observed time series this method is applicable to a wide variety of symmetric unimodal distributions with heavy tails in order to quantify the deviation from a gaussian distribution by analyzing random cascade type processes and superstatistical non gaussian models with power law tails we demonstrate that our method can provide detailed characterization in a wide range of non gaussian fluctuations

analysis of clusters formed by the moving average of a long range correlated time series we analyze the stochastic function c n i ensuremath equiv y i ensuremath y n i where y i is a long range correlated time series of length n mathrm max and y n i ensuremath equiv 1 n ensuremath sum k 0 n ensuremath 1 y i ensuremath k is the moving average with window n we argue that c n i generates a stationary sequence of self affine clusters mathcal c with length mathcal l lifetime ensuremath tau and area s the length and the area are related to the lifetime by the relationships mathcal l ensuremath sim ensuremath tau ensuremath psi mathcal l and s ensuremath sim ensuremath tau ensuremath psi s where ensuremath psi mathcal l 1 and ensuremath psi s 1 h we also find that mathcal l ensuremath tau and s are power law distributed with exponents depending on h p mathcal l ensuremath sim mathcal l ensuremath ensuremath alpha p ensuremath tau ensuremath sim ensuremath tau ensuremath ensuremath beta and p s ensuremath sim s ensuremath ensuremath gamma with ensuremath alpha ensuremath beta 2 ensuremath h and ensuremath gamma 2 1 h these predictions are tested by extensive simulations on series generated by the midpoint displacement algorithm of assigned hurst exponent h ranging from 0 05 to 0 95 of length up to n mathrm max 2 21 and n up to 2 13

estimator of a non gaussian parameter in multiplicative log normal models we study non gaussian probability density functions pdf s of multiplicative log normal models in which the multiplication of gaussian and log normally distributed random variables is considered to describe the pdf of the velocity difference between two points in fully developed turbulent flows the non gaussian pdf model was originally introduced by castaing et al physica d 46 177 1990 in practical applications an experimental pdf is approximated with castaing s model by tuning a single non gaussian parameter which corresponds to the logarithmic variance of the log normally distributed variable in the model in this paper we propose an estimator of the non gaussian parameter based on the qth order absolute moments to test the estimator we introduce two types of stochastic processes within the framework of the multiplicative log normal model one is a sequence of independent and identically distributed random variables the other is a log normal cascade type multiplicative process by analyzing the numerically generated time series we demonstrate that the estimator can reliably determine the theoretical value of the non gaussian parameter scale dependence of the non gaussian parameter in multiplicative log normal models is also studied both analytically and numerically as an application of the estimator we demonstrate that non gaussian pdf s observed in the s p500 index fluctuations are well described by the multiplicative log normal model

effect of nonstationarities on detrended fluctuation analysis detrended fluctuation analysis dfa is a scaling analysis method used to quantify long range power law correlations in signals many physical and biological signals are noisy heterogeneous and exhibit different types of nonstationarities which can affect the correlation properties of these signals we systematically study the effects of three types of nonstationarities often encountered in real data specifically we consider nonstationary sequences formed in three ways i stitching together segments of data obtained from discontinuous experimental recordings or removing some noisy and unreliable parts from continuous recordings and stitching together the remaining parts a cutting procedure commonly used in preparing data prior to signal analysis ii adding to a signal with known correlations a tunable concentration of random outliers or spikes with different amplitudes and iii generating a signal comprised of segments with different properties e g different standard deviations or different correlation exponents we compare the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types of nonstationarities we find that introducing nonstationarities to stationary correlated signals leads to the appearance of crossovers in the scaling behavior and we study how the characteristics of these crossovers depend on a the fraction and size of the parts cut out from the signal b the concentration of spikes and their amplitudes c the proportion between segments with different standard deviations or different correlations and d the correlation properties of the stationary signal we show how to develop strategies for preprocessing raw data prior to analysis which will minimize the effects of nonstationarities on the scaling properties of the data and how to interpret the results of dfa for complex signals with different local characteristics

nonlinear filtering properties of detrended fluctuation analysis abstract detrended fluctuation analysis dfa has been widely used for quantifying long range correlation and fractal scaling behavior in dfa to avoid spurious detection of scaling behavior caused by a nonstationary trend embedded in the analyzed time series a detrending procedure using piecewise least squares fitting has been applied however it has been pointed out that the nonlinear filtering properties involved with detrending may induce instabilities in the scaling exponent estimation to understand this issue we investigate the adverse effects of the dfa detrending procedure on the statistical estimation we show that the detrending procedure using piecewise least squares fitting results in the nonuniformly weighted estimation of the root mean square deviation and that this property could induce an increase in the estimation error in addition for comparison purposes we investigate the performance of a centered detrending moving average analysis with a linear detrending filter and sliding window dfa and show that these methods have better performance than the standard dfa

detrending fluctuation analysis based on moving average filtering abstract detrended fluctuation analysis dfa is a scaling method commonly used for detecting long range correlations in nonstationary time series applications range from financial time series to physiological data however as the removal of trends in dfa is based on discontinuous polynomial fitting oscillations in the fluctuation function and significant errors in crossover locations can be introduced to reduce the problems induced by discontinuous fitting moving average ma methods have been proposed previously by alesio et al eur j phys b 27 2002 197 in this work a variant of such ma methods is studied specifically the performance and characteristics of a ma method based on central differences is studied some important properties of this ma method are analyzed and illustrated with several artificial and real time series