34 research outputs found
Long-range memory model of trading activity and volatility
Earlier we proposed the stochastic point process model, which reproduces a
variety of self-affine time series exhibiting power spectral density S(f)
scaling as power of the frequency f and derived a stochastic differential
equation with the same long range memory properties. Here we present a
stochastic differential equation as a dynamical model of the observed memory in
the financial time series. The continuous stochastic process reproduces the
statistical properties of the trading activity and serves as a background model
for the modeling waiting time, return and volatility. Empirically observed
statistical properties: exponents of the power-law probability distributions
and power spectral density of the long-range memory financial variables are
reproduced with the same values of few model parameters.Comment: 12 pages, 5 figure
Point process model of 1/f noise versus a sum of Lorentzians
We present a simple point process model of noise, covering
different values of the exponent . The signal of the model consists of
pulses or events. The interpulse, interevent, interarrival, recurrence or
waiting times of the signal are described by the general Langevin equation with
the multiplicative noise and stochastically diffuse in some interval resulting
in the power-law distribution. Our model is free from the requirement of a wide
distribution of relaxation times and from the power-law forms of the pulses. It
contains only one relaxation rate and yields spectra in a wide
range of frequency. We obtain explicit expressions for the power spectra and
present numerical illustrations of the model. Further we analyze the relation
of the point process model of noise with the Bernamont-Surdin-McWhorter
model, representing the signals as a sum of the uncorrelated components. We
show that the point process model is complementary to the model based on the
sum of signals with a wide-range distribution of the relaxation times. In
contrast to the Gaussian distribution of the signal intensity of the sum of the
uncorrelated components, the point process exhibits asymptotically a power-law
distribution of the signal intensity. The developed multiplicative point
process model of noise may be used for modeling and analysis of
stochastic processes in different systems with the power-law distribution of
the intensity of pulsing signals.Comment: 23 pages, 10 figures, to be published in Phys. Rev.
Quantum Trajectory method for the Quantum Zeno and anti-Zeno effects
We perform stochastic simulations of the quantum Zeno and anti-Zeno effects
for two level system and for the decaying one. Instead of simple projection
postulate approach, a more realistic model of a detector interacting with the
environment is used. The influence of the environment is taken into account
using the quantum trajectory method. The simulation of the measurement for a
single system exhibits the probabilistic behavior showing the collapse of the
wave-packet. When a large ensemble is analysed using the quantum trajectory
method, the results are the same as those produced using the density matrix
method. The results of numerical calculations are compared with the analytical
expressions for the decay rate of the measured system and a good agreement is
found. Since the analytical expressions depend on the duration of the
measurement only, the agreement with the numerical calculations shows that
otherparameters of the model are not important.Comment: 12 figures, accepted for publication in Phys. Rev. A replaced with
single-spaced versio
The class of nonlinear stochastic models as a background for the bursty behavior in financial markets
We investigate large changes, bursts, of the continuous stochastic signals,
when the exponent of multiplicativity is higher than one. Earlier we have
proposed a general nonlinear stochastic model which can be transformed into
Bessel process with known first hitting (first passage) time statistics. Using
these results we derive PDF of burst duration for the proposed model. We
confirm analytical expressions by numerical evaluation and discuss bursty
behavior of return in financial markets in the framework of modeling by
nonlinear SDE.Comment: 9 pages, 5 figure
1/f noise from correlations between avalanches in self-organized criticality
We show that large, slowly driven systems can evolve to a self-organized
critical state where long range temporal correlations between bursts or
avalanches produce low frequency noise. The avalanches can occur
instantaneously in the external time scale of the slow drive, and their event
statistics are described by power law distributions. A specific example of this
behavior is provided by numerical simulations of a deterministic ``sandpile''
model.Comment: Completely revised version: 4 pages (revtex), 3 eps figure
Influence of the detector's temperature on the quantum Zeno effect
In this paper we study the quantum Zeno effect using the irreversible model
of the measurement. The detector is modeled as a harmonic oscillator
interacting with the environment. The oscillator is subjected to the force,
proportional to the energy of the measured system. We use the Lindblad-type
master equation to model the interaction with the environment. The influence of
the detector's temperature on the quantum Zeno effect is obtained. It is shown
that the quantum Zeno effect becomes stronger (the jump probability decreases)
when the detector's temperature increases
Influence of measurement on the life-time and the line-width of unstable systems
We investigate the quantum Zeno effect in the case of electron tunneling out
of a quantum dot in the presence of continuous monitoring by a detector. It is
shown that the Schr\"odinger equation for the whole system can be reduced to
Bloch-type rate equations describing the combined time-development of the
detector and the measured system. Using these equations we find that continuous
measurement of the unstable system does not affect its exponential decay to a
reservoir with a constant density of states. The width of the energy
distribution of the tunneling electron, however, is not equal to the inverse
life-time -- it increases due to the decoherence generated by the detector. We
extend the analysis to the case of a reservoir described by an energy dependent
density of states, and we show that continuous measurement of such quantum
systems affects both the exponential decay rate and the energy distribution.
The decay does not always slow down, but might be accelerated. The energy
distribution of the tunneling electron may reveal the lines invisible before
the measurement.Comment: 13 pages, 8 figures, comments and references added; to appear in
Phys. Rev.
Weak measurement of arrival time
The arrival time probability distribution is defined by analogy with the
classical mechanics. The difficulty of requirement to have the values of
non-commuting operators is circumvented using the concept of weak measurements.
The proposed procedure is suitable to the free particles and to the particles
subjected to an external potential, as well. It is shown that such an approach
imposes an inherent limitation to the accuracy of the arrival time
determination.Comment: 3 figure
Demagnetization via Nucleation of the Nonequilibrium Metastable Phase in a Model of Disorder
We study both analytically and numerically metastability and nucleation in a
two-dimensional nonequilibrium Ising ferromagnet. Canonical equilibrium is
dynamically impeded by a weak random perturbation which models homogeneous
disorder of undetermined source. We present a simple theoretical description,
in perfect agreement with Monte Carlo simulations, assuming that the decay of
the nonequilibrium metastable state is due, as in equilibrium, to the
competition between the surface and the bulk. This suggests one to accept a
nonequilibrium "free-energy" at a mesoscopic/cluster level, and it ensues a
nonequilibrium "surface tension" with some peculiar low-T behavior. We
illustrate the occurrence of intriguing nonequilibrium phenomena, including:
(i) Noise-enhanced stabilization of nonequilibrium metastable states; (ii)
reentrance of the limit of metastability under strong nonequilibrium
conditions; and (iii) resonant propagation of domain walls. The cooperative
behavior of our system may also be understood in terms of a Langevin equation
with additive and multiplicative noises. We also studied metastability in the
case of open boundaries as it may correspond to a magnetic nanoparticle. We
then observe burst-like relaxation at low T, triggered by the additional
surface randomness, with scale-free avalanches which closely resemble the type
of relaxation reported for many complex systems. We show that this results from
the superposition of many demagnetization events, each with a well- defined
scale which is determined by the curvature of the domain wall at which it
originates. This is an example of (apparent) scale invariance in a
nonequilibrium setting which is not to be associated with any familiar kind of
criticality.Comment: 26 pages, 22 figure