dynamics of stock prices we show that the dynamics of stock prices can be accurately described as a continuous time random walk with a time dependent diffusion coefficient the time evolution of the diffusion coefficient can be derived from tick by tick databases provided the stock price is characterized in terms of a couple of values describing the best ask and the best bid we are then led to a finding and namely that the transition rate of the random walk process is different from the frequency of transactions our results allow us to obtain a fast and reliable determination of the diffusion coefficient and precisely confirm that fat tails in the distribution of price variations are due to volatility fluctuations

multifractal hurst analysis of crude oil prices abstract daily records of international crude oil prices are studied using multifractal analysis methods rescaled range hurst analysis provides evidence that the crude oil market is a persistent process with long run memory effects on the other hand height height correlation analysis reveals evidence of multifractal structures in the sense that the crude oil dynamics displays mixing of rough hurst exponents the existence of two characteristic time scales in the order of weeks and quarters is discovered and the corresponding prices dynamics are extracted using moving average based filtering these results seem to demonstrate that the crude oil market is consistent with the random walk assumption only at time scales of the order of days to weeks a plausible oil price formation mechanism is discussed in terms of the market dynamics at three different time scales

schwinger dyson operators as invariant vector fields on a matrix model analog of the group of loops for a class of large n multimatrix models we identify a group g that plays the same role as the group of loops on space time does for yang mills theory g is the spectrum of a commutative shuffle deconcatenation hopf algebra that we associate with correlations g is the exponential of the free lie algebra the generating series of correlations is a function on g and satisfies quadratic equations in convolution these factorized schwinger dyson or loop equations involve a collection of schwinger dyson operators which are shown to be right invariant vector fields on g one for each linearly independent primitive of the hopf algebra a large class of formal matrix models satisfying these properties are identified including as special cases the zero momentum limits of the gaussian chern simons and yang mills field theories moreover the schwinger dyson operators of the continuum yang mills action are shown to be right invariant derivations of the shuffle deconcatenation hopf algebra generated by source

crossing of two mobile averages a method for measuring the roughness exponent self affine signals characterized by a defined hurst roughness exponent h can be investigated through mobile averages the density ensuremath rho of crossing points between any two moving averages is a measure of long range power law correlations in the signal the method is compared to the detrended fluctuation analysis we take advantage of our findings in order to propose a practically powerful and accurate technique for determining h and apply it right away to cases with persistent or antipersistent correlations

statistical properties of the yuan exchange rate index abstract we choice the yuan exchange rate index based on a basket of currencies as the effective exchange rate of the yuan and investigate the statistical properties of the yuan exchange rate index after china s exchange rate system reform on the 21st july 2005 after dividing the time series into two parts according to the change in the yuan exchange rate regime in july 2008 we compare the statistical properties of the yuan exchange rate index during these two periods we find that the distribution of the two return series has the exponential form we also perform the detrending moving average analysis dma and the multifractal detrending moving average analysis mfdma the two periods possess different degrees of long range correlations and the multifractal nature is also unveiled in these two time series significant difference is found in the scaling exponents q and singularity spectra f of the two periods obtained from the mfdma analysis besides in order to detect the sources of multifractality shuffling and phase randomization procedures are applied to destroy the long range temporal correlation and fat tailed distribution of the yuan exchange rate index respectively we find that the fat tailedness plays a critical role in the sources of multifractality in the first period while the long memory is the major cause in the second period the results suggest that the change in china s exchange rate regime in july 2008 gives rise to the different multifractal properties of the yuan exchange rate index in these two periods and thus has an effect on the effective exchange rate of the yuan after the exchange rate reform on the 21st july 2005

inequalities for bessel functions we derive some inequalities for the cylinder function c x defined by cv x jv x cos yv x sin 0 where j x and y x denote the bessel functions of the first and second kind respectively our chief tools are three recurrence relations for c x and an integral formula for j x

scale invariant distribution and multifractality of volatility multipliers in stock markets abstract the statistical properties of the multipliers of the absolute returns are investigated using 1 min high frequency data of financial time series the multiplier distribution is found to be independent of the box size s when s is larger than some crossover scale providing direct evidence of the existence of scale invariance in financial data the multipliers with base a 2 are well approximated by a normal distribution and the most probable multiplier scales as a power law with respect to the base a we unravel that the volatility multipliers possess multifractal nature which is independent of construction of the multipliers that is the values of s and a

the tetrahedron algebra the onsager algebra and the sl2 loop algebra let k denote a field with characteristic 0 and let t denote an indeterminate we give a presentation for the three point loop algebra sl2 k t t 1 t 1 1 via generators and relations this presentation displays s4 symmetry using this presentation we obtain a decomposition of the above loop algebra into a direct sum of three subalgebras each of which is isomorphic to the onsager algebra

multifractality in the stock market price increments versus waiting times abstract by applying the multifractal detrended fluctuation analysis to the high frequency tick by tick data from deutsche borse both in the price and in the time domains we investigate multifractal properties of the time series of logarithmic price increments and inter trade intervals of time we show that both quantities reveal multiscaling and that this result holds across different stocks the origin of the multifractal character of the corresponding dynamics is among others the long range correlations in price increments and in inter trade time intervals as well as the non gaussian distributions of the fluctuations since the transaction to transaction price increments do not strongly depend on or are almost independent of the inter trade waiting times both can be sources of the observed multifractal behaviour of the fixed delay returns and volatility the results presented also allow one to evaluate the applicability of the multifractal model of asset returns in the case of tick by tick data

onsager s algebra and the dolan grady condition in the non self dual case in this paper it is shown that a pair of operators that satisfy the dolan grady conditions generate an onsager algebra even in the case that they are not self dual briefly considered will be the implications for the spectra of superintegrable spin chains in various sectors

