Trading activity as driven Poisson process: comparison with empirical data

We propose the point process model as the Poissonian-like stochastic sequence with slowly diffusing mean rate and adjust the parameters of the model to the empirical data of trading activity for 26 stocks traded on NYSE. The proposed scaled stochastic differential equation provides the universal description of the trading activities with the same parameters applicable for all stocks.Comment: 9 pages, 5 figures, proceedings of APFA

Nonlinear stochastic models of 1/f noise and power-law distributions

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^{\beta}$noise and$1/x^{\lambda}$distribution density of the signal intensity with different values of$\beta$and$\lambda$. The processes with$1/f^{\beta}$ 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.Comment: 6 pages, 6 figures, presented at the 3rd NEXT-SigmaPhi International Conference (13-18 August 2005, Kolymbari CRETE

Modelling financial markets by the multiplicative sequence of trades

We introduce the stochastic multiplicative point process modelling trading activity of financial markets. Such a model system exhibits power-law spectral density S(f) ~ 1/f**beta, scaled as power of frequency for various values of beta between 0.5 and 2. Furthermore, we analyze the relation between the power-law autocorrelations and the origin of the power-law probability distribution of the trading activity. The model reproduces the spectral properties of trading activity and explains the mechanism of power-law distribution in real markets.Comment: 6 pages, 2 figure

Quantum anti-Zeno effect

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

Fluctuation analysis of the three agent groups herding model

We derive a system of stochastic differential equations simulating the dynamics of the three agent groups with herding interaction. Proposed approach can be valuable in the modeling of the complex socio-economic systems with similar composition of the agents. We demonstrate how the sophisticated statistical features of the absolute return in the financial markets can be reproduced by extending the herding interaction of the agents and introducing the third agent state. As well we consider possible extension of proposed herding model introducing additional exogenous noise. Such consistent microscopic and macroscopic model precisely reproduces empirical power law statistics of the return in the financial markets.Comment: 9 pages, 2 figure

Spurious memory in non-equilibrium stochastic models of imitative behavior

The origin of the long-range memory in the non-equilibrium systems is still an open problem as the phenomenon can be reproduced using models based on Markov processes. In these cases a notion of spurious memory is introduced. A good example of Markov processes with spurious memory is stochastic process driven by a non-linear stochastic differential equation (SDE). This example is at odds with models built using fractional Brownian motion (fBm). We analyze differences between these two cases seeking to establish possible empirical tests of the origin of the observed long-range memory. We investigate probability density functions (PDFs) of burst and inter-burst duration in numerically obtained time series and compare with the results of fBm. Our analysis confirms that the characteristic feature of the processes described by a one-dimensional SDE is the power-law exponent $3/2$ of the burst or inter-burst duration PDF. This property of stochastic processes might be used to detect spurious memory in various non-equilibrium systems, where observed macroscopic behavior can be derived from the imitative interactions of agents.Comment: 11 pages, 5 figure

Control of the socio-economic systems using herding interactions

Collective behavior of the complex socio-economic systems is heavily influenced by the herding, group, behavior of individuals. The importance of the herding behavior may enable the control of the collective behavior of the individuals. In this contribution we consider a simple agent-based herding model modified to include agents with controlled state. We show that in certain case even the smallest fixed number of the controlled agents might be enough to control the behavior of a very large system.Comment: 8 pages, 3 figure