NONEXTENSIVE STATISTICAL MECHANICS DISTRIBUTIONS AND DYNAMICS OF FINANCIAL OBSERVABLES FROM THE NONLINEAR STOCHASTIC DIFFERENTIAL EQUATIONS

摘要

We present nonlinear stochastic differential equations, generating processes with thenq-exponential and q-Gaussian distributions of the observables, i.e. with the long-rangenpower-law autocorrelations and 1/fβ power spectral density. Similarly, the Tsallisnq-distributions may be obtained in the superstatistical framework as a superpositionnof different local dynamics at different time intervals. In such approach, the average ofnthe stochastic variable is generated by the nonlinear stochastic process, while the localndistribution of the signal is exponential or Gaussian one, conditioned by the slow average.nFurther we analyze relevance of the generalized and adapted equations for modelingnthe financial processes. We model the inter-trade durations, the trading activity and thennormalized return using the superstatistical approaches with the exponential and normalndistributions of the local signals driven by the nonlinear stochastic process