bounds for modified bessel functions abstract in this paper we derive inequalities for the modified bessel functions i v t and k v t in some cases our results are completely new in other cases we improve or extend known results similar inequalities are proved for the function w v ti v t i v 1 t which is useful in some problems of finite elasticity

power spectrum and detrended fluctuation analysis application to daily temperatures the variability measures of fluctuation analysis fa and detrended fluctuation analysis dfa are expressed in terms of the power spectral density and of the autocovariance of a given process the diagnostic potential of these methods is tested on several model power spectral densities in particular we find that both fa and dfa reveal an algebraic singularity of the power spectral density at small frequencies corresponding to an algebraic decay of the autocovariance a scaling behavior of the power spectral density in an intermediate frequency regime is better reflected by dfa than by fa we apply fa and dfa to ambient temperature data from the 20th century with the primary goal to resolve the controversy in literature whether the low frequency behavior of the corresponding power spectral densities are better described by a power law or a stretched exponential as a third possible model we suggest a weibull distribution however it turns out that neither fa nor dfa can reliably distinguish between the proposed models

continuous time random walk model for financial distributions we apply the formalism of the continuous time random walk to the study of financial data the entire distribution of prices can be obtained once two auxiliary densities are known these are the probability densities for the pausing time between successive jumps and the corresponding probability density for the magnitude of a jump we have applied the formalism to data on the u s dollar deutsche mark future exchange finding good agreement between theory and the observed data

size dependent standard deviation for growth rates empirical results and theoretical modeling we study annual logarithmic growth rates r of various economic variables such as exports imports and foreign debt for each of these variables we find that the distributions of r can be approximated by double exponential laplace distributions in the central parts and power law distributions in the tails for each of these variables we further find a power law dependence of the standard deviation sigma r on the average size of the economic variable with a scaling exponent surprisingly close to that found for the gross domestic product gdp phys rev lett 81 3275 1998 by analyzing annual logarithmic growth rates r of wages of 161 different occupations we find a power law dependence of the standard deviation sigma r on the average value of the wages with a scaling exponent beta 0 14 close to those found for the growth of exports imports debt and the growth of the gdp in contrast to these findings we observe for payroll data collected from 50 states of the usa that the standard deviation sigma r of the annual logarithmic growth rate r increases monotonically with the average value of payroll however also in this case we observe a power law dependence of sigma r on the average payroll with a scaling exponent beta 0 08 based on these observations we propose a stochastic process for multiple cross correlated variables where for each variable i the distribution of logarithmic growth rates decays exponentially in the central part ii the distribution of the logarithmic growth rate decays algebraically in the far tails and iii podobnik the standard deviation of the logarithmic growth rate depends algebraically on the average size of the stochastic variable

autoregressive conditional duration a new model for irregularly spaced transaction data this paper proposes a new statistical model for the analysis of data which arrive at irregular intervals the model treats the time between events as a stochastic process and proposes a new class of point processes with dependent arrival rates the conditional intensity is developed and compared with other self exciting processes because the model focuses on the expected duration between events it is called the autoregressive conditional duration acd model asymptotic properties of the quasi maximum likelihood estimator are developed as a corollary to arch model results strong evidence is provided for duration clustering for the financial transaction data analyzed both deterministic time of day effects and stochastic effects are important the model is applied to the arrival times of trades and therefore is a model of transaction volume and also to the arrival of other events such as price changes models for the volatility of prices are estimated with price based durations and examined from a market microstructure point of view

multifractality in stock indexes fact or fiction abstract multifractal analysis and extensive statistical tests are performed upon intraday minutely data within individual trading days for four stock market indexes including hsi szsc s p 500 and nasdaq to check whether the indexes instead of the returns possess multifractality we find that the mass exponent q is linear and the singularity q is close to 1 for all trading days and all indexes furthermore we find strong evidence showing that the scaling behaviors of the original data sets cannot be distinguished from those of shuffled time series hence the so called multifractality in the intraday stock market indexes is merely an illusion

cross correlations between volume change and price change in finance one usually deals not with prices but with growth rates r defined as the difference in logarithm between two consecutive prices here we consider not the trading volume but rather the volume growth rate r the difference in logarithm between two consecutive values of trading volume to this end we use several methods to analyze the properties of volume changes r and their relationship to price changes r we analyze 14 981 daily recordings of the standard and poor s s p 500 index over the 59 year period 1950 2009 and find power law cross correlations between r and r by using detrended cross correlation analysis dcca we introduce a joint stochastic process that models these cross correlations motivated by the relationship between r and r we estimate the tail exponent of the probability density function p r r 1 for both the s p 500 index as well as the collection of 1819 constituents of the new york stock exchange composite index on 17 july 2009 as a new method to estimate we calculate the time intervals q between events where r q we demonstrate that q the average of q obeys q q we find 3 furthermore by aggregating all q values of 28 global financial indices we also observe an approximate inverse cubic law

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

bounds for modified struve functions of the first kind and their ratios abstract we obtain a simple two sided inequality for the ratio l x l 1 x in terms of the ratio i x i 1 x where l x is the modified struve function of the first kind and i x is the modified bessel function of the first kind this result allows one to use the extensive literature on bounds for i x i 1 x to immediately deduce bounds for l x l 1 x we note some consequences and obtain further bounds for l x l 1 x by adapting techniques used to bound the ratio i x i 1 x we apply these results to obtain new bounds for the condition numbers x l x l x the ratio l x l y and the modified struve function l x itself amongst other results we obtain two sided inequalities for x l x l x and l x l y that are given in terms of x i x i x and i x i y respectively which again allows one to exploit the substantial literature on bounds for these quantities the results obtained in this paper complement and improve existing bounds in the literature

