Dariusz Grech

Can One Make Any Crash Prediction in Finance Using the Local Hurst Exponent Idea

We apply the Hurst exponent idea for investigation of DJIA index time-series data. The behavior of the local Hurst exponent prior to drastic changes in financial series signal is analyzed. The optimal length of the time-window over which this exponent can be calculated in order to make some meaningful predictions is discussed. Our prediction hypothesis is verified with examples of 1929 and 1987 crashes, as well as with more recent phenomena in stock market from the period 1995 to 2003. Some interesting agreements are found.

On the scaling ranges of detrended fluctuation analysis for long-term memory correlated short series of data

We examine the scaling regime for the detrended fluctuation analysis (DFA)—the most popular method used to detect the presence of long-term memory in data and the fractal structure of time series. First, the scaling range for DFA is studied for uncorrelated data as a function of time series length L and the correlation coefficient of the linear regression R2 at various confidence levels. Next, a similar analysis for artificial short series of data with long-term memory is performed. In both cases the scaling range λ is found to change linearly—both with L and R2. We show how this dependence can be generalized to a simple unified model describing the relation λ=λ(L,R2,H) where H (1/2≤H≤1) stands for the Hurst exponent of the long range autocorrelated signal. Our findings should be useful in all applications of DFA technique, particularly for instantaneous (local) DFA where a huge number of short time series has to be analyzed at the same time, without possibility of checking the scaling range in each of them separately.

On the multifractal effects generated by monofractal signals

We study quantitatively the level of false multifractal signal one may encounter while analyzing multifractal phenomena in time series within multifractal detrended fluctuation analysis (MF-DFA). The investigated effect appears as a result of finite length of used data series and is additionally amplified by the long-term memory the data eventually may contain. We provide the detailed quantitative description of such apparent multifractal background signal as a threshold in spread of generalized Hurst exponent values Δh or a threshold in the width of multifractal spectrum Δα below which multifractal properties of the system are only apparent, i.e. do not exist, despite Δα≠0 or Δh≠0. We find this effect quite important for shorter or persistent series and we argue it is linear with respect to autocorrelation exponent γ. Its strength decays according to power law with respect to the length of time series. The influence of basic linear and nonlinear transformations applied to initial data in finite time series with various levels of long memory is also investigated. This provides additional set of semi-analytical results. The obtained formulas are significant in any interdisciplinary application of multifractality, including physics, financial data analysis or physiology, because they allow to separate the ‘true’ multifractal phenomena from the apparent (artificial) multifractal effects. They should be a helpful tool of the first choice to decide whether we do in particular case with the signal with real multiscaling properties or not.

The amazing cases of motion with friction

The paper describes the behaviour of a simple mechanical system, which should help students (or teachers) to understand and clarify the importance of relative motion of two surfaces when kinetic friction is present. We show that despite the simplicity of this system, the peculiar interplay between friction forces, tension forces and gravity leads to physical solutions exceeding in many cases most intuitive expectations. These are discussed in detail. The problem is intended to be solved in a theoretical framework as an example, which helps to understand better the physical background of kinetic friction phenomena.

Global and Local Approaches Describing Critical Phenomena on the Developing and Developed Financial Markets

We define and confront global and local methods to analyze the financial crash-like events on the financial markets from the critical phenomena point of view. These methods are based respectively on the analysis of log-periodicity and on the local fractal properties of financial time series in the vicinity of phase transitions (crashes). The log-periodicity analysis is made in a daily time horizon, for the whole history (1991–2008) of Warsaw Stock Exchange Index (WIG) connected with the largest developing financial market in Europe. We find that crash-like events on the Polish financial market are described better by the log-divergent price model decorated with log-periodic behavior than by the power-law-divergent price model usually discussed in log-periodic scenarios for developed markets. Predictions coming from log-periodicity scenario are verified for all main crashes that took place in WIG history. It is argued that crash predictions within log-periodicity model strongly depend on the amount of data taken to make a fit and therefore are likely to contain huge inaccuracies. Next, this global analysis is confronted with the local fractal description. To do so, we provide calculation of the so-called local (time dependent) Hurst exponent H loc for the WIG time series and for main US stock market indices like DJIA and S&P 500. We point out dependence between the behavior of the local fractal properties of financial time series and the crashes appearance on the financial markets. We conclude that local fractal method seems to work better than the global approach – both for developing and developed markets. The very recent situation on the market, particularly related to the Fed intervention in September 2007 and the situation immediately afterwards is also analyzed within fractal approach. It is shown in this context how the financial market evolves through different phases of fractional Brownian motion. Finally, the current situation on American market is analyzed in fractal language. This is to show how far we still are from the end of recession and from the beginning of a new boom on US financial market or on other world leading stocks.

