Memory effect and multifractality of cross-correlations in financial markets

We introduce an instantaneous and an average instantaneous cross-correlation function to detect the temporal cross-correlations between individual stocks based on the daily data of the United States and the Chinese stock markets. The memory effect of the instantaneous cross-correlations is investigated by applying the detrended fluctuation analysis (DFA), where the DFA exponents can be partly explained by the correlation function from the common sense. Long-range memory is observed for the average instantaneous cross-correlations, and persists up to a month magnitude of timescale for the United States stock market and half a month magnitude of timescale for the Chinese stock market. In addition, multifractal nature is investigated by a multifractal detrended fluctuation analysis.     

Two-dimensional random-random walks: Dynamical exponents in a quenched directed model

This paper presents a study of the dynamics of a particle undergoing a directed random walk in a two-dimensional disordered square lattice. We derive the asymptotical behaviors of the coordinate and of the mean square displacement. All the dynamical exponents are calculated both in the normal and the anomalous regimes. It is shown that, as contrasted to the one-dimensional case, the so-called quenched and annealed diffusion constants indeed coincide.     

Volatilities, traded volumes, and the hypothesis of price increments in derivative securities

A detrended fluctuation analysis (DFA) is applied to the statistics of Korean treasury bond (KTB) futures from which the logarithmic increments, volatilities, and traded volumes are estimated over a specific time lag. In this study, the logarithmic increment of futures prices has no long-memory property, while the volatility and the traded volume exhibit the existence of the long-memory property. To analyze and calculate whether the volatility clustering is due to a inherent higher-order correlation not detected by with the direct application of the DFA to logarithmic increments of KTB futures, it is of importance to shuffle the original tick data of future prices and to generate a geometric Brownian random walk with the same mean and standard deviation. It was found from a comparison of the three tick data that the higher-order correlation inherent in logarithmic increments leads to volatility clustering. Particularly, the result of the DFA on volatilities and traded volumes can be supported by the hypothesis of price changes.     

Non-linear characteristics and long-range correlations in Asian stock markets

We test several non-linear characteristics of Asian stock markets, which indicates the failure of efficient market hypothesis and shows the essence of fractal of the financial markets. In addition, by using the method of detrended fluctuation analysis (DFA) to investigate the long range correlation of the volatility in the stock markets, we find that the crossover phenomena exist in the results of DFA. Further, in the region of small volatility, the scaling behavior is more complicated; in the region of large volatility, the scaling exponent is close to 0.5, which suggests the market is more efficient. All these results may indicate the possibility of characteristic multifractal scaling behaviors of the financial markets.     

On statistical properties of traded volume in financial markets

In this article we study the dependence degree of the traded volume of the Dow Jones 30 constituent equities by using a nonextensive generalised form of the Kullback-Leibler information measure. Our results show a slow decay of the dependence degree as a function of the lag. This feature is compatible with the existence of non-linearities in this type time series. In addition, we introduce a dynamical mechanism whose associated stationary probability density function (PDF) presents a good agreement with the empirical results.     

Detrended fluctuation analysis on spot and futures markets of West Texas Intermediate crude oil

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.     

Symmetry alteration of ensemble return distribution in crash and rally days of financial markets

We select the n stocks traded in the New York Stock Exchange and we form a statistical ensemble of daily stock returns for each of the k trading days of our database from the stock price time series. We study the ensemble return distribution for each trading day and we find that the symmetry properties of the ensemble return distribution drastically change in crash and rally days of the market. In crash and rally days, the distribution becomes asymmetric. In particular for crashes the positive tail is steeper than the negative one whereas the reverse is observed in rally days. Received 25 February 2000     

Temporal-spatial diversities of long-range correlation for relative humidity over China

Long-range correlations of daily relative humidity anomaly records from 191 weather stations over China during 1951-2000 are analyzed by means of fluctuation analysis (FA) and detrended fluctuation analysis (DFA). The information about trends in the relative humidity records can be obtained by comparing the FA curve with DFA curves. The daily relative humidity fluctuations are found to be power-law correlated and their average scaling exponent is higher than that of the temperature fluctuations, indicating that the relative humidity fluctuations take different statistical behavior from other meteorological quantities and there exists a stronger persistence in the relative humidity fluctuations. Furthermore, it is also found that these power-law scaling properties vary from station to station and show both spatial and temporal diversities, which may be explained by a proposed mechanism.     