multifractal formalism for fractal signals the structure function approach versus the wavelet transform modulus maxima method several attempts have been made recently to generalize the multifractal formalism originally introduced for singular measures to fractal signals we report on a systematic comparison between the structure function approach pioneered by parisi and frisch in 2 proceedings of the international school on turbulence and predictability in geophysical fluid dynamics and climate dynamics edited by m ghil r benzi and g parisi north holland amsterdam 1985 p 84 to account for the multifractal nature of fully developed turbulent signals and an alternative method we have developed within the framework of the wavelet transform analysis we comment on the intrinsic limitations of the structure function approach this technique has fundamental drawbacks and does not provide a full characterization of the singularities of a signal in many cases we demonstrate that our method based on the wavelet transform modulus maxima representation works in most situations and is likely to be the ground of a unified multifractal description of self affine distributions our theoretical considerations are both illustrated on pedagogical examples and supported by numerical simulations

detrended fluctuation analysis on spot and futures markets of west texas intermediate crude oil abstract in this paper we study the auto correlations and cross correlations of west texas intermediate wti crude oil spot and futures return series employing detrended fluctuation analysis dfa and detrended cross correlation analysis dcca scaling analysis shows that for time scales smaller than a month the auto correlations and cross correlations are persistent for time scales larger than a month but smaller than a year the correlations are anti persistent while for time scales larger than a year the series are neither auto correlated nor cross correlated indicating the efficient operation of the crude oil markets moreover for small time scales the degree of short term cross correlations is higher than that of auto correlations using the multifractal extension of dfa and dcca we find that for small time scales the correlations are strongly multifractal while for large time scales the correlations are nearly monofractal analyzing the multifractality of shuffled and surrogated series we find that both long range correlations and fat tail distributions make important contributions to the multifractality our results have important implications for market efficiency and asset pricing models

poisson brackets of normal ordered wilson loops we formulate yang mills theory in terms of the large n limit viewed as a classical limit of gauge invariant dynamical variables which are closely related to wilson loops via deformation quantization we obtain a poisson algebra of these dynamical variables corresponding to normal ordered quantum at a finite value of operators comparing with a poisson algebra one of us introduced in the past for weyl ordered quantum operators we find using ideas closely related to topological graph theory that these two poisson algebras are roughly speaking the same more precisely speaking there exists an invertible poisson morphism between them

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

long term correlations and multifractal nature in the intertrade durations of a liquid chinese stock and its warrant abstract the intertrade duration of equities is an important financial measure characterizing trading activities it is defined as the waiting time between successive trades of an equity using the ultrahigh frequency data of a liquid chinese stock and its associated warrant we perform a comparative investigation of the statistical properties of their intertrade duration time series the distributions of the two equities can be better described by the shifted power law form than the weibull form and their scaled distributions do not collapse onto a single curve although the intertrade durations of the two equities have very different magnitude their intraday patterns exhibit very similar shapes both detrended fluctuation analysis dfa and detrending moving average analysis dma show that the 1 min intertrade duration time series of the two equities are strongly correlated in addition both multifractal detrended fluctuation analysis mfdfa and multifractal detrending moving average analysis mfdma unveil that the 1 min intertrade durations possess multifractal nature however the difference between the two singularity spectra of the two equities obtained from the mfdma is much smaller than that from the mfdfa

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

multifractal detrended fluctuation analysis of derivative and spot markets abstract we investigate the multifractal properties of price increments in the cases of derivative and spot markets through the multifractal detrended fluctuation analysis we estimate the generalized hurst and the renyi exponents for price fluctuations by deriving the singularity spectrum from the above exponents we quantify the multifractality of a financial time series and compare the multifractal properties of two different markets the different behavior of each agent group in transactions is also discussed in order to identify the nature of the underlying multifractality we apply the method of surrogate data to both sets of financial data it is shown that multifractality due to a fat tailed distribution is significant

fractal heterogeneous media a method is presented for generating compact fractal disordered media by generalizing the random midpoint displacement algorithm the obtained structures are invasive stochastic fractals with the hurst exponent varying as a continuous parameter as opposed to lacunar deterministic fractals such as the menger sponge by employing the detrending moving average algorithm a carbone phys rev e 76 056703 2007 the hurst exponent of the generated structure can be subsequently checked the fractality of such a structure is referred to a property defined over a three dimensional topology rather than to the topology itself consequently in this framework the hurst exponent should be intended as an estimator of compactness rather than of roughness applications can be envisaged for simulating and quantifying complex systems characterized by self similar heterogeneity across space for example exploitation areas range from the design and control of multifunctional self assembled artificial nanostructures and microstructures to the analysis and modeling of complex pattern formation in biology environmental sciences geomorphological sciences etc

detrended fluctuation analysis of intertrade durations abstract the intraday pattern long memory and multifractal nature of the intertrade durations which are defined as the waiting times between two consecutive transactions are investigated based upon the limit order book data and order flows of 23 liquid chinese stocks listed on the shenzhen stock exchange in 2003 an inverse u shaped intraday pattern in the intertrade durations with an abrupt drop in the first minute of the afternoon trading is observed based on a detrended fluctuation analysis we find a crossover of power law scaling behaviors for small box sizes trade numbers in boxes and large box sizes and strong evidence in favor of long memory in both regimes in addition the multifractal nature of intertrade durations in both regimes is confirmed by a multifractal detrended fluctuation analysis for individual stocks with a few exceptions in the small duration regime the intraday pattern has little influence on the long memory and multifractality

statistical properties of daily ensemble variables in the chinese stock markets abstract we study dynamical behavior of the chinese stock markets by investigating the statistical properties of daily ensemble return and variety defined respectively as the mean and the standard deviation of the ensemble daily price return of a portfolio of stocks traded in china s stock markets on a given day the distribution of the daily ensemble return has an exponential form in the center and power law tails while the variety distribution is lognormal in the bulk followed by a power law tail for large variety based on detrended fluctuation analysis r s analysis and modified r s analysis we find evidence of long memory in the ensemble return and strong evidence of long memory in the evolution of variety

establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes stochastic fractal signals can be characterized by the hurst coefficient h which is related to the exponents of various power law statistics characteristic of these processes two techniques widely used to estimate h are spectral analysis and detrended fluctuation analysis dfa this paper examines the analytical link between these two measures and shows that they are related through an integral transform numerical simulations confirm this relationship for ideal synthesized fractal signals their performance as estimators of h is compared based on a mean square error criterion and found to be similar dfa measures are derived for physiological signals of heartbeat r ensuremath r intervals through the integral transform of a spectral density estimate these agree with directly calculated dfa estimates indicating that the relationship holds for signals with nonideal fractal properties it is concluded that dfa and spectral measures provide equivalent characterizations of stochastic signals with long term correlation

inequalities for some special functions abstract inequalities for bessel functions modified bessel functions and of their ratios are obtained these results are either sharper or hold under weaker conditions than had been known earlier similar inequalities are proved for the function w v t ti v t i v 1 t which is useful in some problems of finite elasticity in the sequel inequalities for confluent hypergeometric functions and of their ratios are established

detrended fluctuation analysis for fractals and multifractals in higher dimensions one dimensional detrended fluctuation analysis dfa and multifractal detrended fluctuation analysis mfdfa are widely used in the scaling analysis of fractal and multifractal time series because they are accurate and easy to implement in this paper we generalize the one dimensional dfa and mfdfa to higher dimensional versions the generalization works well when tested with synthetic surfaces including fractional brownian surfaces and multifractal surfaces the two dimensional mfdfa is also adopted to analyze two images from nature and experiment and nice scaling laws are unraveled

efficiency and multifractality analysis of csi 300 based on multifractal detrending moving average algorithm abstract we apply the multifractal detrending moving average mfdma to investigate and compare the efficiency and multifractality of 5 min high frequency china securities index 300 csi 300 the results show that the csi 300 market becomes closer to weak form efficiency after the introduction of csi 300 future we find that the csi 300 is featured by multifractality and there are less complexity and risk after the csi 300 index future was introduced with the shuffling surrogating and removing extreme values procedures we unveil that extreme events and fat distribution are the main origin of multifractality besides we discuss the knotting phenomena in multifractality and find that the scaling range and the irregular fluctuations for large scales in the f q s vs s plot can cause a knot

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

singularity spectrum of fractal signals from wavelet analysis exact results the multifractal formalism for singular measures is revisited using the wavelet transform for bernoulli invariant measures of some expanding markov maps the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima the generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures it is demonstrated that the hausdorff dimensiond h of the set of singularities of holder exponenth can be directly determined from the wavelet transform modulus maxima the singularity spectrum so obtained is shown to be not disturbed by the presence in the signal of a superimposed polynomial behavior of ordern provided one uses an analyzing wavelet that possesses at leastn n vanishing moments however it is shown that ac behavior generally induces a phase transition in thed h singularity spectrum that somewhat masks the weakest singularities this phase transition actually depends on the numbern of vanishing moments of the analyzing wavelet its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal these theoretical results are illustrated with numerical examples they are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional brownian motions and turbulent velocity signals

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

multifractal moving average analysis and test of multifractal model with tuned correlations we address two common major problems in the study of time series characterizing fluctuations in complex systems multifractal analysis and multifractal modeling specifically we introduce a multi fractal centered moving average mf cma analysis which is computationally easier but equivalently performing compared with the well established multi fractal detrended fluctuation analysis mf dfa with linear detrending in addition we study in detail a generalized binomial multi fractal model gb mfm to conveniently and reliably generate multifractal surrogate data with arbitrary singularity strengths and arbitrary long term persistence we use the data generated by this model as well as realistic by construction monofractal data series with crossovers and trends to test and compare the multifractal analysis methods and discuss finite size effects as well as limitations due to spurious multifractality

poisson brackets of wilson loops and derivations of free algebras we describe a finite analog of the poisson algebra of wilson loops in yang mills theory it is shown that this algebra arises in an apparently completely different context as a lie algebra of vector fields on a noncommutative space this suggests that noncommutative geometry plays a fundamental role in the manifestly gauge invariant formulation of yang mills theory we also construct the deformation of the algebra of loops induced by quantization in the large nc limit

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

investigating existence of chaos in short and long term dynamics of moroccan exchange rates abstract this paper proposes a new methodology to investigate presence of chaos in exchange rate time series by combining wavelet transform and lyapunov exponent estimation in particular stationary wavelet transform swt is applied to exchange rate original time series for decomposition purpose as a result approximation and details coefficients are extracted they are used to represent long and short term dynamics of the original exchange rate time series then largest lyapunov exponent is estimated for each type of dynamics to check for presence of chaos our methodology is applied to several moroccan exchange rate time series the empirical results show that in general the hypothesis of chaotic structure is accepted for currency levels but it is rejected for currency returns on both long and short dynamics in addition long and short dynamics exhibit different chaotic patterns in some exchange rate time series our approach may be useful to understand chaotic behaviour in original exchange rate time series

multifractal analysis of chinese stock volatilities based on the partition function approach abstract we have performed a detailed multifractal analysis on the 1 min volatility of two indexes and 1139 stocks in the chinese stock markets based on the partition function approach the partition function q s scales as a power law with respect to the box size s the scaling exponents q form a nonlinear function of q statistical tests based on bootstrapping show that the extracted multifractal nature is significant at the 1 significance level the individual securities can be well modeled by the p model in turbulence with p 0 40 0 02 based on the idea of ensemble averaging including quenched and annealed average we treat each stock exchange as a whole and confirm the existence of multifractal nature in the chinese stock markets