NUMERICAL REEXAMINATION OF SUPERSYMMETRIC AND NONSUPERSYMMETRIC SU(5)-LIKE GRAND UNIFIED MODELS

The significance of numerical analysis in both nonsupersymmetric and supersymmetric Grand Unified Theories is pointed out. The exact analytical and numerical analysis we present shows a need of larger corrections to the values of unifying parameters, i.e. sin2 θw, Mx, τp than those often quoted in literature. When an unmodified nonsupersymmetric version of SU(5) is considered we show that numerical computation allows some of the models still to be experimentally admissible. The difference between analytical and numerical results for the supersymmetric SU(5) model is also stressed. In particular, corrections due to the mass threshold of additional generations or supersymmetric particles are calculated both analytically and numerically at the two-loop level. We found them far more important for the final values of sin2 θw, Mx and τp than the effects of Higgs-Yukawa couplings between scalars and fermions.

A PLACE FOR TECHNICOLOR IN GUT

A possibility of incorporation of the extra QCD-like forces into the unified models of interactions is analysed. It is shown that this is possible only for models with the naive number of fermion generations gN=0, but thanks to the new QCD-like symmetry, the observable number of them can be as large as gR=8. All such models are listed. Then the asymptotic behavior of such theories including the effects of decoupling is investigated. It is found that in none of the unified models the additional strong forces are asymptotically free. This excludes them as possible candidates for technicolor interactions causing dynamical breaking of the standard SU(3)c× SU(2)L×U(1)y model.

Numerical simulation of the Perrin-like experiments

A simple model of the random Brownian walk of a spherical mesoscopic particle in viscous liquids is proposed. The model can be solved analytically and simulated numerically. The analytic solution gives the known Einstein–Smoluchowski diffusion law r2 = 2Dt, where the diffusion constant D is expressed by the mass and geometry of a particle, the viscosity of a liquid and the average effective time between consecutive collisions of the tracked particle with liquid molecules. The latter allows us to make a simulation of the Perrin experiment and to verify in the detailed study the influence of the statistics on the expected theoretical results. To avoid the problem of small statistics causing departures from the diffusion law we introduce in the second part of the paper the idea of the so-called artificially increased statistics (AIS), and we prove that, within this method of experimental data analysis, one can confirm the diffusion law and get a good prediction for the diffusion constant even if trajectories of just a few particles immersed in a liquid are considered.

SECOND-LOOP CORRECTIONS FROM SUPERHEAVY GAUGE SECTOR TO GAUGE COUPLING UNIFICATION

We deal with extensions of the Standard Model adding horizontal interactions between particle generations. We calculate two-loop corrections caused by the presence of coupling between hypothetical horizontal gauge bosons and matter field at high energy. It is shown that the coupling of such extra bosons does not affect up to two-loop level the positive features of unified and extended Standard Model with horizontal symmetry discussed in former publications. Corrections from bosonic horizontal sector make up about tenth part of those caused by fermionic sector. Although small, they are larger than the accuracy of some electroweak measurements and therefore they might be important for future verification of various proposed horizontal models.

Construction of grand unified theories

Abstract Systematic treatment of the theoretical and aesthetic requirements employed in the construction of unifying groups and their representations is given. Formulating the extensive set of conditions and giving reasons for them we derive a much more compact class of unified models than the one already discussed in the literature. It is shown that most models based on nonunitary groups and those with exotic representations are excluded. On the other hand, the permissible class of unitary unifying models is large but well ordered, thus making them usefull for future applications.