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

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.