multifractal detrended cross correlation analysis for two nonstationary signals we propose a method called multifractal detrended cross correlation analysis to investigate the multifractal behaviors in the power law cross correlations between two time series or higher dimensional quantities recorded simultaneously which can be applied to diverse complex systems such as turbulence finance ecology physiology geophysics and so on the method is validated with cross correlated one and two dimensional binomial measures and multifractal random walks as an example we illustrate the method by analyzing two financial time series

multifractal analysis of managed and independent float exchange rates abstract we investigate multifractal properties of daily price changes in currency rates using the multifractal detrended fluctuation analysis mf dfa we analyze managed and independent floating currency rates in eight countries and determine the changes in multifractal spectrum when transitioning between the two regimes we find that after the transition from managed to independent float regime the changes in multifractal spectrum position of maximum and width indicate an increase in market efficiency the observed changes are more pronounced for developed countries that have a well established trading market after shuffling the series we find that the multifractality is due to both probability density function and long term correlations for managed float regime while for independent float regime multifractality is in most cases caused by broad probability density function

a multifractal detrended fluctuation description of iranian rial us dollar exchange rate the miltifractal properties and scaling behaviour of the exchange rate variations of the iranian rial against the us dollar from a daily perspective is numerically investigated for this purpose the multifractal detrended fluctuation analysis mf dfa is used through multifractal analysis the scaling exponents generalized hurst exponents generalized fractal dimensions and singularity spectrum are derived moreover contribution of two major sources of multifractality that is fat tailed probability distributions and nonlinear temporal correlations are studied

financial multifractality and its subtleties an example of dax abstract detailed study of multifractal characteristics of the financial time series of asset values and of its returns is performed using a collection of the high frequency deutsche aktienindex data the tail index the renyi exponents based on the box counting algorithm for the graph d q and the generalized hurst exponents h q are computed in parallel for short and daily return times the results indicate a more complicated nature of the stock market dynamics than just consistent multifractal

are crude oil markets multifractal evidence from mf dfa and mf ssa perspectives abstract in this article we investigated the multifractality and its underlying formation mechanisms in international crude oil markets namely brent and wti which are the most important oil pricing benchmarks globally we attempt to find the answers to the following questions 1 are those different markets multifractal 2 what are the dynamical causes for multifractality in those markets if any to answer these questions we applied both multifractal detrended fluctuation analysis mf dfa and multifractal singular spectrum analysis mf ssa based on the partition function two widely used multifractality detecting methods we found that both markets exhibit multifractal properties by means of these methods furthermore in order to identify the underlying formation mechanisms of multifractal features we destroyed the underlying nonlinear temporal correlation by shuffling the original time series thus we identified that the causes of the multifractality are influenced mainly by a nonlinear temporal correlation mechanism instead of a non gaussian distribution at last by tracking the evolution of left and right half multifractal spectra we found that the dynamics of the large price fluctuations is significantly different from that of the small ones our main contribution is that we not only provided empirical evidence of the existence of multifractality in the markets but also the sources of multifractality and plausible explanations to current literature furthermore we investigated the different dynamical price behaviors influenced by large and small price fluctuations

inhomogeneous scaling behaviors in malaysian foreign currency exchange rates in this paper we investigate the fractal scaling behaviors of foreign currency exchange rates with respect to malaysian currency ringgit malaysia these time series are examined piecewise before and after the currency control imposed in 1st september 1998 using the monofractal model based on fractional brownian motion the global hurst exponents are determined using the r s analysis the detrended fluctuation analysis and the method of second moment using the correlation coefficients the limitation of these monofractal analyses is discussed the usual multifractal analysis reveals that there exists a wide range of hurst exponents in each of the time series a new method of modelling the multifractal time series based on multifractional brownian motion with time varying hurst exponents is studied

on the wavelet transformation of fractal objects the wavelet transformation is briefly presented it is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half plane near the boundary of its domain of analyticity as an example a weierstrass like fractal function is considered for which the wavelet transform is related to a jacobi theta function some of the scalings of this theta function are analyzed and give some information about the scaling behavior of this fractal

long range correlation properties of coding and noncoding dna sequences genbank analysis an open question in computational molecular biology is whether long range correlations are present in both coding and noncoding dna or only in the latter to answer this question we consider all 33301 coding and all 29453 noncoding eukaryotic sequences each of length larger than 512 base pairs bp in the present release of the genbank to dtermine whether there is any statistically significant distinction in their long range correlation properties standard fast fourier transform fft analysis indicates that coding sequences have practically no correlations in the range from 10 bp to 100 bp spectral exponent beta 0 00 0 04 where the uncertainty is two standard deviations in contrast for noncoding sequences the average value of the spectral exponent beta is positive 0 16 0 05 which unambiguously shows the presence of long range correlations we also separately analyze the 874 coding and the 1157 noncoding sequences that have more than 4096 bp and find a larger region of power law behavior we calculate the probability that these two data sets coding and noncoding were drawn from the same distribution and we find that it is less than 10 10 we obtain independent confirmation of these findings using the method of detrended fluctuation analysis dfa which is designed to treat sequences with statistical heterogeneity such as dna s known mosaic structure patchiness arising from the nonstationarity of nucleotide concentration the near perfect agreement between the two independent analysis methods fft and dfa increases the confidence in the reliability of our conclusion

probability distribution function and multiscaling properties in the korean stock market abstract we consider the probability distribution function pdf and the multiscaling properties of the index and the traded volume in the korean stock market we observed the power law of the pdf at the fat tail region for the return volatility the traded volume and changes of the traded volume we also investigate the multifractality in the korean stock market we consider the multifractality by the detrended fluctuation analysis mfdfa we observed the multiscaling behaviors for index return traded volume and the changes of the traded volume we apply mfdfa method for the randomly shuffled time series to observe the effects of the autocorrelations the multifractality is strongly originated from the long time correlations of the time series

