Interplay between positive feedbacks in the generalized CEV process

The dynamics of the generalized CEV process dXt=aXtndt+bXtmdWt(gCEV) is due to an interplay of two feedback mechanisms: State-to-Drift and State-to-Diffusion, whose degrees are n and m respectively. We particularly show that the gCEV, in which both feedback mechanisms are positive, i.e. n,m>1, admits a stationary probability distribution P provided that n<2m−1. In this case the stationary pdf asymptotically decays as a power law P(x)∼1xμ with tail exponent μ=2m>2. Furthermore the power spectral density obeys S(f)∼1fβ, where β=2−1+ϵ2(m−1), ϵ>0. The tail behavior of the stationary pdf as well as of the power-spectral density thus are both independent of the drift feedback degree n but governed by the diffusion feedback degree m. Bursting behavior of the gCEV is investigated numerically. Burst intensity S and burst duration T are shown to be related by S∼T2.