Dynamical properties of a logistic growth model with cross-correlated noises

A logistic growth model driven by additive and multiplicative noises which are correlated with each other is investigated. Using the Novikov theorem and the projection operator method, we obtain the analytic expressions of the stationary probability distribution p_st(x), the relaxation time T_c, and the normalized correlation function C(s) of this system. The computational results show that the relaxation time T_c increases as the cross-correlated time τ increases, but decreases while the cross-correlated strength λ increases. The relationship between the relaxation time C(s) and the decay time s is given. Correlation time τ and correlation strength λ play an opposite role on dynamic properties in this logistic growth model.     

Fractal dimensionality and microscopic analysis of the dynamics in simple liquids

Summary The idea to evaluate the ?average fractal dimension? of the motion in a condensed phase system, on the basis of the analysis of the fractal properties of the trajectories of its microscopic components has been further developed. The fractal dimension of particle trajectories, evaluated through the correlation density integral, is formally related with the self-part of the dynamic structure factorG _s(r, t) (the self-part of the Van Hove function); so far a bridge between fractal and thermodynamical properties has been built up.

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Riassunto L’idea di stimare la ?dimensione frattale media? del moto in un sistema nella fase condensata è ulteriormente sviluppata mediante l’analisi delle proprietà frattali delle traiettorie delle componenti microscopiche. La dimensione frattale delle traiettorie delle particelle, stimata attraverso le correlazioni di densità, obbedisce ad una relazione formale con la self-part del fattore di struttura dinamicoG _s(r, t); in questo modo si mostra un legame tra le proprietà frattali e quelle termodinamiche.

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Резюме Развивается идея оценить ?среднюю фрактальную размерность? движения в конденсированной фазовой системе на основе анализа фрактальных свойств частицы траекторий для микроскопических компонент. Фрактальная размерность траекторий, определенная через корреляционный интеграл плотности, подчиняется формальному соотношеию с собтвенной частью динамического структурного фактораG _s(r,t) (собственная часть функции Ван Хава). Указывается связь между фрактальнымк и термодинамическими свойствами.

     

Unified approach to quantum fluctuations in tunnelling including dissipation

Summary A simple procedure for evaluating quantum fluctuations at zero temperature has been applied to derive the decay rate for a metastable state in strongly anharmonic potentials (quartic and cubic). We also derive the tunnelling splitting and the energy shift in symmetrical and nearly symmetrical double-well potentials. Dissipation is then considered for the decay of a metastable state, both in the limit of weak and strong damping.

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Riassunto Un metodo semplificato per la valutazione delle fluttuazioni quantistiche, a temperatura zero, è stato impiegato per ottenere la velocità di decadimento di uno stato metastabile in potenziali fortemente anarmonici (quartico e cubico). Si ricavano pure la separazione per tunnelling e lo spostamento di energia nel caso di potenziale a doppio pozzo, simmetrico e quasi simmetrico. Si considerano poi effetti dissipati per il decadimento di uno stato metastabile, sia nel limite di debole che di forte smorzamento.

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Резюме Простая процедура оценки квантовых флуктуаций при нулевой температуре применяется для вывода скорости распада метастабильного состояния в сильно ангармонических потенциалах (четвертого и кубичкского порядков). Мы также выводим расщепление за счет туннелирования и сдвиг энергии в потенциалах симметричной и почти симметричной двойной ямы. Затем рассматривается влияние диссипации при распаде метастабильного состояния в пределе слабого и сильного затухания.

     

Diffusion on a random comb: Distribution function of the survival probability

We investigate the distribution functionQ(P) describing the survival probability on a comb consisting of a backbone with lateral, randomly disconnected infinite branches. Two different regimes are analyzed in some detail: (i) at short times,Q(P) is shown to have a self-similar structure (devil's staircase); (ii) at large times, this function becomes smooth and tends toward a rather well-defined unit step function. The disorder-averaged survival probability <p ₀(t)> is expected to decrease ast ^–3/4 at large times, whereas the relative fluctuations of the sample-dependentp ₀(t) display a very slow decay in time, going to zero liket ^–1/8.     