multifractal detrending moving average cross correlation analysis there are a number of situations in which several signals are simultaneously recorded in complex systems which exhibit long term power law cross correlations the multifractal detrended cross correlation analysis mfdcca approaches can be used to quantify such cross correlations such as the mfdcca based on the detrended fluctuation analysis mfxdfa method we develop in this work a class of mfdcca algorithms based on the detrending moving average analysis called mfxdma the performances of the proposed mfxdma algorithms are compared with the mfxdfa method by extensive numerical experiments on pairs of time series generated from bivariate fractional brownian motions two component autoregressive fractionally integrated moving average processes and binomial measures which have theoretical expressions of the multifractal nature in all cases the scaling exponents h xy extracted from the mfxdma and mfxdfa algorithms are very close to the theoretical values for bivariate fractional brownian motions the scaling exponent of the cross correlation is independent of the cross correlation coefficient between two time series and the mfxdfa and centered mfxdma algorithms have comparative performances which outperform the forward and backward mfxdma algorithms for two component autoregressive fractionally integrated moving average processes we also find that the mfxdfa and centered mfxdma algorithms have comparative performances while the forward and backward mfxdma algorithms perform slightly worse for binomial measures the forward mfxdma algorithm exhibits the best performance the centered mfxdma algorithms performs worst and the backward mfxdma algorithm outperforms the mfxdfa algorithm when the moment order q 0 we apply these algorithms to the return time series of two stock market indexes and to their volatilities for the returns the centered mfxdma algorithm gives the best estimates of h xy q since its h xy 2 is closest to 0 5 as expected and the mfxdfa algorithm has the second best performance for the volatilities the forward and backward mfxdma algorithms give similar results while the centered mfxdma and the mfxdfa algorithms fail to extract rational multifractal nature

scaling in the distribution of intertrade durations of chinese stocks abstract the distribution of intertrade durations defined as the waiting times between two consecutive transactions is investigated based upon the limit order book data of 23 liquid chinese stocks listed on the shenzhen stock exchange in the whole year 2003 a scaling pattern is observed in the distributions of intertrade durations where the empirical density functions of the normalized intertrade durations of all 23 stocks collapse onto a single curve the scaling pattern is also observed in the intertrade duration distributions for filled and partially filled trades and in the conditional distributions the ensemble distributions for all stocks are modeled by the weibull and the tsallis q exponential distributions maximum likelihood estimation shows that the weibull distribution outperforms the q exponential for not too large intertrade durations which account for more than 98 5 of the data alternatively nonlinear least squares estimation selects the q exponential as a better model in which the optimization is conducted on the distance between empirical and theoretical values of the logarithmic probability densities the distribution of intertrade durations is weibull followed by a power law tail with an asymptotic tail exponent close to 3

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

a characteristic time scale in dollar yen exchange rates abstract here we analyze tick data of yen dollar exchange using random walk methods we find that there exists a characteristic time scale approximately at 10 min according to the results at time scales shorter than 10 min the market exhibits anti persistence meaning that it self organizes so as to restore a given tendency for time scales longer than 10 min the market approaches a behavior appropriate to pure brownian motion

asymmetric multifractal detrending moving average analysis in time series of pm2 5 concentration abstract in this paper we propose the asymmetric multifractal detrending moving average analysis a mfdma method to explore the asymmetric correlation in non stationary time series the proposed method is applied to explore the asymmetric correlation of pm2 5 daily average concentration with uptrends or downtrends in china in addition shuffling and phase randomization procedures are applied to detect the sources of multifractality the results show that existences of asymmetric correlations and the asymmetric correlations are multifractal further the multifractal scaling behavior in the chinese pm2 5 is caused not only by long range correlation but also by fat tailed distribution but the major source of multifractality is fat tailed distribution

nonextensive statistical mechanics and economics abstract ergodicity this is to say dynamics whose time averages coincide with ensemble averages naturally leads to boltzmann gibbs bg statistical mechanics hence to standard thermodynamics this formalism has been at the basis of an enormous success in describing among others the particular stationary state corresponding to thermal equilibrium there are however vast classes of complex systems which accommodate quite badly or even not at all within the bg formalism such dynamical systems exhibit in one way or another nonergodic aspects in order to be able to theoretically study at least some of these systems a formalism was proposed 14 years ago which is sometimes referred to as nonextensive statistical mechanics we briefly introduce this formalism its foundations and applications furthermore we provide some bridging to important economical phenomena such as option pricing return and volume distributions observed in the financial markets and the fascinating and ubiquitous concept of risk aversion one may summarize the whole approach by saying that bg statistical mechanics is based on the entropy s bg k i p i ln p i and typically provides exponential laws for describing stationary states and basic time dependent phenomena while nonextensive statistical mechanics is instead based on the entropic form s q k 1 i p i q q 1 with s 1 s bg and typically provides for the same type of description asymptotic power laws

variety and volatility in financial markets we study the price dynamics of stocks traded in a financial market by considering the statistical properties of both a single time series and an ensemble of stocks traded simultaneously we use the n stocks traded on the new york stock exchange to form a statistical ensemble of daily stock returns for each trading day of our database we study the ensemble return distribution we find that a typical ensemble return distribution exists in most of the trading days with the exception of crash and rally days and of the days following these extreme events we analyze each ensemble return distribution by extracting its first two central moments we observe that these moments fluctuate in time and are stochastic processes themselves we characterize the statistical properties of ensemble return distribution central moments by investigating their probability density functions and temporal correlation properties in general time averaged and portfolio averaged price returns have different statistical properties we infer from these differences information about the relative strength of correlation between stocks and between different trading days last we compare our empirical results with those predicted by the single index model and we conclude that this simple model cannot explain the statistical properties of the second moment of the ensemble return distribution

