To be published in Phys. Rev. E (2005).We present a simple point process model of 1/f β noise, covering different values of the exponent

Starting from the developed generalized point process model of 1/f noise (B. Kaulakys et al, Phys. Rev. E 71 (2005) 051105; cond-mat/0504025) we derive the nonlinear stochastic differential equations for the signal exhibiting 1/f β noise and 1/x λ distribution density of the signal intensity with different values of β and λ. The processes with 1/f β are demonstrated by the numerical solution of the derived equations with the appropriate restriction of the diffusion of the signal in some finite interval. The proposed consideration may be used for modeling and analysis of stochastic processes in different systems with the power-law distributions, long-range memory or with the elements of self-organization.

Prevention of a quantum system's time evolution by repetitive, frequent measurements of the system's state has been called the quantum Zeno effect (or paradox). Here we investigate theoretically and numerically the effect of repeated measurements on the quantum dynamics of the multilevel systems that exhibit the quantum localization of the classical chaos. The analysis is based on the wave function and Schroedinger equation, without introduction of the density matrix. We show how the quantum Zeno effect in simple few-level systems can be recovered and understood by formal modeling the measurement effect on the dynamics by randomizing the phases of the measured states. Further the similar analysis is extended to investigate of the dynamics of multilevel systems driven by an intense external force and affected by frequent measurement. We show that frequent measurements of such quantum systems results in the delocalization of the quantum suppression of the classical chaos. This result is the opposite of the quantum Zeno effect. The phenomenon of delocalization of the quantum suppression and restoration of the classical-like time evolution of these quasiclassical systems, owing to repetitive frequent measurements, can therefore be called the 'quantum anti-Zeno effect'. From this analysis we furthermore conclude that frequently or continuously observable quasiclassical systems evolve basically in a classical manner.Comment: 12 pages with 2 figure

We are looking for the agent-based treatment of the financial markets considering necessity to build bridges between microscopic, agent based, and macroscopic, phenomenological modeling. The acknowledgment that agent-based modeling framework, which may provide qualitative and quantitative understanding of the financial markets, is very ambiguous emphasizes the exceptional value of well defined analytically tractable agent systems. Herding as one of the behavior peculiarities considered in the behavioral finance is the main property of the agent interactions we deal with in this contribution. Looking for the consentaneous agent-based and macroscopic approach we combine two origins of the noise: exogenous one, related to the information flow, and endogenous one, arising form the complex stochastic dynamics of agents. As a result we propose a three state agent-based herding model of the financial markets. From this agent-based model we derive a set of stochastic differential equations, which describes underlying macroscopic dynamics of agent population and log price in the financial markets. The obtained solution is then subjected to the exogenous noise, which shapes instantaneous return fluctuations. We test both Gaussian and q-Gaussian noise as a source of the short term fluctuations. The resulting model of the return in the financial markets with the same set of parameters reproduces empirical probability and spectral densities of absolute return observed in New York, Warsaw and NASDAQ OMX Vilnius Stock Exchanges. Our result confirms the prevalent idea in behavioral finance that herding interactions may be dominant over agent rationality and contribute towards bubble formation.

We extend Kirman's model by introducing variable event time scale. The proposed flexible time scale is equivalent to the variable trading activity observed in financial markets. Stochastic version of the extended Kirman's agent based model is compared to the non-linear stochastic models of long-range memory in financial markets. Agent based model providing matching macroscopic description serves as a microscopic reasoning of the earlier proposed stochastic model exhibiting power law statistics.

The motion of an electron of a classical hydrogenic atom in an oscillating electric field is studied theoretically. An analysis is provided, based on the iterative (mapping) forms of the classical equations of motion in perturbation theory and the adiabatic approximation. This greatly facilitates the numerical investigation of stochasticity and the ionisation process and allows the approximate analytical estimation of the threshold field strengths for the onset of chaos and of the diffusion coefficient of the electron in energy space. The method is asymptotically exact at high field frequencies and gives a good approximation for medium and low frequencies. The adiabatic approximation describes well the approach of the stochastic ionisation threshold field strength to the static field ionisation threshold. From the quantum mechanical point of view the ionisation is a result of the great number of one-photon transitions in the strongly perturbed spectrum of the atom. This results in the diffusion of the electron in energy space identical to the diffusion due to stochastic classical motion. The estimation of the mean time of diffusive ionisation is also given.

We present a nonlinear stochastic differential equation (SDE) which mimics the probability density function (PDF) of the return and the power spectrum of the absolute return in financial markets. Absolute return as a measure of market volatility is considered in the proposed model as a long-range memory stochastic variable. The SDE is obtained from the analogy with earlier proposed model of trading activity in the financial markets and generalized within the nonextensive statistical mechanics framework. The proposed stochastic model generates time series of the return with two power law statistics, i.e., the PDF and the power spectral density, reproducing the empirical data for the one minute trading return in the NYSE.Comment: 9 pages, 3 figure

Signals consisting of a sequence of pulses show that inherent origin of the 1/f noise is a Brownian fluctuation of the average interevent time between subsequent pulses of the pulse sequence. In this paper we generalize the model of interevent time to reproduce a variety of self-affine time series exhibiting power spectral density S(f) scaling as a power of the frequency f. Furthermore, we analyze the relation between the power-law correlations and the origin of the power-law probability distribution of the signal intensity. We introduce a stochastic multiplicative model for the time intervals between point events and analyze the statistical properties of the signal analytically and numerically. Such model system exhibits power-law spectral density S(f)~1/f**beta for various values of beta, including beta=1/2, 1 and 3/2. Explicit expressions for the power spectra in the low frequency limit and for the distribution density of the interevent time are obtained. The counting statistics of the events is analyzed analytically and numerically, as well. The specific interest of our analysis is related with the financial markets, where long-range correlations of price fluctuations largely depend on the number of transactions. We analyze the spectral density and counting statistics of the number of transactions. The model reproduces spectral properties of the real markets and explains the mechanism of power-law distribution of trading activity. The study provides evidence that the statistical properties of the financial markets are enclosed in the statistics of the time interval between trades. A multiplicative point process serves as a consistent model generating this statistics.Comment: 10 pages, 3 figure