Identifying financial crises in real time

Following the thermodynamic formulation of a multifractal measure that was shown to enable the detection of large fluctuations at an early stage, here we propose a new index which permits us to distinguish events like financial crises in real time. We calculate the partition function from which we can obtain thermodynamic quantities analogous to the free energy and specific heat. The index is defined as the normalized energy variation and it can be used to study the behavior of stochastic time series, such as financial market daily data. Famous financial market crashes–Black Thursday (1929), Black Monday (1987) and the subprime crisis (2008)–are identified with clear and robust results. The method is also applied to the market fluctuations of 2011. From these results it appears as if the apparent crisis of 2011 is of a different nature to the other three. We also show that the analysis has forecasting capabilities.     

Missing ordinal patterns in correlated noises

Recent research aiming at the distinction between deterministic or stochastic behavior in observational time series has looked into the properties of the “ordinal patterns” [C. Bandt, B. Pompe, Phys. Rev. Lett. 88 (2002) 174102]. In particular, new insight has been obtained considering the emergence of the so-called “forbidden ordinal patterns” [J.M. Amigó, S. Zambrano, M.A. F Sanjuán, Europhys. Lett. 79 (2007) 50001]. It was shown that deterministic one-dimensional maps always have forbidden ordinal patterns, in contrast with time series generated by an unconstrained stochastic process in which all the patterns appear with probability one. Techniques based on the comparison of this property in an observational time series and in white Gaussian noise were implemented. However, the comparison with correlated stochastic processes was not considered. In this paper we used the concept of “missing ordinal patterns” to study their decay rate as a function of the time series length in three stochastic processes with different degrees of correlation: fractional Brownian motion, fractional Gaussian noise and, noises with f^−k power spectrum. We show that the decay rate of “missing ordinal patterns” in these processes depend on their correlation structures. We finally discuss the implications of the present results for the use of these properties as a tool for distinguishing deterministic from stochastic processes.     

Blocks adjustment—reduction of bias and variance of detrended fluctuation analysis using Monte Carlo simulation

The length of minimal and maximal blocks equally distant on log-log scale versus fluctuation function considerably influences bias and variance of DFA. Through a number of extensive Monte Carlo simulations and different fractional Brownian motion/fractional Gaussian noise generators, we found the pair of minimal and maximal blocks that minimizes the sum of mean-squared error of estimated Hurst exponents for the series of length . Sensitivity of DFA to sort-range correlations was examined using ARFIMA(p,d,q) generator. Due to the bias of the estimator for anti-persistent processes, we narrowed down the range of Hurst exponent to      

Injected Power Fluctuations in Langevin Equation

In this paper, we consider the Langevin equation from an unusual point of view, that is as an archetype for a dissipative system driven out of equilibrium by an external excitation. Using path integral method, we compute exactly the probability density function of the power (averaged over a time interval of length ) injected (and dissipated) by the random force into a Brownian particle driven by a Langevin equation. The resulting distribution, as well as the associated large deviation function, display strong asymmetry, whose origin is explained. Connections with the so-called Fluctuation Theorem are thereafter discussed. Finally, considering Langevin equations with a pinning potential, we show that the large deviation function associated with the injected power is completely insensitive to the presence of a potential.     

Space-magnitude dependent scaling behaviour in seismic interevent series revealed by detrended fluctuation analysis

The scaling behaviour of the 1981-2007 seismicity data in central Italy, which is one of the most seismically active areas in Italy is investigated. In particular we examined the earthquakes located in a circular area centred on the epicentre of the strongest event, occurred in September 26, 1997 (duration magnitude M_D=5.8). On the base of the detrended fluctuation analysis (DFA), we found that in the magnitude range between 2.5 and 2.9 the scaling exponents fall into disjoint sets for events relatively close and far from the epicentre of the strongest event.     

Long-range dependence in tree-ring width time series of Austrocedrus Chilensis revealed by means of the detrended fluctuation analysis

By using the detrended fluctuation analysis and detrended moving average method, 823 time series of tree-ring widths in Austrocedrus Chilensis in Patagonia were analyzed. The tree-ring widths of A. Chilensis have been widely used for climatological studies. The results point out to the presence of significant scaling in the temporal fluctuations of tree-ring, which is not due to singular probability density function of the widths but due to the presence of long-range correlations. Such results are in good agreement with those concerning the evidence of long-range dependencies in weather time series.