scaling properties of foreign exchange volatility in this paper we investigate the scaling properties of foreign exchange volatility our methodology is based on a wavelet multi scaling approach which decomposes the variance of a time series and the covariance between two time series on a scale by scale basis through the application of a discrete wavelet transformation it is shown that foreign exchange rate volatilities follow different scaling laws at different horizons particularly there is a smaller degree of persistence in intra day volatility as compared to volatility at one day and higher scales therefore a common practice in the risk management industry to convert risk measures calculated at shorter horizons into longer horizons through a global scaling parameter may not be appropriate this paper also demonstrates that correlation between the foreign exchange volatilities is the lowest at the intra day scales but exhibits a gradual increase up to a daily scale the correlation coefficient stabilizes at scales one day and higher therefore the benefit of currency diversification is the greatest at the intra day scales and diminishes gradually at higher scales lower frequencies the wavelet cross correlation analysis also indicates that the association between two volatilities is stronger at lower frequencies

a lie algebra for closed strings spin chains and gauge theories we consider quantum dynamical systems whose degrees of freedom are described by n n matrices in the planar limit n examples are gauge theories and the m atrix theory of strings states invariant under u n are closed strings modeled by traces of products of matrices we have discovered that the u n invariant operators acting on both open and closed string states form a remarkable new lie algebra which we will call the heterix algebra the simplest special case with one degree of freedom is an extension of the virasoro algebra by the infinite dimensional general linear algebra furthermore these operators acting on closed string states only form a quotient algebra of the heterix algebra we will call this quotient algebra the cyclix algebra we express the hamiltonian of some gauge field theories like those with adjoint matter fields and dimensionally reduced pure qcd models as elements of this lie algebra finally we apply this cyclix algebra to establish an isomorphism between certain pl

universal features in the growth dynamics of complex organizations we analyze the fluctuations in the gross domestic product gdp of 152 countries for the period 1950 1992 we find that i the distribution of annual growth rates for countries of a given gdp decays with fatter tails than for a gaussian and ii the width of the distribution scales as a power law of gdp with a scaling exponent b 0 15 both findings are in surprising agreement with results on firm growth these results are consistent with the hypothesis that the evolution of organizations with complex structure is governed by similar growth mechanisms s0031 9007 98 07339 6 pacs numbers 02 50 ey 01 75 m 05 40 j 05 45 b in the study of physical systems the analysis of the scaling properties of the fluctuations has been shown to give important information regarding the underlying processes responsible for the observed macroscopic behavior in contrast most studies on the time evolution of economic time series have concentrated on average growth rates 1 18 here we investigate the possibility that the study of fluctuations in economics may also lead to a better understanding of the mechanisms responsible for the observed dynamics 19 23 we therefore analyze the fluctuations in the growth rate of the gross domestic product gdp of 152 countries during the period 1950 1992 24 we will show that i the distribution of annual growth rates for countries of a given gdp is consistent for a certain range with an exponential decay and ii the width of the distribution scales as a power law of gdp with a scaling exponent b 0 15 both findings are in surprising agreement with

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

multifractal geometry in stock market time series it has been recently noticed that time series of returns in stock markets are of multifractal multiscaling character in that context multifractality has been always evidenced by its statistical signature i e the scaling exponents associated to a related variable however a direct geometrical framework much more revealing about the underlying dynamics is possible in this paper we present the techniques allowing the multifractal decomposition we will show that there exists a particular fractal component the most singular manifold msm which contains the relevant information about the dynamics of the series it is possible to reconstruct the series at a given precision from the msm we analyze the dynamics of the msm which shows revealing features about the evolution of this type of series

activity spectrum from waiting time distribution in high frequency financial data not only returns but also waiting times between trades are random variables in this work we analyze the spectra of the waiting time processes for tick by tick trades the numerical problem strictly related with the real inversion of laplace transforms is analyzed by using tikhonov s regularization method we also analyze these spectra by a rough method using a comb of dirac s delta functions

detrending moving average algorithm for multifractals the detrending moving average dma algorithm is a widely used technique to quantify the long term correlations of nonstationary time series and the long range correlations of fractal surfaces which contains a parameter ensuremath theta determining the position of the detrending window we develop multifractal detrending moving average mfdma algorithms for the analysis of one dimensional multifractal measures and higher dimensional multifractals which is a generalization of the dma method the performance of the one dimensional and two dimensional mfdma methods is investigated using synthetic multifractal measures with analytical solutions for backward ensuremath theta 0 centered ensuremath theta 0 5 and forward ensuremath theta 1 detrending windows we find that the estimated multifractal scaling exponent ensuremath tau q and the singularity spectrum f ensuremath alpha are in good agreement with the theoretical values in addition the backward mfdma method has the best performance which provides the most accurate estimates of the scaling exponents with lowest error bars while the centered mfdma method has the worse performance it is found that the backward mfdma algorithm also outperforms the multifractal detrended fluctuation analysis the one dimensional backward mfdma method is applied to analyzing the time series of shanghai stock exchange composite index and its multifractal nature is confirmed

multifractal behavior of the korean stock market index kospi abstract we investigate multifractality in the korean stock market index kospi the generalized qth order height height correlation function shows multiscaling properties there are two scaling regimes with a crossover time around t c 40 min we consider the original data sets and the modified data sets obtained by removing the daily jumps which occur due to the difference between the closing index and the opening index to clarify the origin of the multifractality we also smooth the data through convolution with a gaussian function after convolution we observe that the multifractality disappears in the short time scaling regime t t c but remains in the long time scaling regime t t c regardless of whether or not the daily jumps are removed we suggest that multifractality in the short time scaling regime is caused by the local fluctuations of the stock index but the multifractality in the long time scaling regime appears to be due to the intrinsic trading properties such as herding behavior information outside the market the long memory of the volatility and the nonlinear dynamics of the stock market

