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

Abstract

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...

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