Interplay between positive feedbacks in the generalized CEV process

TL;DR

This paper analyzes the generalized CEV process with positive feedback mechanisms, revealing conditions for stationary distributions, power-law tails, and spectral properties, supported by numerical investigation of bursting behavior.

Contribution

It introduces a comprehensive analysis of the generalized CEV process with dual positive feedbacks, deriving conditions for stationarity, tail behavior, and spectral characteristics, and explores bursting dynamics numerically.

Findings

Stationary distribution exists if n<2m-1.

Power-law tail with exponent 2m for the distribution.

Spectral density follows a 1/f^β law with β depending on parameters.

Abstract

The dynamics of the {\em generalized} CEV process $dX_t = aX_t^n dt+ bX_t^m dW_t$ $(gCEV)$ 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 {\sc positive}, i.e. $n,m>1$, admits a stationary probability distribution $P$ provided that $n<2m-1$, which asymptotically decays as a power law $P(x) \sim \frac{1}{x^\mu}$ with tail exponent $\mu = 2m > 2$. Furthermore the power spectral density obeys $S(f) \sim \frac{1}{f^\beta}$, where $\beta = 2 - \:\frac{1+\epsilon}{2(m-1)}$, $\epsilon>0$. Bursting behavior of the gCEV is investigated numerically. Burst intensity $S$ and burst duration $T$ are shown to be related by $S\sim T^2$.