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

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

estimating the distribution of volatility of realized stock returns and exchange rate changes abstract realized stock return volatility is modelled with a distribution based on the laplace distribution the moment properties of suggested volatility distribution are derived the properties of distribution correspond to the empirical regularities found in the finance literature ml estimator for is also provided the advantage of laplace approach lies in estimating from returns distribution f x directly instead of volatility distribution based on bias sensitive standard deviation estimates the goodness to fit tests with 5 day standard deviations of daily hex closing price returns in period 3 1 1983 4 3 2003 daily s p500 closing stock index returns in period 1 3 1950 27 3 2003 and daily usd euro exchange rate changes in period 28 12 1978 28 2 2003 support the suggested volatility distribution model

fitting the empirical distribution of intertrade durations based on the analysis of a tick by tick data set used in the previous work by one of the authors djia stocks traded at nyse in october 1999 in this paper we reject the hypothesis that tails of the empirical intertrade distribution are described by a power law we further argue that the tsallis q exponentials are a viable tool for fitting and describing the unconditional distribution of empirical intertrade durations and they compare well to the weibull distribution

scaling self similarity and multifractality in fx markets abstract this paper presents an empirical investigation of scaling and multifractal properties of us dollar deutschemark usd dem returns the data set is ten years of 5 min returns the cumulative return distributions of positive and negative tails at different time intervals are linear in the double logarithmic space this presents strong evidence that the usd dem returns exhibit power law scaling in the tails to test the multifractal properties of usd dem returns the mean moment of the absolute returns as a function of time intervals is plotted for different powers of absolute returns these moments show different slopes for these powers of absolute returns the nonlinearity of the scaling exponent indicates that the returns are multifractal

anti correlation and multifractal features of spain electricity spot market we use multifractal detrended fluctuation analysis mf dfa to numerically investigate correlation persistence multifractal properties and scaling behavior of the hourly spot prices for the spain electricity exchange compania o peradora del mercado de electricidad omel through multifractal analysis fluctuations behavior the scaling exponents and generalized hurst exponents are studied moreover contribution of fat tailed probability distributions and nonlinear temporal correlations to multifractality is studied

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

algorithm to estimate the hurst exponent of high dimensional fractals we propose an algorithm to estimate the hurst exponent of high dimensional fractals based on a generalized high dimensional variance around a moving average low pass filter as working examples we consider rough surfaces generated by the random midpoint displacement and by the cholesky levinson factorization algorithms the surrogate surfaces have hurst exponents ranging from 0 1 to 0 9 with step 0 1 and different sizes the computational efficiency and the accuracy of the algorithm are also discussed

differentiating intraday seasonalities through wavelet multi scaling it is well documented that strong intraday seasonalities may induce distortions in the estimation of volatility models these seasonalities are also the dominant source for the underlying misspecifications of the various volatility models therefore an obvious route is to filter out the underlying intraday seasonalities from the data in this paper we propose a simple method for intraday seasonality extraction that is free of model selection parameters which may affect other intraday seasonality filtering methods our methodology is based on a wavelet multi scaling approach which decomposes the data into its low and high frequency components through the application of a non decimated discrete wavelet transform it is simple to calculate does not depend on a particular model selection criterion or model specific parameter choices the proposed filtering method is translation invariant has the ability to decompose an arbitrary length series without boundary adjustments is associated with a zero phase filter and is circular being circular helps to preserve the entire sample unlike other two sided filters where data loss occurs from the beginning and the end of the studied sample

common scaling patterns in intertrade times of u s stocks we analyze the sequence of time intervals between consecutive stock trades of thirty companies representing eight sectors of the u s economy over a period of 4 yrs for all companies we find that i the probability density function of intertrade times may be fit by a weibull distribution ii when appropriately rescaled the probability densities of all companies collapse onto a single curve implying a universal functional form iii the intertrade times exhibit power law correlated behavior within a trading day and a consistently greater degree of correlation over larger time scales in agreement with the correlation behavior of the absolute price returns for the corresponding company and iv the magnitude series of intertrade time increments is characterized by long range power law correlations suggesting the presence of nonlinear features in the trading dynamics while the sign series is anticorrelated at small scales our results suggest that independent of industry sector market capitalization and average level of trading activity the series of intertrade times exhibit possibly universal scaling patterns which may relate to a common mechanism underlying the trading dynamics of diverse companies further our observation of long range power law correlations and a parallel with the crossover in the scaling of absolute price returns for each individual stock support the hypothesis that the dynamics of transaction times may play a role in the process of price formation

detrending moving average algorithm a closed form approximation of the scaling law abstract the hurst exponent h of long range correlated series can be estimated by means of the detrending moving average dma method the computational tool on which the algorithm is based is the generalized variance dma 2 1 n n i n n y i y n i 2 with y n i 1 n k 0 n y i k being the average over the moving window n and n the dimension of the stochastic series y i the ability to yield h relies on the property of dma 2 to vary as n 2 h over a wide range of scales e alessio a carbone g castelli v frappietro eur j phys b 27 2002 197 here we give a closed form proof that dma 2 is equivalent to c h n 2 h and provide an explicit expression for c h we furthermore compare the values of c h with those obtained by applying the dma algorithm to artificial self similar signals

introducing false eur and false eur exchange rates abstract the euro eur is a new currency introduced by the european community its exchange rate is very puzzling we have invented a false euro feur dating back to 1993 and have derived the exchange rates of the feur with respect to currencies not belonging to the eur i e dkk chf jpy and usd this allows us to search for correlations between the fluctuations preexisting to the introduction of eur and present ones in such financial data the detrended fluctuation analysis dfa statistical method is used this leads to assume a power law behavior i e a scaling hypothesis through an exponent the latter has demonstrated its usefulness for the investigations of long range power law correlations in several types of financial sequences our findings show that the exponent interestingly characterizes fractional brownian motion of the currency exchange rates between eur and dkk over a 25 day interval and usual brownian motion otherwise and for the three other investigated exchange rates we can devise an investment strategy based on the local technique and obtain appreciable gains for the time being