Interplay between positive feedbacks in the generalized CEV process

Abstract

The dynamics of the {\em generalized} CEV process $d{X}_t=a{X}_t^{n}dt+b{X}_t^{m}d{W}_t$ $(gCEV)$ 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 $S\sim T^2$.

8 pages, 9 figures

References in corpus (3)

• 1/f noise from nonlinear stochastic differential equations
• Modeling scaled processes and 1/f^b noise by the nonlinear stochastic differential equations
• Agent based reasoning for the non-linear stochastic models of long-range memory

Cited by in corpus (5)

• Agent based reasoning for the non-linear stochastic models of long-range memory
• The class of nonlinear stochastic models as a background for the bursty behavior in financial markets
• Agent-based Versus Macroscopic Modeling of Competition and Business Processes in Economics and Finance
• Agent-based Versus Macroscopic Modeling of Competition and Business Processes in Economics
• Deterministic versus stochastic aspects of superexponential population growth models