Modeling scaled processes and 1/f^b noise by the nonlinear stochastic differential equations

We present and analyze stochastic nonlinear differential equations generating signals with the power-law distributions of the signal intensity, 1/f^b noise, power-law autocorrelations and second order structural (height-height correlation) functions. Analytical expressions for such characteristics are derived and the comparison with numerical calculations is presented. The numerical calculations reveal links between the proposed model and models where signals consist of bursts characterized by the power-law distributions of burst size, burst duration and the interburst time, as in a case of avalanches in self-organized critical (SOC) models and the extreme event return times in long-term memory processes. The presented approach may be useful for modeling the long-range scaled processes exhibiting 1/f noise and power-law distributions.