Modeling long-memory processes by stochastic difference equations and superstatistical approach

2011

Phys. Rev. E

Self Cite

Probability distributions that emerge from the formalism of nonextensive statistical mechanics have been applied to a variety of problems. In this article we unite modeling of such distributions with the model of widespread 1/f noise. We propose a class of nonlinear stochastic differential equations giving both the q-exponential or q-Gaussian distributions of signal intensity, revealing long-range correlations and 1/f(β) behavior of the power spectral density. The superstatistical framework to get 1/f(β) noise with q-exponential and q-Gaussian distributions of the signal intensity is proposed, as well.

“…More reasoning of the correlation of these exponents, and of the type of equations like (5) in general, have been given in Refs. [9,10,49,50,52,53].…”

Section: Nonlinear Stochastic Differential Equation Generating Asmentioning

confidence: 99%

Section: A Q-exponential Distributionmentioning

confidence: 99%

Tsallis distributions and1/fnoise from nonlinear stochastic differential equations

2011

Phys. Rev. E

Self Cite

“…Some motivations of Eq. (1) have been given in [33,30,32,29,28,54]. In papers [53,36] the nonlinear SDE of type (1) has been obtained starting from a simple agent-based model describing the herding behavior.…”

Section: Introductionmentioning

confidence: 99%

Nonextensive Statistical Mechanics Distributions and Dynamics of Financial Observables From the Nonlinear Stochastic Differential Equations

Gontis

2012

Advs. Complex Syst.

We present nonlinear stochastic differential equations, generating processes with the q-exponential and q-Gaussian distributions of the observables, i.e. with the long-range power-law autocorrelations and 1/f β power spectral density. Similarly, the Tsallis q-distributions may be obtained in the superstatistical framework as a superposition of different local dynamics at different time intervals. In such approach, the average of the stochastic variable is generated by the nonlinear stochastic process, while the local distribution of the signal is exponential or Gaussian one, conditioned by the slow average. Further we analyze relevance of the generalized and adapted equations for modeling the financial processes. We model the inter-trade durations, the trading activity and the normalized return using the superstatistical approaches with the exponential and normal distributions of the local signals driven by the nonlinear stochastic process.Many complex systems show large fluctuations that follow non-Gaussian, heavytailed distributions with the power-law temporal correlations, scaling and the fractal features [42,44,41]. These distributions, scaling, self-similarity and fractality are often related with the Tsallis nonextensive statistical mechanics [60,48,61,62,57] and with 1/f β noise (see, e.g. [41,30,29,14,37,54], and references herein). Often nonextensive statistical mechanics represents a consistent theoretical background for the investigation of complex systems, [61,62,57]. On the other hand, a usual way to describe stochastic evolution and the properties of complex systems is by the stochastic differential equations (SDE) [16,52,13,64,65]. Such nondeterministic equations of motion are used for modeling the financial systems, as well [38,44,27,20,19,21,22].There are empirically established facts that the trading activity, trading volume, and volatility are stochastic variables with the long-range correlation [10,46,15]. However, these aspects are not accounted for in some widely used models of the financial systems. Moreover, the trading volume and the trading activity are positively correlated with the market volatility, while the trading volume and volatility show the same type of the long memory behavior [40], including 1/f β noise [21,22].The purpose of this article is to model the inter-trade durations, the trading activity and the normalized return using the superstatistical approaches with the exponential and normal distributions of the local signals driven by the nonlinear stochastic process. We present a class of nonlinear stochastic differential equations giving the power-law behavior of the probability density function (PDF) of the signal intensity and of the power spectral density (1/f β noise) in any desirably wide range of frequency. Modifications of these equations by introducing an additional parameter yields Tsallis distributions preserving 1/f β behavior of the power spectral density. The superstatistical framework [4,59,1,2,63,24,5] using a fast dynamics with the slowly changing parameter...

“…Equation, similar to (19) has been obtained in Ref. [41] by considering a relaxing linear system driven by the white noise.…”

Section: A Special Casesmentioning

confidence: 54%

“…Equation (19) shows that the shape of the power spectrum depends on the behavior of the eigenfunctions and eigenvalues in terms of function X 2 λ D(λ). The SDE (3) considered in this article gives the density of eigenvalues D(λ) proportional to 1/ √ λ.…”

Section: A Special Casesmentioning

confidence: 99%

Solutions of nonlinear stochastic differential equations with 1/ƒ noise power spectrum

2011 21st International Conference on Noise and Fluctuations

2011

The special nonlinear stochastic differential equations generating power-law distributed signals and 1/f noise are considered. The models involve the generalized Constant Elasticity of Variance (CEV) process, the Bessel process, the Squared Bessel process, and the Cox-Ingersoll-Ross (CIR) process, which are applied for modeling the financial markets, as well. In the paper, 1/f β behavior of the power spectral density is derived directly from the nonlinear stochastic differential equations and the exact solutions for the particular CEV process are presented.